Algebraic structures of weakened lattices and applications

Abstract

<p>Ako je&nbsp; Lalgebarska mreža i&nbsp; a kodistributivan elemenat u L, onda sve klase kongruencije&nbsp; (pa indukovane homomorfizmom&nbsp; ma : xi&mdash; &gt;&nbsp; a&nbsp;&nbsp; A x imaju najveće elemente. Najveći elemenat klase kojoj&nbsp; x E Lpripada je označen sa&nbsp; x.Ako je&nbsp; *a binarna op&shy;eracija definisana sa x *ay = (x A y)&nbsp; V&nbsp; (x A y),onda je istraživana struktura&nbsp; (L, *a), i odgovarajući poset ( L, &lt;&raquo;). Kao primer takve strukture posmatrana je algebra slabih kongruencija&nbsp; (CwA, *a), gde je *a specijalna grafička kompozicija. Dobijeni rezultati daju prirodne posledice u strukturi slabih kongruencija. Data je primena ovih rezultata u univerzalnoj algebri. Njihovom primenom karakterizuje se CEP i Hamiltonovo svojstvo. Dat je potreban i dovoljan uslov da poset (L, &lt; -) bude mreža i ovi rezultati su primenjeni na mrežu slabih kongruencija.</p>

<p>If&nbsp; Lis an algebraic lattice and&nbsp; a codistributive element in&nbsp; L,then all the classes of the congruences&nbsp; 4&gt;a determined by the homomorphism&nbsp; ma :&nbsp; x&nbsp; \&mdash; &gt;&nbsp; a Ax&nbsp;&nbsp; have top elements. The top element of the class which to belongs an&nbsp; x&nbsp;&nbsp; &euro;&nbsp; Lis denoted by&nbsp; x.&nbsp;&nbsp; If *a is a binary operation defined by&nbsp; x&nbsp; *ay=&nbsp; (xA y)&nbsp; V&nbsp; (xA y),then we investigate the<br />structure&nbsp; (L,*a), and the corresponding poset&nbsp; (L, &lt; t ). Asan example of such a structure we observe an algebra of weak congruences ( C w A , * a),where *a is a special graphical composition. We obtain natural conse&shy; quences of the mentioned results to the structure weak congruences. An application in universal algebra is presented, for example, we characterized CEP and Hamiltonian property. Necessary and sufficient conditions for a poset&nbsp; (L,&lt;*) to be a lattice are given, and the results are applied in the case of weak congruence lattices.</p>