Special elements in lattices and applications

Abstract

<p>Data je karakterizacija raznih tipova specijalnih elemenata mreže, kao &scaron;to su kodistributivni, neutralni, skrativi, standardni, izuzetni, neprekidni, beskonačno distributivni i drugi i ti rezultati su primenjeni u strukturnim ispitivanjima algebri, posebno u mrežama kongruencija, podalgebri i slabih kongruencija algebri.&nbsp; Specijalni elementi su posebno proučavani i u bipolumrežama i dobijene su nove teoreme reprezentacije za bipolumreže. Ispitana je kolekcija svih mreža sa istim skupom i-nerazloživih elemenata, pokazano je da je ta kolekcija i sama mreža u odnosu na inkluziju i daju se karakterizacije te mreže.&nbsp; Re&scaron;avan je problem preno&scaron;enja mrežnih identiteta sa mreže podalgebri i kongruencija na mrežu slabih kongruencija. Proučavane su osobine svojstva preseka kongruencija i svojstva pro&scaron;irenja kongruencija i neke varijante tih svojstava u vezi sa mrežama slabih kongruencija. Date su karakterizacije mreže slabih kongruencija nekih posebnih klasa algebri i varijeteta, kao &scaron;to su unarne algebra, mreže, grupe, Hamiltonove algebra i druge.</p>

<p>A characterization of various types of special elements in lattices: codistributive,&nbsp; neutral, cancellable, standard, exceptional, continuous, infinitely distributive and others are given, and the results are applied in structural investigations in algebras, in particular in lattices of subalgebras, congruences and weak congruences. Special elements are investigated also in bi-semilattices and new representation theorems for bisemilattices are obtained. The collection of all lattices with the same poset of meet-irreducible elements is studied and it is proved that this collection is a lattice under inclusion and characterizations of this lattice is given.&nbsp; A problem of transferability of lattice identities from lattices of subalgebras and congruences to&nbsp; lattices of weak congruencse of&nbsp; algebras is solved. The congruence intersection property and the congruence extension property as well as various alternations of these properties are investigated in connection with weak congruence lattices. Characterizations of weak congruence lattices of special classes of algebras and varieties, as unary algebras, lattices, groups, Hamiltonian algebras and others are given.</p>