Gerald Bourgeois

The Matrix Equation <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.572001pt" height="13.9461pt" version="1.1" viewBox="-0.0498162 -10.3741 113.203 13.9461" width="113.203pt"><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" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="765" vert-adv-y="765" /></g><g transform="matrix(.0135,0,0,-0.0135,9.957,0)"><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(.0135,0,0,-0.0135,22.505,0)"><path id="g117-33" d="M535 230V280H52V230H535Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587" /></g><g transform="matrix(.0135,0,0,-0.0135,33.486,0)"><use xlink:href="#g113-66"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="703" vert-adv-y="703" /></g><g transform="matrix(.0135,0,0,-0.0135,43.006,0)"><use xlink:href="#g113-89"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="765" vert-adv-y="765" /></g><g transform="matrix(.0135,0,0,-0.0135,57.155,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587" /></g><g transform="matrix(.0135,0,0,-0.0135,68.893,0)"><use xlink:href="#g113-89"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="765" vert-adv-y="765" /></g><g transform="matrix(.0095,0,0,-0.0095,79.258,-5.984)"><path id="g50-223" d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z" /><glyph.data ascent="3443" descent="-2856" horiz-adv-x="582" vert-adv-y="582" /></g><g transform="matrix(.0135,0,0,-0.0135,85.296,0)"><path id="g113-104" d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="569" vert-adv-y="569" /></g><g transform="matrix(.0135,0,0,-0.0135,93.006,0)"><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,-0.0135,97.694,0)"><use xlink:href="#g113-89"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="765" vert-adv-y="765" /></g><g transform="matrix(.0135,0,0,-0.0135,108.058,0)"><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> over Fields or Rings

We introduce the concept of almost semiprime submodules
of unitary modules over a commutative ring with nonzero identity. We investigate
some basic properties of almost semiprime and weakly semiprime
submodules and give some characterizations of them, especially for (finitely
generated faithful) multiplication modules.

Farkhonde Farzalipour

On Almost Semiprime Submodules

Swan’s theorem determines the parity of the number of irreducible factors of a binary trinomial. In this work, we study the parity of the number of irreducible factors for a special binary pentanomial with even degree <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3626003pt" id="M1" height="10.7367pt" version="1.1" viewBox="-0.0498162 -10.3741 128.894 10.7367" width="128.894pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/></g><g 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xlink:href="#g113-111"/></g><g transform="matrix(.0095,0,0,-0.0095,56.15,3.264)"><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><g transform="matrix(.0135,0,0,-0.0135,65.069,0)"><use xlink:href="#g117-91"/></g><g transform="matrix(.0135,0,0,-0.0135,76.806,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0095,0,0,-0.0095,83.54,3.264)"><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,-0.0135,92.459,0)"><path id="g117-93" d="M531 71V127L115 310L531 494V550L57 335V285L531 71ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.0135,0,0,-0.0135,104.197,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 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23V0H405C417 45 427 89 440 131L412 140Z"/></g></svg>, and exactly one of  <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.31356pt" id="M3" height="9.52372pt" version="1.1" viewBox="-0.0498162 -6.21016 29.2566 9.52372" width="29.2566pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0095,0,0,-0.0095,6.733,3.264)"><use xlink:href="#g50-50"/></g><g transform="matrix(.0135,0,0,-0.0135,11.868,0)"><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,-0.0135,17.227,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0095,0,0,-0.0095,23.961,3.264)"><use xlink:href="#g50-51"/></g></svg>,  and  <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42779pt" id="M4" height="9.63795pt" version="1.1" viewBox="-0.0498162 -6.21016 11.9923 9.63795" width="11.9923pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0095,0,0,-0.0095,6.733,3.264)"><use xlink:href="#g50-52"/></g></svg> is odd. This kind of irreducible pentanomials can be used for a fast implementation of trace and square root computations in finite fields of characteristic 2.

Ryul Kim

Yun Mi Kim

The Reducibility of a Special Binary Pentanomial

We consider unions of parafree Lie algebras and we prove that such unions are again parafree under some conditions.

Naime Ekici

Zehra Velio&#x11f;lu

Unions of Parafree Lie Algebras

Two special classes of symmetric coefficient matrices were defined based on characteristics matrix; meanwhile, the expressions of the solution to inverse problems are given and the conditions for the solvability of these problems are studied relying on researching. Finally, the optimal approximation solution of these problems is provided.

Rui Huang

Xiaodong Wu

Ruihe Wang

Hui Li

A Study of Inverse Problems Based on Two Kinds of Special Matrix Equations in Euclidean Space

We introduce the notion of Demazure descent data on a triangulated category <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1881pt;width:13.675px;" id="M1" height="12.0625" version="1.1" viewBox="0 0 13.675 12.0625" width="13.675">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.738)"><path id="x1D49E" d="M449 645l-12 -19q-55 31 -136 31q-52 0 -103 -17t-85.5 -54.5t-34.5 -84.5q0 -60 41.5 -96.5t109.5 -43.5q81 125 216 216q137 92 240 92q46 0 71 -19.5t25 -55.5q0 -60 -67 -123q-66 -63 -177 -105q-109 -42 -225 -42q-69 -100 -69 -199q0 -49 31.5 -79.5t85.5 -30.5&#xA;q79 0 141 50q61 49 61 108q0 35 -20 54.5t-51 19.5q-53 0 -103 -53q-47 -52 -55 -131l-22 2q0 91 58 158t139 67q55 0 86.5 -26.5t31.5 -76.5q0 -81 -96 -148q-78 -54 -174 -54q-86 0 -141 46q-56 47 -56 127q0 81 48 171q-72 12 -121 55t-49 115q0 75 71 131q72 57 185 57&#xA;q91 0 156 -42zM331 355l1 -1q122 0 244 62q73 37 118.5 85.5t45.5 90.5q0 40 -50 40q-73 0 -174 -79q-102 -80 -185 -198z"></path></g>

		
	
</svg> and define the descent category for such data. We illustrate the definition by our basic example. Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.15048pt;width:12.4375px;" id="M2" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z"></path></g>

		
	
</svg> be a reductive algebraic group with a Borel subgroup <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:10.6125px;" id="M3" height="11.175" version="1.1" viewBox="0 0 10.6125 11.175" width="10.6125">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z"></path></g>

		
	
</svg>. Demazure functors form Demazure descent data on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.72116pt;width:80.737503px;" id="M4" height="20.8375" version="1.1" viewBox="0 0 80.737503 20.8375" width="80.737503">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><path id="x1D437" d="M780 372q0 -191 -178 -297q-128 -75 -335 -75h-250l7 27q61 7 77 22t28 76l77 398q11 62 0 76.5t-77 22.5l6 28h292q164 0 258.5 -67.5t94.5 -210.5zM678 375q0 119 -73 180.5t-201 61.5q-62 0 -82 -13q-17 -7 -25 -53l-80 -427q-8 -47 -4 -62t26 -20q24 -6 80 -6&#xA;q166 0 262.5 95t96.5 244z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,13.712,8.537)"><path id="x1D44F" d="M452 333q0 -82 -46 -164t-116 -128q-81 -53 -158 -53q-26 0 -51 13t-37 32q-26 41 -6 134l91 431q7 40 2.5 47t-34.5 7h-33l2 25q36 4 72.5 13t58.5 15.5t28 6.5q10 0 4 -29l-91 -391h2q65 77 130 116.5t105 39.5q36 0 56.5 -32t20.5 -83zM365 316q0 76 -35 76&#xA;q-31 0 -93.5 -49t-107.5 -110q-12 -37 -17 -70q-9 -65 12 -96.5t59 -31.5q31 0 56 14q53 29 89.5 105t36.5 162z"></path></g>

			<g transform="matrix(.017,-0,0,-.017,19.912,16.7)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z"></path></g><g transform="matrix(.017,-0,0,-.017,25.794,16.7)"><path id="x52" d="M631 18l-4 -26q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -63 14 -77.5t74 -19.5v-28h-262v28q62 5 76 19.5t14 77.5v402q0 62 -13 76t-73 19v28h248q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172q27 -52 89 -144&#xA;q49 -71 81 -104q31 -36 72 -47zM212 316h48q73 0 108 29q55 42 55 124q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247z"></path></g><g transform="matrix(.017,-0,0,-.017,36.401,16.7)"><path id="x65" d="M380 106l15 -22q-42 -52 -87 -74t-76 -22q-89 0 -142 62t-53 151q0 103 66 175q67 73 146 73h1q61 0 105.5 -46.5t44.5 -105.5q0 -16 -4.5 -22.5t-18.5 -9.5q-54 -10 -262 -17q1 -95 46 -143.5t104 -48.5q61 0 115 50zM225 412q-35 0 -64.5 -32.5t-40.5 -90.5q83 0 170 3&#xA;q14 0 19 4.5t5 15.5q1 41 -23.5 70.5t-65.5 29.5z"></path></g><g transform="matrix(.017,-0,0,-.017,43.626,16.7)"><path id="x70" d="M169 380l92 53q26 16 49 16q80 0 128.5 -56t48.5 -133q0 -108 -72.5 -180.5t-176.5 -91.5q-38 1 -69 18v-131q0 -57 13 -70t71 -18v-27h-236v27q49 5 61 17t12 61v452q0 48 -9.5 58.5t-56.5 16.5v24q66 11 145 43v-79zM169 346v-270q37 -39 94 -39q63 0 101.5 50&#xA;t38.5 134q0 78 -38.5 123t-95.5 45q-45 0 -100 -43z"></path></g><g transform="matrix(.017,-0,0,-.017,52.533,16.7)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,58.415,16.7)"><use xlink:href="#x1D435"></use></g><g transform="matrix(.017,-0,0,-.017,68.903,16.7)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z"></path></g><g transform="matrix(.017,-0,0,-.017,74.784,16.7)"><use xlink:href="#x29"></use></g>

		
	
</svg> and the descent category is equivalent to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.72116pt;width:82.550003px;" id="M5" height="20.8375" version="1.1" viewBox="0 0 82.550003 20.8375" width="82.550003">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><use xlink:href="#x1D437"></use></g>

			<g transform="matrix(.012,-0,0,-.012,13.712,8.537)"><use xlink:href="#x1D44F"></use></g>

			<g transform="matrix(.017,-0,0,-.017,19.912,16.7)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,25.794,16.7)"><use xlink:href="#x52"></use></g><g transform="matrix(.017,-0,0,-.017,36.401,16.7)"><use xlink:href="#x65"></use></g><g transform="matrix(.017,-0,0,-.017,43.626,16.7)"><use xlink:href="#x70"></use></g><g transform="matrix(.017,-0,0,-.017,52.533,16.7)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,58.415,16.7)"><use xlink:href="#x1D43A"></use></g><g transform="matrix(.017,-0,0,-.017,70.722,16.7)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,76.603,16.7)"><use xlink:href="#x29"></use></g>

		
	
</svg>.

Sergey Arkhipov

Tina Kanstrup

Demazure Descent and Representations of Reductive Groups

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.55815pt;width:14.1375px;" id="M1" height="15.4125" version="1.1" viewBox="0 0 14.1375 15.4125" width="14.1375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.875)"><path id="x210C" d="M626 695l19 -13q-96 -135 -219 -135q-71 0 -133 50q-41 -33 -41 -91q0 -78 57 -139q161 94 224 127q113 -26 176.5 -100t63.5 -179q0 -75 -22 -120q-22 -46 -67 -92q-40 -42 -127 -104q-87 -63 -117 -103h-13q-60 62 -132 62q-69 0 -114 -56l-20 9q60 136 185 136&#xA;q71 0 129 -64q105 66 145 150q36 75 36 139q0 91 -52 156q-51 66 -133 66q-63 0 -143 -54l44 -56q35 -46 35 -82q0 -44 -38 -93q-40 -49 -139 -131q-23 18 -61 37q-43 24 -56 28q-33 -31 -46 -53l-24 15l104 147l76 -45q55 -33 59 -33q17 15 23 26t7 36q0 36 -79 121&#xA;q-32 36 -55 86t-23 97q0 73 75 145q31 31 61 57q27 24 66 53q58 -53 131 -53q88 0 138 53z"></path></g>

		
	
</svg> be a class of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3135pt;width:34.724998px;" id="M2" height="8.7124996" version="1.1" viewBox="0 0 34.724998 8.7124996" width="34.724998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,8.438)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z"></path></g><g transform="matrix(.017,-0,0,-.017,12.37,8.438)"><path id="xD7" d="M528 54l-36 -38l-198 201l-198 -201l-36 38l197 200l-197 201l36 38l198 -202l198 202l36 -38l-197 -201z"></path></g><g transform="matrix(.017,-0,0,-.017,26.122,8.438)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> Hankel matrices <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.13504pt;width:23.9375px;" id="M3" height="15.7" version="1.1" viewBox="0 0 23.9375 15.7" width="23.9375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path id="x1D407" d="M759 25v-25h-339v25q54 6 71.5 20.5t17.5 52.5v228h-241v-228q0 -40 17 -54q16 -13 72 -19v-25h-336v25q54 8 69 20q16 13 16 53v479q0 41 -16.5 55t-68.5 19v25h336v-25q-54 -4 -71.5 -18.5t-17.5 -55.5v-204h241v204q0 41 -17.5 55.5t-71.5 18.5v25h339v-25&#xA;q-51 -4 -69.5 -19t-18.5 -55v-479q0 -40 16 -53q15 -12 72 -20z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,14.538,15.637)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z"></path></g>

		
	
</svg> whose entries, depending on a given matrix <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:12.4px;" id="M4" height="11.85" version="1.1" viewBox="0 0 12.4 11.85" width="12.4">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.788)"><path id="x1D400" d="M689 25v-25h-322v25l32 3q46 4 46 33q0 31 -46 136h-225l-13 -34q-31 -78 -31 -97q0 -36 81 -41v-25h-202v25q36 6 53 28q18 22 46 92l221 545h28l222 -522q36 -87 58 -117q21 -26 52 -26zM189 236h195l-101 243z"></path></g>

		
	
</svg>, are linear forms in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:8.6625004px;" id="M5" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> variables with coefficients in a finite field <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.74141pt;width:18.862499px;" id="M6" height="18.3125" version="1.1" viewBox="0 0 18.862499 18.3125" width="18.862499">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D53D" d="M613 503l-29 -3q-13 55 -28 80q-13 21 -35.5 28.5t-78.5 7.5h-76q-22 0 -27.5 -5t-5.5 -27v-238h103q66 0 82.5 12t25.5 62h29v-193h-29q-9 54 -25.5 67t-82.5 13h-103v-181q0 -64 15 -78.5t84 -19.5v-28h-394v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28&#xA;h565q1 -40 10 -147zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5t7.5 79.5z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,12.337,15.188)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z"></path></g>

		
	
</svg>. For every matrix in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.55815pt;width:14.1375px;" id="M7" height="15.4125" version="1.1" viewBox="0 0 14.1375 15.4125" width="14.1375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.875)"><use xlink:href="#x210C"></use></g>

		
	
</svg>, it is shown that the varieties specified by the leading minors of orders from 1 to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:34.349998px;" id="M8" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><path id="x2212" d="M535 230h-483v50h483v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z"></path></g>

		
	
</svg> have the same number <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.29482pt;width:27.762501px;" id="M9" height="20.2125" version="1.1" viewBox="0 0 27.762501 20.2125" width="27.762501">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D45E"></use></g>

			<g transform="matrix(.012,-0,0,-.012,8.413,7.613)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.012,-0,0,-.012,14.386,7.613)"><use xlink:href="#x2212"></use></g><g transform="matrix(.012,-0,0,-.012,21.37,7.613)"><use xlink:href="#x31"></use></g>

		
	
</svg> of points in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.74141pt;width:19px;" id="M10" height="20.75" version="1.1" viewBox="0 0 19 20.75" width="19">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.55)"><use xlink:href="#x1D53D"></use></g>

			<g transform="matrix(.012,-0,0,-.012,12.337,5.388)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.012,-0,0,-.012,12.337,17.638)"><use xlink:href="#x1D45E"></use></g>

		
	
</svg>. Further properties are derived, which show that sets of varieties, tied to a given Hankel matrix, resemble a set of hyperplanes as regards the number of points of their intersections.

Edoardo Ballico

Michele Elia

On Determinantal Varieties of Hankel Matrices

We establish new results concerning endomorphisms of a finite chain, if the cardinality of the image of such endomorphism is no more than some fixed number. The semiring of all such endomorphisms can be seen as a simplex whose vertices are the constant endomorphisms. We explore the properties of these simplices.

Ivan Trendafilov

Simplices in the Endomorphism Semiring of a Finite Chain

We discuss several partial solutions to the so-called &#x201c;coquecigrue problem&#x201d; of Loday; these solutions parallel, but also generalize in several directions, the classical Lie group-Lie algebra correspondence. Our study highlights some clear similarities between the split and nonsplit cases and leads us to a general unifying scheme that provides an answer to the problem of the algebraic structure of a coquecigrue.

Juan Monterde

Fausto Ongay

On Integral Manifolds for Leibniz Algebras

We present a method to iteratively construct new bent functions of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.51414pt;width:34.349998px;" id="M3" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><use xlink:href="#x2B"></use></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><use xlink:href="#x32"></use></g>

		
	
</svg> variables from a bent function of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:8.6625004px;" id="M4" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> variables and its cyclic shift permutations using minterms of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:8.6625004px;" id="M5" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> variables and minterms of 2 variables. In addition, we provide the number of bent functions of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.51414pt;width:34.349998px;" id="M6" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><use xlink:href="#x2B"></use></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><use xlink:href="#x32"></use></g>

		
	
</svg> variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions.

Joan-Josep Climent

Francisco J. Garc&#xed;a

Ver&#xf3;nica Requena

A Construction of Bent Functions of <svg style="vertical-align:-0.51414pt" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg> Variables from a Bent Function of <svg style="vertical-align:-0.10033pt" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"/></g></svg> Variables and Its Cyclic Shifts

Based on the definition of vertex coalgebra introduced by Hubbard, 2009, we
prove that this notion can be reformulated using coskew symmetry, coassociator and
cocommutator formulas without restrictions on the grading. We also prove that a
vertex coalgebra can be defined in terms of dual versions of the axioms of Lie conformal
algebra and differential algebra.

Florencia Orosz Hunziker

Jos&#xe9; I. Liberati

Vertex Coalgebras, Coassociator, and Cocommutator Formulas

We introduce the notion of ordered quasi-<svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M1" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x393" d="M485 501l-30 -4q-15 64 -35 90q-12 16 -33.5 22t-68.5 6h-74q-21 0 -26.5 -5t-5.5 -26v-458q0 -62 15.5 -78t83.5 -20v-28h-270v28q58 4 72 20t14 78v399q0 61 -13.5 77t-71.5 20v28h433q0 -51 10 -149z" /></g>

		
	
</svg>-ideals of regular ordered <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M2" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-semigroups and study the basic properties of ordered quasi-<svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M3" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideals of ordered <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M4" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-semigroups. We also characterize regular ordered <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M5" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-semigroups in terms of their ordered quasi-<svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M6" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideals, ordered right <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M7" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideals, and left <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M8" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideals. Finally, we have shown that (i) a partially ordered <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M9" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-semigroup <svg     style="vertical-align:-0.23206pt;width:8.2875004px;" id="M10" height="11.75" version="1.1" viewBox="0 0 8.2875004 11.75" width="8.2875004"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g>

		
	
</svg> is regular if and only if for every ordered bi-<svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M11" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideal <svg     style="vertical-align:-0.0pt;width:10.6125px;" id="M12" height="11.175" version="1.1" viewBox="0 0 10.6125 11.175" width="10.6125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g>

		
	
</svg>, every ordered <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M13" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideal <svg     style="vertical-align:-0.0pt;width:7.5500002px;" id="M14" height="11.175" version="1.1" viewBox="0 0 7.5500002 11.175" width="7.5500002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43C" d="M414 650l-6 -28q-63 -4 -78.5 -17.5t-27.5 -71.5l-77 -416q-11 -58 0.5 -71.5t77.5 -17.5l-5 -28h-275l8 28q63 4 79.5 18.5t27.5 70.5l80 416q11 57 -2 71t-81 18l6 28h273z" /></g>

		
	
</svg>, and every ordered quasi-<svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M15" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideal <svg     style="vertical-align:-2.2932pt;width:13.175px;" id="M16" height="14.3125" version="1.1" viewBox="0 0 13.175 14.3125" width="13.175"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D444" d="M745 361q0 -134 -83.5 -233t-214.5 -130l16 -11q97 -67 250 -132l-8 -23q-76 3 -131 16q-81 19 -242 125l-20 13q-129 8 -209 91t-80 208q0 160 116 271t289 111q136 0 226.5 -83t90.5 -223zM645 356q0 127 -57.5 201.5t-169.5 74.5q-126 0 -210.5 -104.5t-84.5 -248.5&#xA;q0 -97 46 -166.5t129 -87.5l84 15l29 -19q104 21 169 121.5t65 213.5z" /></g>

		
	
</svg>, we have <svg     style="vertical-align:-2.3205pt;width:145.53751px;" id="M17" height="15.0875" version="1.1" viewBox="0 0 145.53751 15.0875" width="145.53751"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D435"/></g><g transform="matrix(.017,-0,0,-.017,14.324,12.138)"><path id="x2229" d="M535 0h-58v354q0 68 -54.5 113.5t-128.5 45.5t-128.5 -45.5t-54.5 -113.5v-354h-58v354q0 90 71.5 149.5t169.5 59.5t169.5 -59.5t71.5 -149.5v-354z" /></g><g transform="matrix(.017,-0,0,-.017,28.076,12.138)"><use xlink:href="#x1D43C"/></g><g transform="matrix(.017,-0,0,-.017,39.279,12.138)"><use xlink:href="#x2229"/></g><g transform="matrix(.017,-0,0,-.017,53.031,12.138)"><use xlink:href="#x1D444"/></g><g transform="matrix(.017,-0,0,-.017,70.811,12.138)"><path id="x2286" d="M531 95h-193q-116 0 -198.5 66.5t-82.5 160.5t82.5 161t198.5 67h193v-50h-192q-93 0 -158.5 -52t-65.5 -126q0 -73 65.5 -125t158.5 -52h192v-50zM531 -40h-445v50h445v-50z" /></g><g transform="matrix(.017,-0,0,-.017,85.515,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,91.397,12.138)"><use xlink:href="#x1D435"/></g><g transform="matrix(.017,-0,0,-.017,101.885,12.138)"><use xlink:href="#x393"/></g><g transform="matrix(.017,-0,0,-.017,110.503,12.138)"><use xlink:href="#x1D43C"/></g><g transform="matrix(.017,-0,0,-.017,117.932,12.138)"><use xlink:href="#x393"/></g><g transform="matrix(.017,-0,0,-.017,126.55,12.138)"><use xlink:href="#x1D444"/></g><g transform="matrix(.017,-0,0,-.017,139.605,12.138)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g>

		
	
</svg> and (ii) a partially ordered <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M18" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-semigroup <svg     style="vertical-align:-0.23206pt;width:8.2875004px;" id="M19" height="11.75" version="1.1" viewBox="0 0 8.2875004 11.75" width="8.2875004"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D446"/></g>

		
	
</svg> is regular if and only if for every ordered quasi-<svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M20" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideal <svg     style="vertical-align:-2.2932pt;width:13.175px;" id="M21" height="14.3125" version="1.1" viewBox="0 0 13.175 14.3125" width="13.175"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D444"/></g>

		
	
</svg>, every ordered left <svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M22" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideal <svg     style="vertical-align:-0.0pt;width:9.8249998px;" id="M23" height="11.175" version="1.1" viewBox="0 0 9.8249998 11.175" width="9.8249998"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g>

		
	
</svg>, and every ordered right-<svg     style="vertical-align:-0.0pt;width:8.7375002px;" id="M24" height="11.175" version="1.1" viewBox="0 0 8.7375002 11.175" width="8.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x393"/></g>

		
	
</svg>-ideal <svg     style="vertical-align:-0.1092pt;width:11.075px;" id="M25" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D445" d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117&#xA;t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" /></g>

		
	
</svg>, we have that <svg     style="vertical-align:-2.3205pt;width:151.0125px;" id="M26" height="15.0875" version="1.1" viewBox="0 0 151.0125 15.0875" width="151.0125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D445"/></g><g transform="matrix(.017,-0,0,-.017,14.783,12.138)"><use xlink:href="#x2229"/></g><g transform="matrix(.017,-0,0,-.017,28.535,12.138)"><use xlink:href="#x1D444"/></g><g transform="matrix(.017,-0,0,-.017,45.364,12.138)"><use xlink:href="#x2229"/></g><g transform="matrix(.017,-0,0,-.017,59.116,12.138)"><use xlink:href="#x1D43F"/></g><g transform="matrix(.017,-0,0,-.017,73.548,12.138)"><use xlink:href="#x2286"/></g><g transform="matrix(.017,-0,0,-.017,88.252,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,94.134,12.138)"><use xlink:href="#x1D445"/></g><g transform="matrix(.017,-0,0,-.017,105.081,12.138)"><use xlink:href="#x393"/></g><g transform="matrix(.017,-0,0,-.017,113.699,12.138)"><use xlink:href="#x1D444"/></g><g transform="matrix(.017,-0,0,-.017,126.754,12.138)"><use xlink:href="#x393"/></g><g transform="matrix(.017,-0,0,-.017,135.373,12.138)"><use xlink:href="#x1D43F"/></g><g transform="matrix(.017,-0,0,-.017,145.079,12.138)"><use xlink:href="#x5D"/></g>

		
	
</svg>.

M. Y. Abbasi

Abul Basar

On Ordered Quasi-Gamma-Ideals of Regular Ordered Gamma-Semigroups

We introduce the notions of the commutative and bounded BE-Algebras. We give some related properties of them.

Zekiye &#xc7;ilo&#x11f;lu

Y&#x131;lmaz &#xc7;even

Commutative and Bounded BE-algebras

The notion of <svg style="vertical-align:-0.1638pt;width:9.7749996px;" id="M2" height="9.0625" version="1.1" viewBox="0 0 9.7749996 9.0625" width="9.7749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.625,8.238)"><use xlink:href="#x1D45B" /></g></svg>-ary semimodules is introduced so that the Jacobson radical of an <svg style="vertical-align:-2.3205pt;width:41.662498px;" id="M3" height="16.200001" version="1.1" viewBox="0 0 41.662498 16.200001" width="41.662498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.625,12.687)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,6.507,12.687)"><use xlink:href="#x1D45A" /></g><g transform="matrix(.017,-0,0,-.017,19.919,12.687)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,26.616,12.687)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,35.149,12.687)"><use xlink:href="#x29" /></g></svg>-semiring is studied and some well-known results concerning the Jacobson radical of a ring (a semiring or a ternary semiring) are generalized to an <svg style="vertical-align:-2.3205pt;width:41.662498px;" id="M4" height="16.200001" version="1.1" viewBox="0 0 41.662498 16.200001" width="41.662498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.625,12.687)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,6.507,12.687)"><use xlink:href="#x1D45A" /></g><g transform="matrix(.017,-0,0,-.017,19.919,12.687)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,26.616,12.687)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,35.149,12.687)"><use xlink:href="#x29" /></g></svg>-semiring.

Yongwen Zhu

On the Jacobson Radical of an <svg style="vertical-align:-2.3205pt" height="15.0875" version="1.1" viewBox="0 0 40.537498 15.0875" width="40.537498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" /></g><g transform="matrix(.017,-0,0,-.017,19.356,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,26.054,12.138)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,34.587,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg>-Semiring

Using the idea of the new sort of fuzzy subnear-ring of a near-ring, fuzzy subgroups, and their generalizations defined by various researchers, we try to introduce the notion of (<svg style="vertical-align:-2.29482pt;width:47.125px;" id="M2" height="13.8125" version="1.1" viewBox="0 0 47.125 13.8125" width="47.125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,9.375)"><path id="x1D716" d="M401 397q0 -24 -24 -40q-18 -13 -31 3q-36 45 -93 45q-48 0 -86.5 -43t-51.5 -112l206 6l-8 -48l-203 6v-25q0 -68 26.5 -107t73.5 -39q73 0 148 64l16 -22q-40 -44 -91 -70.5t-102 -26.5q-75 0 -116.5 45t-41.5 136q0 125 99 208q36 30 87 50.5t93 20.5q46 0 72.5 -16&#xA;t26.5 -35z" /></g><g transform="matrix(.017,-0,0,-.017,7.27,9.375)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,13.967,9.375)"><use xlink:href="#x1D716" /></g><g transform="matrix(.017,-0,0,-.017,24.949,9.375)"><path id="x2228" d="M574 548l-249 -548h-62l-249 548h62l217 -479h2l217 479h62z" /></g><g transform="matrix(.017,-0,0,-.017,38.718,9.375)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" /></g></svg>)-fuzzy ideals of <svg style="vertical-align:-0.0pt;width:14.9875px;" id="M3" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D441" /></g></svg>-groups. These fuzzy ideals are characterized by their level ideals, and some other related properties are investigated.

B. Davvaz

O. Ratnabala Devi

A Study on Fuzzy Ideals of <svg style="vertical-align:-0.0pt" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153&#xA;l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g></svg>-Groups

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Algebra