The so-called ideal and subalgebra and some additional concepts of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M2" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 49.1759 11.5564" width="49.1759pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.013,0,0,-0.013,10.91,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,15.408,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,21.648,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,26.791,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,33.031,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,38.174,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,44.414,0)"><use xlink:href="#g113-42"/></g></svg> algebras are discussed. A partial order and congruence relations on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M3" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 49.1759 11.5564" width="49.1759pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.013,0,0,-0.013,10.91,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,15.408,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,21.648,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,26.791,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,33.031,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,38.174,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,44.414,0)"><use xlink:href="#g113-42"/></g></svg> algebras are also proposed, and some properties are investigated.

Fang-an Deng

Lu Chen

ShouHeng Tuo

ShengZhang Ren

Characterizations of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 49.1759 11.5564" width="49.1759pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z" /></g><g transform="matrix(.013,0,0,-0.013,10.91,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" /></g><g transform="matrix(.013,0,0,-0.013,15.408,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" /></g><g transform="matrix(.013,0,0,-0.013,21.648,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" /></g><g transform="matrix(.013,0,0,-0.013,26.791,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" /></g><g transform="matrix(.013,0,0,-0.013,33.031,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" /></g><g transform="matrix(.013,0,0,-0.013,38.174,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" /></g><g transform="matrix(.013,0,0,-0.013,44.414,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" /></g></svg> Algebras

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M2" height="8.64656pt" version="1.1" viewBox="-0.0498162 -8.59674 7.41997 8.64656" width="7.41997pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-77" d="M567 167L534 181C505 122 482 94 454 70C428 46 385 42 330 42C288 42 248 44 235 55C216 70 213 91 220 130L295 528C310 609 314 611 401 618L409 650H140L131 618C222 611 224 609 209 528L133 126C118 47 108 41 25 33L18 0H497C514 33 554 126 567 167Z"/></g></svg> be a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M3" height="9.01192pt" version="1.1" viewBox="-0.0498162 -8.80553 8.93363 9.01192" width="8.93363pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-68" d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"/></g></svg>-lattice and let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M4" height="8.64656pt" version="1.1" viewBox="-0.0498162 -8.59674 13.0831 8.64656" width="13.0831pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-78" d="M1006 650H829L492 147L357 650H174L166 618C270 609 275 601 247 502L194 315C158 189 136 127 122 96C104 56 80 37 18 32L12 0H256L263 32C181 40 176 51 183 98C188 132 197 179 226 281L299 541H301L433 6H458L809 551H811L753 141C739 41 734 39 650 32L642 0H917L923 32C836 39 827 41 839 142L887 535C896 610 901 612 997 618L1006 650Z"/></g></svg> be a lattice module over <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M5" height="8.64656pt" version="1.1" viewBox="-0.0498162 -8.59674 7.41997 8.64656" width="7.41997pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-77"/></g></svg>. Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40338pt" id="M6" height="12.3655pt" version="1.1" viewBox="-0.0498162 -8.96212 68.7053 12.3655" width="68.7053pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-243"/></g><g transform="matrix(.013,0,0,-0.013,11.164,0)"><path id="g50-59" d="M120 417C85 417 60 392 60 357C60 320 85 295 119 295S178 320 178 357C178 392 153 417 120 417ZM119 -12C154 -12 179 12 179 49C179 84 154 110 120 110S61 84 61 49C61 12 86 -12 119 -12Z"/></g><g transform="matrix(.013,0,0,-0.013,17.921,0)"><use xlink:href="#g50-78"/></g><g transform="matrix(.013,0,0,-0.013,36.739,0)"><path id="g54-148" d="M935 263C895 299 821 372 745 456L719 439L824 289H72V236H823L719 87L745 70C821 154 895 227 935 263Z"/></g><g transform="matrix(.013,0,0,-0.013,55.489,0)"><use xlink:href="#g50-78"/></g></svg> be a function. A proper element <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.7022505pt" id="M7" height="9.29899pt" version="1.1" viewBox="-0.0498162 -8.59674 36.1054 9.29899" width="36.1054pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-81" d="M622 479C622 589 545 650 400 650H146L140 618C229 609 230 602 215 527L135 132C117 43 112 41 27 32L18 0H305L313 32C213 40 205 44 222 132L251 284L324 262H365C484 269 622 333 622 479ZM528 477C528 356 420 305 344 305C300 305 276 314 259 321L303 554C311 597 313 600 331 606C345 611 361 614 379 614C463 614 528 580 528 477Z"/></g><g transform="matrix(.013,0,0,-0.013,11.766,0)"><path id="g54-173" d="M496 -1V52H376C259 52 161 132 147 236H496V289H147C162 392 259 473 376 473H496V526H374C213 526 83 408 83 262S213 -1 374 -1H496Z"/></g><g transform="matrix(.013,0,0,-0.013,22.969,0)"><use xlink:href="#g50-78"/></g></svg> is said to be <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40338pt" id="M8" height="12.3655pt" version="1.1" viewBox="-0.0498162 -8.96212 7.6157 12.3655" width="7.6157pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-243"/></g></svg>-absorbing primary if, for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.16066pt" id="M9" height="11.7574pt" version="1.1" viewBox="-0.0498162 -8.59674 91.2586 11.7574" width="91.2586pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-121" d="M545 403C545 425 527 451 498 451C448 451 400 404 312 286L290 341C262 412 246 451 219 451C182 451 138 404 92 345L112 323C152 368 169 378 181 378C192 378 201 359 216 321L257 214C184 115 142 69 113 69C102 69 93 73 88 79S78 89 68 89C47 89 24 61 24 40C24 8 49 -12 75 -12C125 -12 175 31 271 175L312 64C330 15 357 -12 382 -12C421 -12 475 35 515 98L496 121C466 88 439 62 420 62C402 62 384 92 363 147L325 247C349 280 374 309 397 333C422 361 446 381 462 381C471 381 481 374 488 366C493 360 499 357 505 357C521 357 545 380 545 403Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.332,3.001)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.013,0,0,-0.013,12.278,0)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/></g><g transform="matrix(.013,0,0,-0.013,17.565,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,24.897,3.001)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.013,0,0,-0.013,29.843,0)"><use xlink:href="#g50-45"/></g><g transform="matrix(.013,0,0,-0.013,35.129,0)"><path id="g50-47" d="M117 -12C153 -12 178 12 178 50C178 84 153 110 118 110C84 110 59 84 59 50C59 12 84 -12 117 -12Z"/></g><g transform="matrix(.013,0,0,-0.013,40.416,0)"><use xlink:href="#g50-47"/></g><g transform="matrix(.013,0,0,-0.013,45.702,0)"><use xlink:href="#g50-47"/></g><g transform="matrix(.013,0,0,-0.013,51.023,0)"><use xlink:href="#g50-45"/></g><g transform="matrix(.013,0,0,-0.013,56.309,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,63.641,3.001)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.013,0,0,-0.013,72.491,0)"><use xlink:href="#g54-173"/></g><g transform="matrix(.013,0,0,-0.013,83.693,0)"><use xlink:href="#g50-77"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.7022505pt" id="M10" height="9.29899pt" version="1.1" viewBox="-0.0498162 -8.59674 39.1066 9.29899" width="39.1066pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-79" d="M870 650H622L613 619C694 615 708 606 708 559C708 531 702 477 686 396L636 136H633L281 650H127L120 619C180 616 198 611 212 592C227 572 232 558 220 503L166 254C148 169 139 131 126 89C112 43 85 35 33 32L24 0H280L286 32C208 37 189 44 189 93C189 124 198 179 212 250L267 526H271L622 -8H655L732 396C747 477 762 532 770 556C785 600 798 616 862 619L870 650Z"/></g><g transform="matrix(.013,0,0,-0.013,14.754,0)"><use xlink:href="#g54-173"/></g><g transform="matrix(.013,0,0,-0.013,25.957,0)"><use xlink:href="#g50-78"/></g></svg>, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.16066pt" id="M11" height="11.7574pt" version="1.1" viewBox="-0.0498162 -8.59674 89.9015 11.7574" width="89.9015pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,7.332,3.001)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,12.278,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,19.61,3.001)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,26.762,0)"><path id="g50-46" d="M178 262C178 297 153 323 118 323C84 323 59 297 59 262C59 225 84 201 118 201C153 201 178 225 178 262Z"/></g><g transform="matrix(.013,0,0,-0.013,32.048,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,37.334,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,42.655,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,49.987,3.001)"><use xlink:href="#g50-111"/></g><g transform="matrix(.013,0,0,-0.013,55.162,0)"><use xlink:href="#g50-79"/></g><g transform="matrix(.013,0,0,-0.013,69.917,0)"><path id="g54-93" d="M552 72V131L122 319L552 507V566L59 344V293L552 72ZM552 -41V12H59V-41H552Z"/></g><g transform="matrix(.013,0,0,-0.013,81.535,0)"><use xlink:href="#g50-81"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40337pt" id="M12" height="12.6917pt" version="1.1" viewBox="-0.0498162 -9.28833 106.761 12.6917" width="106.761pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,7.332,3.001)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,12.278,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,19.61,3.001)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,26.762,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,32.048,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,37.334,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,42.655,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,49.987,3.001)"><use xlink:href="#g50-111"/></g><g transform="matrix(.013,0,0,-0.013,55.162,0)"><use xlink:href="#g50-79"/></g><g transform="matrix(.013,0,0,-0.013,69.917,0)"><path id="g54-101" d="M552 -41V12H260L312 179L552 72V131L328 229L392 437L552 507V566L412 503L472 696L414 714L338 470L59 344V293L256 205L197 12H59V-41H181L140 -172L198 -190L244 -41H552ZM318 405L272 254L122 319L318 405Z"/></g><g transform="matrix(.013,0,0,-0.013,81.535,0)"><use xlink:href="#g50-243"/></g><g transform="matrix(.013,0,0,-0.013,89.024,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.013,0,0,-0.013,93.678,0)"><use xlink:href="#g50-81"/></g><g transform="matrix(.013,0,0,-0.013,101.769,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg> together imply <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.16067pt" id="M13" height="12.449pt" version="1.1" viewBox="-0.0498162 -9.28833 114.511 12.449" width="114.511pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,7.332,3.001)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,12.278,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,19.61,3.001)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,26.762,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,32.048,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,37.334,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,42.655,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,49.987,3.001)"><use xlink:href="#g50-111"/></g><g transform="matrix(.013,0,0,-0.013,58.838,0)"><use xlink:href="#g54-93"/></g><g transform="matrix(.013,0,0,-0.013,70.456,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.013,0,0,-0.013,75.11,0)"><use xlink:href="#g50-81"/></g><g transform="matrix(.013,0,0,-0.013,86.876,0)"><use xlink:href="#g50-59"/></g><g transform="matrix(.013,0,0,-0.013,93.633,0)"><use xlink:href="#g50-50"/></g><g transform="matrix(.0091,0,0,-0.0091,99.964,3.001)"><use xlink:href="#g50-78"/></g><g transform="matrix(.013,0,0,-0.013,109.55,0)"><use xlink:href="#g50-42"/></g></svg> or <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.2702pt" id="M14" height="15.2613pt" version="1.1" viewBox="-0.0498162 -11.9911 158.549 15.2613" width="158.549pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,7.332,3.001)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,12.278,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,19.61,3.001)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,26.762,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,32.048,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,37.334,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,42.655,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,49.987,3.001)"><path id="g50-106" d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"/></g><g transform="matrix(.0091,0,0,-0.0091,52.536,3.001)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g><g transform="matrix(.0091,0,0,-0.0091,58.096,3.001)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,63.072,0)"><use xlink:href="#g50-121"/></g><g transform="matrix(.0091,0,0,-0.0091,70.404,3.001)"><use xlink:href="#g50-106"/></g><g transform="matrix(.0091,0,0,-0.0091,72.952,3.001)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g><g transform="matrix(.0091,0,0,-0.0091,78.512,3.001)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,85.694,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,90.981,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,96.267,0)"><use xlink:href="#g50-46"/></g><g transform="matrix(.013,0,0,-0.013,101.588,0)"><use xlink:href="#g50-121"/></g><g 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We study some basic properties of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40338pt" id="M16" height="12.3655pt" version="1.1" viewBox="-0.0498162 -8.96212 7.6157 12.3655" width="7.6157pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-243"/></g></svg>-absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40338pt" id="M17" height="12.3655pt" version="1.1" viewBox="-0.0498162 -8.96212 7.6157 12.3655" width="7.6157pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-243"/></g></svg>-absorbing elements and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40338pt" id="M18" height="12.3655pt" version="1.1" viewBox="-0.0498162 -8.96212 7.6157 12.3655" width="7.6157pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-243"/></g></svg>-absorbing primary elements are unified.

Sachin Ballal

Vilas Kharat

On <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40338pt" height="12.3655pt" version="1.1" viewBox="-0.0498162 -8.96212 7.6157 12.3655" width="7.6157pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-243" d="M552 243C552 372 474 431 385 468C378 471 378 492 382 510L419 675L401 683L345 637L224 54C173 71 118 128 118 218C118 323 174 382 270 423L255 458C130 420 24 339 24 204C24 97 93 14 213 -8L188 -131C170 -220 197 -250 217 -257L224 -254L278 -7C427 -3 552 98 552 243ZM478 228C478 124 408 37 287 40L363 405C401 395 478 336 478 228Z"/></g></svg>-Absorbing Primary Elements in Lattice Modules

We introduce a new class of extension rings called the <i>generalized Malcev-Neumann series ring</i>&#x2009;<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M1" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 58.8607 12.0395" width="58.8607pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.0135,0,0,-0.0135,8.495,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,13.183,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,17.87,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g><g transform="matrix(.0135,0,0,-0.0135,24.265,0)"><path id="g113-60" d="M114 412C81 412 58 388 58 355C58 321 81 297 113 297S169 321 169 355C169 388 145 412 114 412ZM95 130C70 130 46 114 46 88C46 72 54 65 59 64C93 56 121 33 121 -3C121 -41 93 -68 45 -88L56 -118C117 -99 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.0135,0,0,-0.0135,29.625,0)"><path id="g113-240" d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"/></g><g transform="matrix(.0135,0,0,-0.0135,37.282,0)"><use xlink:href="#g113-60"/></g><g transform="matrix(.0135,0,0,-0.0135,42.642,0)"><path id="g113-241" d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"/></g><g transform="matrix(.0135,0,0,-0.0135,49.189,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.0135,0,0,-0.0135,53.877,0)"><use xlink:href="#g113-42"/></g></svg> with
coefficients in a ring <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M2" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 8.62618 9.21094" width="8.62618pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-83"/></g></svg> and exponents in a strictly ordered monoid <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M3" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 6.51834 9.21094" width="6.51834pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-84"/></g></svg>
which extends the usual construction of Malcev-Neumann series rings. Ouyang et al. in 2014 introduced the modules with the Beachy-Blair condition
as follows: A right <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M4" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 8.62618 9.21094" width="8.62618pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-83"/></g></svg>-module satisfies the right Beachy-Blair condition
if each of its faithful submodules is cofaithful. In this paper, we study
the relationship between the right Beachy-Blair condition of a right <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M5" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 8.62618 9.21094" width="8.62618pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-83"/></g></svg>-module <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427751pt" id="M6" height="12.4257pt" version="1.1" viewBox="-0.0498162 -8.99795 19.435 12.4257" width="19.435pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g><g transform="matrix(.0095,0,0,-0.0095,12.735,3.264)"><path id="g50-83" d="M636 22C611 29 589 38 560 74C536 104 518 134 495 173L430 300C518 328 592 392 592 492C592 555 563 594 528 618C491 643 435 650 377 650H140L131 618C222 610 223 601 208 518L137 136C121 43 116 40 27 32L18 0H295L300 32C212 40 206 42 223 138L248 279H296C334 279 350 272 369 230C393 174 413 140 441 94C482 24 523 1 626 -8L636 22ZM497 480C497 422 468 374 417 343C387 324 353 315 304 315H255L298 556C303 583 309 598 319 603C331 610 348 613 367 613C448 613 497 573 497 480Z"/></g></svg> and its Malcev-Neumann series module extension <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.8747pt" id="M7" height="14.5526pt" version="1.1" viewBox="-0.0498162 -9.6779 78.2136 14.5526" width="78.2136pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-78"/></g><g transform="matrix(.0135,0,0,-0.0135,13.347,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,18.035,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,22.723,0)"><use xlink:href="#g113-84"/></g><g transform="matrix(.0135,0,0,-0.0135,29.118,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,33.805,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0095,0,0,-0.0095,38.493,3.264)"><use xlink:href="#g50-83"/></g><g transform="matrix(.0095,0,0,-0.0095,44.525,3.264)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.0095,0,0,-0.0095,47.92,3.264)"><use xlink:href="#g50-41"/></g><g transform="matrix(.0095,0,0,-0.0095,51.315,3.264)"><path id="g50-84" d="M463 633C453 636 427 644 417 648C388 660 349 667 316 667C183 667 100 592 100 483C100 398 176 335 223 305L259 282C321 242 358 203 358 149C358 66 301 22 234 22C110 22 70 131 57 208L24 205L28 48C51 23 128 -17 215 -17C342 -17 446 60 446 174C446 262 394 310 317 361L284 383C229 420 183 455 183 515C183 569 222 628 298 628C384 628 418 563 428 481L461 487C459 527 458 593 463 633Z"/></g><g transform="matrix(.0095,0,0,-0.0095,55.848,3.264)"><path id="g50-60" d="M120 417C85 417 60 392 60 356S85 295 119 295S178 320 178 356S153 417 120 417ZM100 135C73 135 48 117 48 91C48 73 56 65 62 64C96 56 125 33 125 -4C125 -42 97 -68 46 -89L59 -124C123 -105 195 -59 195 22C195 94 137 135 100 135Z"/></g><g transform="matrix(.0095,0,0,-0.0095,58.096,3.264)"><path id="g50-240" d="M558 454L532 470C512 451 497 442 465 442C421 442 376 450 318 450C146 450 24 313 24 164C24 46 91 -12 183 -12C317 -12 439 105 439 247C439 306 428 346 382 385L383 389C414 384 450 379 485 379C517 379 539 412 558 454ZM357 273C357 191 313 28 205 28C149 28 111 79 111 160C111 266 175 396 280 396C298 396 328 386 341 357C353 331 357 315 357 273Z"/></g><g transform="matrix(.0095,0,0,-0.0095,63.56,3.264)"><use xlink:href="#g50-60"/></g><g transform="matrix(.0095,0,0,-0.0095,65.808,3.264)"><path id="g50-241" d="M487 458L455 460C436 434 425 433 397 433C331 433 279 437 225 437C103 437 52 374 24 338L49 312C100 363 152 379 229 374C210 308 184 213 163 117C158 94 156 74 156 58C156 6 180 -12 211 -12C261 -12 313 27 348 68L332 99C307 76 285 64 267 64C252 64 236 86 250 148L301 373C320 373 344 369 364 369C399 369 435 370 450 372C458 386 470 411 487 458Z"/></g><g transform="matrix(.0095,0,0,-0.0095,70.542,3.264)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g><g transform="matrix(.0095,0,0,-0.0095,73.938,3.264)"><use xlink:href="#g50-42"/></g></svg>.

Mohamed A. Farahat

Generalized Malcev-Neumann Series Modules with the Beachy-Blair Condition

For each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in
standard generators.

Chris Bates

Peter Rowley

Paul Taylor

Involutions in the Automorphism Groups of Small Sporadic Simple Groups

Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings.

Allen Herman

Gurmail Singh

On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427751pt" id="M2" height="12.4257pt" version="1.1" viewBox="-0.0498162 -8.99795 13.7806 12.4257" width="13.7806pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-70" d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z"/></g><g transform="matrix(.0095,0,0,-0.0095,8.278,3.264)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g></svg> be the ring of Eisenstein integers modulo <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M3" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 6.92631 6.42315" width="6.92631pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g></svg>. In this paper we study the zero divisor graph <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427799pt" id="M4" height="13.1057pt" version="1.1" viewBox="-0.0498162 -9.6779 30.0857 13.1057" width="30.0857pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-132" d="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z"/></g><g transform="matrix(.0135,0,0,-0.0135,6.873,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,11.561,0)"><use xlink:href="#g113-70"/></g><g transform="matrix(.0095,0,0,-0.0095,19.839,3.264)"><use xlink:href="#g50-111"/></g><g transform="matrix(.0135,0,0,-0.0135,25.211,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>. We find the diameters and girths for such zero divisor graphs and characterize <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M5" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 6.92631 6.42315" width="6.92631pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g></svg> for which the graph <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427799pt" id="M6" height="13.1057pt" version="1.1" viewBox="-0.0498162 -9.6779 30.0857 13.1057" width="30.0857pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-132"/></g><g transform="matrix(.0135,0,0,-0.0135,6.873,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,11.561,0)"><use xlink:href="#g113-70"/></g><g transform="matrix(.0095,0,0,-0.0095,19.839,3.264)"><use xlink:href="#g50-111"/></g><g transform="matrix(.0135,0,0,-0.0135,25.211,0)"><use xlink:href="#g113-42"/></g></svg> is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

Osama Alkam

Emad Abu Osba

Zero Divisor Graph for the Ring of Eisenstein Integers Modulo <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 6.92631 6.42315" width="6.92631pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="502" vert-adv-y="502" /></g></svg>

The concepts of a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129898pt" id="M3" height="9.89089pt" version="1.1" viewBox="-0.0498162 -9.6779 6.9399 9.89089" width="6.9399pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-108"/></g></svg>-idempotent <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M4" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 6.99431 9.04777" width="6.99431pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-132"/></g></svg>-semiring, a right <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129898pt" id="M5" height="9.89089pt" version="1.1" viewBox="-0.0498162 -9.6779 6.9399 9.89089" width="6.9399pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-108"/></g></svg>-weakly regular <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M6" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 6.99431 9.04777" width="6.99431pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-132"/></g></svg>-semiring, and a right pure <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129898pt" id="M7" height="9.89089pt" version="1.1" viewBox="-0.0498162 -9.6779 6.9399 9.89089" width="6.9399pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-108"/></g></svg>-ideal of a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M8" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 6.99431 9.04777" width="6.99431pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-132"/></g></svg>-semiring are introduced. Several characterizations of them are furnished.

R. D. Jagatap

Right <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129898pt" height="9.89089pt" version="1.1" viewBox="-0.0498162 -9.6779 6.9399 9.89089" width="6.9399pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="503" vert-adv-y="503" /></g></svg>-Weakly Regular <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 6.99431 9.04777" width="6.99431pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-132" d="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="507" vert-adv-y="507" /></g></svg>-Semirings

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M1" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 10.5029 9.04777" width="10.5029pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"/></g></svg> be a topological space. The semigroup of all the &#xe9;tale mappings of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M2" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 10.5029 9.04777" width="10.5029pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-89"/></g></svg> (the local homeomorphisms <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M3" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 45.9916 9.04777" width="45.9916pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0135,0,0,-0.0135,16.421,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><g transform="matrix(.0135,0,0,-0.0135,35.402,0)"><use xlink:href="#g113-89"/></g></svg>) is denoted by <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M4" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 29.8406 12.0395" width="29.8406pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g190-102" d="M380 106C343 72 306 56 265 56C195 56 116 112 115 248C235 252 361 262 377 265C396 269 400 277 400 297C400 374 333 449 250 449H249C198 449 144 421 103 376S37 269 37 201C37 88 109 -12 232 -12C263 -12 332 6 395 84L380 106ZM225 412C281 412 315 364 314 312C314 297 308 292 290 292C232 290 176 289 120 289C135 370 180 412 225 412Z"/></g><g transform="matrix(.0135,0,0,-0.0135,5.758,0)"><path id="g190-117" d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z"/></g><g transform="matrix(.0135,0,0,-0.0135,9.927,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,14.615,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0135,0,0,-0.0135,24.98,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>. If <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M5" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 53.8926 12.0395" width="53.8926pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g><g transform="matrix(.0135,0,0,-0.0135,13.052,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.0135,0,0,-0.0135,23.991,0)"><use xlink:href="#g190-102"/></g><g transform="matrix(.0135,0,0,-0.0135,29.749,0)"><use xlink:href="#g190-117"/></g><g transform="matrix(.0135,0,0,-0.0135,33.918,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,38.606,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0135,0,0,-0.0135,48.97,0)"><use xlink:href="#g113-42"/></g></svg>, then the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M6" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 9.40133 9.21094" width="9.40133pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-72"/></g></svg>-right (left) composition operator on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M7" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 29.8406 12.0395" width="29.8406pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g190-102"/></g><g transform="matrix(.0135,0,0,-0.0135,5.758,0)"><use xlink:href="#g190-117"/></g><g transform="matrix(.0135,0,0,-0.0135,9.927,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,14.615,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0135,0,0,-0.0135,24.98,0)"><use xlink:href="#g113-42"/></g></svg> is defined by <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427751pt" id="M8" height="12.4257pt" version="1.1" viewBox="-0.0498162 -8.99795 15.6532 12.4257" width="15.6532pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.0095,0,0,-0.0095,8.427,3.264)"><path id="g50-72" d="M722 297H451L442 267C561 257 561 253 547 182L531 98C521 44 465 24 390 24C228 24 130 134 130 296C130 482 252 628 465 628C592 628 657 579 656 458L692 461C695 540 699 605 702 631C664 642 590 667 492 667C203 667 24 502 24 285C24 93 170 -17 368 -17C464 -17 543 11 617 28C613 43 612 68 620 106L635 182C649 255 652 258 716 268L722 297Z"/></g></svg>&#x2009;&#x2009;<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427799pt" id="M9" height="13.1057pt" version="1.1" viewBox="-0.0498162 -9.6779 120.519 13.1057" width="120.519pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 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xlink:href="#g117-148"/></g><g transform="matrix(.0135,0,0,-0.0135,90.477,0)"><use xlink:href="#g190-102"/></g><g transform="matrix(.0135,0,0,-0.0135,96.235,0)"><use xlink:href="#g190-117"/></g><g transform="matrix(.0135,0,0,-0.0135,100.405,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,105.092,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0135,0,0,-0.0135,115.457,0)"><use xlink:href="#g113-42"/></g></svg>, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427799pt" id="M10" height="13.1057pt" version="1.1" viewBox="-0.0498162 -9.6779 169.127 13.1057" width="169.127pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-83"/></g><g transform="matrix(.0095,0,0,-0.0095,8.427,3.264)"><use xlink:href="#g50-72"/></g><g transform="matrix(.0135,0,0,-0.0135,15.522,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,20.21,0)"><path id="g113-71" d="M584 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xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,86.456,0)"><use xlink:href="#g113-77"/></g><g transform="matrix(.0095,0,0,-0.0095,95.099,3.264)"><use xlink:href="#g50-72"/></g><g transform="matrix(.0135,0,0,-0.0135,102.194,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,106.882,0)"><use xlink:href="#g113-71"/></g><g transform="matrix(.0135,0,0,-0.0135,115.11,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,123.583,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.0135,0,0,-0.0135,135.32,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.0135,0,0,-0.0135,147.616,0)"><use xlink:href="#g117-73"/></g><g transform="matrix(.0135,0,0,-0.0135,155.67,0)"><use xlink:href="#g113-71"/></g><g transform="matrix(.0135,0,0,-0.0135,163.898,0)"><use xlink:href="#g113-42"/></g></svg>. When are the composition operators injective?  The Problem originated in a new approach to study &#xe9;tale polynomial mappings <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2402pt" id="M11" height="12.3183pt" version="1.1" viewBox="-0.0498162 -12.0781 54.6277 12.3183" width="54.6277pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path id="g197-14" d="M632 159C554 79 496 30 378 30C330 30 251 50 213 80V583C258 614 324 632 379 632C497 632 552 583 630 503L661 534C584 627 498 676 375 676C172 676 45 529 45 332C45 140 176 -14 374 -14C497 -14 586 35 663 128L632 159ZM169 118C160 125 155 134 148 145C111 201 89 264 89 332C89 416 122 490 169 558V118Z"/></g><g transform="matrix(.0095,0,0,-0.0095,9.551,-5.984)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.0135,0,0,-0.0135,20.742,0)"><use xlink:href="#g117-148"/></g><g transform="matrix(.0136,0,0,-0.0136,39.722,0)"><use xlink:href="#g197-14"/></g><g transform="matrix(.0095,0,0,-0.0095,49.274,-5.984)"><use xlink:href="#g50-51"/></g></svg> and in particular the two-dimensional Jacobian conjecture. This approach constructs a fractal structure on the semigroup of the (normalized) Keller mappings and outlines a new method of a possible attack on this open problem (in preparation). The construction uses the left composition operator and the injectivity problem is essential. In this paper we will completely solve the injectivity problems of the two composition operators for (normalized) Keller mappings. We will also solve the much easier surjectivity problem of these composition operators.

Ronen Peretz

Injectivity of the Composition Operators of &#xC9;tale Mappings

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M1" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 10.3669 9.04777" width="10.3669pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-76" d="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"/></g></svg> be a commutative field of characteristic <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" id="M2" height="12.2707pt" version="1.1" viewBox="-0.0498162 -8.69876 30.244 12.2707" width="30.244pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g><g transform="matrix(.0135,0,0,-0.0135,11.82,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></g><g transform="matrix(.0135,0,0,-0.0135,23.558,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g></svg> and let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.31355pt" id="M3" height="12.3115pt" version="1.1" viewBox="-0.0498162 -8.99795 67.728 12.3115" width="67.728pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g><g transform="matrix(.0135,0,0,-0.0135,13.052,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,24.79,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.0095,0,0,-0.0095,33.989,3.264)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.0135,0,0,-0.0135,42.151,0)"><path id="g117-42" d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z"/></g><g transform="matrix(.0135,0,0,-0.0135,53.132,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.0095,0,0,-0.0095,62.331,3.264)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.31355pt" id="M4" height="12.3115pt" version="1.1" viewBox="-0.0498162 -8.99795 14.4674 12.3115" width="14.4674pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.0095,0,0,-0.0095,9.199,3.264)"><use xlink:href="#g50-50"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.31355pt" id="M5" height="12.3115pt" version="1.1" viewBox="-0.0498162 -8.99795 14.4674 12.3115" width="14.4674pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.0095,0,0,-0.0095,9.199,3.264)"><use xlink:href="#g50-51"/></g></svg> are two finite cyclic groups. We give some structure results of finitely generated <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2392pt" id="M6" height="11.9171pt" version="1.1" viewBox="-0.0498162 -9.6779 29.0519 11.9171" width="29.0519pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-76"/></g><g transform="matrix(.0135,0,0,-0.0135,10.231,0)"><path id="g113-92" d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z"/></g><g transform="matrix(.0135,0,0,-0.0135,14.905,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.0135,0,0,-0.0135,24.172,0)"><path id="g113-94" d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z"/></g></svg>-modules in the case where the order of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M7" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 9.40133 9.21094" width="9.40133pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-72"/></g></svg> is divisible by <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571919pt" id="M8" height="10.6388pt" version="1.1" viewBox="-0.0498162 -7.06688 8.16381 10.6388" width="8.16381pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-113"/></g></svg>. Extensions of modules are also investigated. Based on these extensions and in the same previous case, we show that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2392pt" id="M9" height="11.9171pt" version="1.1" viewBox="-0.0498162 -9.6779 29.0519 11.9171" width="29.0519pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-76"/></g><g transform="matrix(.0135,0,0,-0.0135,10.231,0)"><use xlink:href="#g113-92"/></g><g transform="matrix(.0135,0,0,-0.0135,14.905,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.0135,0,0,-0.0135,24.172,0)"><use xlink:href="#g113-94"/></g></svg>-modules satisfying some conditions have a fairly simple form.

Ahmed Najim

Mohammed Elhassani Charkani

Finitely Generated Modules over Group Rings of a Direct Product of Two Cyclic Groups

I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any field of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated with any binomial hypersurface over any field of arbitrary positive characteristic is rational. As an example, I also prove the well known fact that for 1-dimensional binomial hypersurfaces the Hilbert-Kunz multiplicity is a positive integer and give a precise account of the integer.

Shyamashree Upadhyay

The Hilbert-Kunz Function for Binomial Hypersurfaces

We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice module <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M1" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 13.4946 9.04777" width="13.4946pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g></svg> and weakly prime element of a multiplicative lattice <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M2" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 7.61985 9.04777" width="7.61985pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/></g></svg>.

C. S. Manjarekar

U. N. Kandale

Residuation Properties and Weakly Primary Elements in Lattice Modules

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M1" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 8.62618 9.21094" width="8.62618pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g></svg> be a ring. We say that a family of maps <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.789499pt" id="M2" height="13.4674pt" version="1.1" viewBox="-0.0498162 -9.6779 65.8862 13.4674" width="65.8862pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-69" d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z"/></g><g transform="matrix(.0135,0,0,-0.0135,14.163,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,25.901,0)"><path id="g113-124" d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z"/></g><g transform="matrix(.0135,0,0,-0.0135,30.602,0)"><path id="g113-101" d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"/></g><g transform="matrix(.0095,0,0,-0.0095,38.054,3.264)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.0135,0,0,-0.0135,43.426,0)"><path id="g113-126" d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g><g transform="matrix(.0095,0,0,-0.0095,48.144,3.264)"><use xlink:href="#g50-111"/></g><g transform="matrix(.0095,0,0,-0.0095,53,3.264)"><path id="g54-173" d="M496 -1V52H376C259 52 161 132 147 236H496V289H147C162 392 259 473 376 473H496V526H374C213 526 83 408 83 262S213 -1 374 -1H496Z"/></g><g transform="matrix(.0095,0,0,-0.0095,58.491,3.264)"><path id="g197-25" d="M628 0V662H584V156H583L237 617L205 662H70V0H92H114V504H116L494 0H628ZM584 44H515L114 578V618H182L584 83V44Z"/></g></svg> is a Jordan higher derivable map (without assumption of additivity) on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M3" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 8.62618 9.21094" width="8.62618pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-83"/></g></svg> if <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427799pt" id="M4" height="13.1057pt" version="1.1" viewBox="-0.0498162 -9.6779 39.7545 13.1057" width="39.7545pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-101"/></g><g transform="matrix(.0095,0,0,-0.0095,7.452,3.264)"><path id="g50-49" d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"/></g><g transform="matrix(.0135,0,0,-0.0135,16.37,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.0135,0,0,-0.0135,28.108,0)"><path id="g113-74" d="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z"/></g><g transform="matrix(.0095,0,0,-0.0095,33.026,3.264)"><path id="g50-83" d="M636 22C611 29 589 38 560 74C536 104 518 134 495 173L430 300C518 328 592 392 592 492C592 555 563 594 528 618C491 643 435 650 377 650H140L131 618C222 610 223 601 208 518L137 136C121 43 116 40 27 32L18 0H295L300 32C212 40 206 42 223 138L248 279H296C334 279 350 272 369 230C393 174 413 140 441 94C482 24 523 1 626 -8L636 22ZM497 480C497 422 468 374 417 343C387 324 353 315 304 315H255L298 556C303 583 309 598 319 603C331 610 348 613 367 613C448 613 497 573 497 480Z"/></g></svg> (the identity map on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M5" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 8.62618 9.21094" width="8.62618pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-83"/></g></svg>) and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-6.051pt" id="M6" height="16.1437pt" version="1.1" viewBox="-0.0498162 -10.0927 289.261 16.1437" width="289.261pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-101"/></g><g transform="matrix(.0095,0,0,-0.0095,7.452,3.264)"><use xlink:href="#g50-111"/></g><g transform="matrix(.0135,0,0,-0.0135,12.824,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,17.511,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g transform="matrix(.0135,0,0,-0.0135,24.369,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g><g transform="matrix(.0135,0,0,-0.0135,33.755,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,44.736,0)"><use xlink:href="#g113-99"/></g><g transform="matrix(.0135,0,0,-0.0135,51.094,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.0135,0,0,-0.0135,57.952,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.0135,0,0,-0.0135,66.425,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.0135,0,0,-0.0135,78.163,.007)"><path id="g119-65" d="M697 -46L668 -39C637 -126 611 -142 545 -142H163L474 298L206 694H483C558 694 584 688 600 668C615 651 627 625 639 580L667 585L655 739H70V714L378 254L53 -203V-228H655L697 -46Z"/></g><g transform="matrix(.0095,0,0,-0.0095,88.362,3.612)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g><g transform="matrix(.0095,0,0,-0.0095,94.024,3.612)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g><g transform="matrix(.0095,0,0,-0.0095,99.819,3.612)"><path id="g50-114" d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z"/></g><g transform="matrix(.0095,0,0,-0.0095,104.552,3.612)"><path id="g54-34" d="M556 331V384H55V331H556ZM556 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transform="matrix(.0135,0,0,-0.0135,179.398,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,190.378,.007)"><use xlink:href="#g119-65"/></g><g transform="matrix(.0095,0,0,-0.0095,200.578,3.612)"><use xlink:href="#g50-113"/></g><g transform="matrix(.0095,0,0,-0.0095,206.24,3.612)"><use xlink:href="#g54-36"/></g><g transform="matrix(.0095,0,0,-0.0095,212.035,3.612)"><use xlink:href="#g50-114"/></g><g transform="matrix(.0095,0,0,-0.0095,216.767,3.612)"><use xlink:href="#g54-34"/></g><g transform="matrix(.0095,0,0,-0.0095,222.562,3.612)"><use xlink:href="#g50-111"/></g><g transform="matrix(.0135,0,0,-0.0135,230.286,0)"><use xlink:href="#g113-101"/></g><g transform="matrix(.0095,0,0,-0.0095,237.738,3.264)"><use xlink:href="#g50-113"/></g><g transform="matrix(.0135,0,0,-0.0135,243.919,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,248.606,0)"><use xlink:href="#g113-99"/></g><g transform="matrix(.0135,0,0,-0.0135,254.965,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,259.653,0)"><use xlink:href="#g113-101"/></g><g transform="matrix(.0095,0,0,-0.0095,267.104,3.264)"><use xlink:href="#g50-114"/></g><g transform="matrix(.0135,0,0,-0.0135,272.352,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,277.04,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.0135,0,0,-0.0135,283.898,0)"><use xlink:href="#g113-42"/></g></svg> hold for all <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.6409pt" id="M7" height="11.3188pt" version="1.1" viewBox="-0.0498162 -9.6779 42.007 11.3188" width="42.007pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.0135,0,0,-0.0135,6.858,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.0135,0,0,-0.0135,12.218,0)"><use xlink:href="#g113-99"/></g><g transform="matrix(.0135,0,0,-0.0135,22.361,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.0135,0,0,-0.0135,33.3,0)"><use xlink:href="#g113-83"/></g></svg> and for each <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6617708pt" id="M8" height="9.71411pt" version="1.1" viewBox="-0.0498162 -9.05234 31.1687 9.71411" width="31.1687pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.0135,0,0,-0.0135,10.586,0)"><use xlink:href="#g117-173"/></g><g transform="matrix(.0136,0,0,-0.0136,21.525,0)"><use xlink:href="#g197-25"/></g></svg>. In this paper, we show that every Jordan higher derivable map on a ring under certain assumptions becomes a higher derivation. As its application, we get that every Jordan higher derivable map on Banach algebra is an additive higher derivation.

Mohammad Ashraf

Nazia Parveen

Jordan Higher Derivable Mappings on Rings

Utpal Dasgupta

Some Properties of Multiplicative <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427751pt" height="12.4257pt" version="1.1" viewBox="-0.0498162 -8.99795 15.4301 12.4257" width="15.4301pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="840" vert-adv-y="840" /></g><g transform="matrix(.0095,0,0,-0.0095,10.5,3.264)"><path id="g185-40" d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z" /><glyph.data ascent="989" descent="-303" horiz-adv-x="447" vert-adv-y="447" /></g></svg>-Rings of Polynomials over Multiplicative Hyperrings

We investigate <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57198pt" id="M3" height="12.9099pt" version="1.1" viewBox="-0.0498162 -9.33792 7.78304 12.9099" width="7.78304pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-243" d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"/></g></svg>-prime and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57198pt" id="M4" height="12.9099pt" version="1.1" viewBox="-0.0498162 -9.33792 7.78304 12.9099" width="7.78304pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-243"/></g></svg>-primary elements in a compactly generated multiplicative lattice <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M5" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 7.61985 9.04777" width="7.61985pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/></g></svg>. By a counterexample, it is shown that a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57198pt" id="M6" height="12.9099pt" version="1.1" viewBox="-0.0498162 -9.33792 7.78304 12.9099" width="7.78304pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-243"/></g></svg>-primary element in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M7" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 7.61985 9.04777" width="7.61985pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-77"/></g></svg> need not be primary. Some characterizations of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57198pt" id="M8" height="12.9099pt" version="1.1" viewBox="-0.0498162 -9.33792 7.78304 12.9099" width="7.78304pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-243"/></g></svg>-primary and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57198pt" id="M9" height="12.9099pt" version="1.1" viewBox="-0.0498162 -9.33792 7.78304 12.9099" width="7.78304pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-243"/></g></svg>-prime elements in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M10" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 7.61985 9.04777" width="7.61985pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-77"/></g></svg> are obtained. Finally, some results for almost prime and almost primary elements in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M11" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 7.61985 9.04777" width="7.61985pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-77"/></g></svg> with characterizations are obtained.

A. V. Bingi

<svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.36796pt" height="12.6651pt" version="1.1" viewBox="-0.0498162 -9.29714 8.09582 12.6651" width="8.09582pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g5-243" d="M567 249C567 371 489 431 403 469C398 471 396 479 400 499L436 676L417 685L355 650L235 60C197 75 150 137 150 221C150 338 202 389 277 428L260 462C128 416 21 344 21 204C21 90 98 9 220 -10L200 -117C183 -207 215 -239 244 -246L250 -243L298 -8C425 2 567 94 567 249ZM478 235C478 131 407 46 313 47L382 397C413 387 478 326 478 235Z" /><glyph.data ascent="3481" descent="-2878" horiz-adv-x="588" vert-adv-y="588" /></g></svg>-Prime and <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.36796pt" height="12.6651pt" version="1.1" viewBox="-0.0498162 -9.29714 8.09582 12.6651" width="8.09582pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g5-243"/><glyph.data ascent="3481" descent="-2878" horiz-adv-x="588" vert-adv-y="588" /></g></svg>-Primary Elements in Multiplicative Lattices

We prove that a free group, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.4278pt" id="M2" height="15.5059pt" version="1.1" viewBox="-0.0498162 -12.0781 61.4336 15.5059" width="61.4336pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-71"/></g><g transform="matrix(.0135,0,0,-0.0135,8.228,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,12.916,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.0095,0,0,-0.0095,23.402,3.264)"><use xlink:href="#g50-100"/></g><g transform="matrix(.0135,0,0,-0.0135,30.841,0)"><use xlink:href="#g117-63"/></g><g transform="matrix(.0135,0,0,-0.0135,41.821,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.0095,0,0,-0.0095,51.341,-5.984)"><use xlink:href="#g50-51"/></g><g transform="matrix(.0135,0,0,-0.0135,56.476,0)"><use xlink:href="#g113-42"/></g></svg>, relative to the variety, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.4278pt" id="M3" height="15.5059pt" version="1.1" viewBox="-0.0498162 -12.0781 43.7685 15.5059" width="43.7685pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.0095,0,0,-0.0095,10.486,3.264)"><use xlink:href="#g50-100"/></g><g transform="matrix(.0135,0,0,-0.0135,17.925,0)"><use xlink:href="#g117-63"/></g><g transform="matrix(.0135,0,0,-0.0135,28.906,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.0095,0,0,-0.0095,38.426,-5.984)"><use xlink:href="#g50-51"/></g></svg>, of all groups simultaneously nilpotent of class at most <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M4" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 5.62081 6.42315" width="5.62081pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-100" d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z"/></g></svg> and metabelian is such that the centralizer of every noncentral element is abelian. We relate that result to the model theory of such groups as well as a quest to find a relative analog in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.4278pt" id="M5" height="15.5059pt" version="1.1" viewBox="-0.0498162 -12.0781 43.7685 15.5059" width="43.7685pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.0095,0,0,-0.0095,10.486,3.264)"><use xlink:href="#g50-100"/></g><g transform="matrix(.0135,0,0,-0.0135,17.925,0)"><use xlink:href="#g117-63"/></g><g transform="matrix(.0135,0,0,-0.0135,28.906,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.0095,0,0,-0.0095,38.426,-5.984)"><use xlink:href="#g50-51"/></g></svg> of a classical theorem of Benjamin Baumslag. We also touch briefly
on similar considerations in the varieties <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427751pt" id="M6" height="12.4257pt" version="1.1" viewBox="-0.0498162 -8.99795 15.0358 12.4257" width="15.0358pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.0095,0,0,-0.0095,10.486,3.264)"><use xlink:href="#g50-100"/></g></svg> of nilpotent groups.

Anthony M. Gaglione

Seymour Lipschutz

Dennis Spellman

The Relatively Free Groups <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.4278pt" height="15.5059pt" version="1.1" viewBox="-0.0498162 -12.0781 61.4336 15.5059" width="61.4336pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="607" vert-adv-y="607" /></g><g transform="matrix(.0135,0,0,-0.0135,8.228,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346" /></g><g transform="matrix(.0135,0,0,-0.0135,12.916,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="839" vert-adv-y="839" /></g><g transform="matrix(.0095,0,0,-0.0095,23.402,3.264)"><path id="g50-100" d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z" /><glyph.data ascent="3443" descent="-2856" horiz-adv-x="411" vert-adv-y="411" /></g><g transform="matrix(.0135,0,0,-0.0135,30.841,0)"><path id="g117-63" d="M574 0L325 548H263L14 0H76L293 479H295L512 0H574Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587" /></g><g transform="matrix(.0135,0,0,-0.0135,41.821,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="703" vert-adv-y="703" /></g><g transform="matrix(.0095,0,0,-0.0095,51.341,-5.984)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" /><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" /></g><g transform="matrix(.0135,0,0,-0.0135,56.476,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346" /></g></svg> Satisfy Noncentral Commutative Transitivity

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Algebra