Implicativity Versus Filtrality, Disjunctivity and Equality Determinants

EasyChair Preprint 3096, version 3

Abstract

The main result of the work is the fact that a [quasi]variety is (restricted )implicative iff it is [relatively ](sub)directly filtral( iff it is [relatively ]filtral) iff it is both [relatively ]semi-simple and [relatively ](sub)directly congruence-distributive, while the class of all its [relatively ]simple and one-element members is either a (universal )first-order model class or (both hereditary and )ultra-closed, if(f) it is [relatively ]semi-simple and has (R)EDP[R]C. Likewise, a [quasi]variety is (finitely )restricted disjunctive iff it is [relatively ]congruence-distributive, while the class of all its [relatively ]finitely-subdirectly-irreducible and one-element members is a universal (first-order )model class, that is, (both ultra-closed and )hereditary. In addition, we prove that a locally finite [quasi]variety is restricted implicative iff it is both (finitely )restricted disjunctive and [relatively ]semi-simple. Moreover, we prove that the quasivariety generated by a class of lattice expansions, non-one-element finite subalgebras of which are all simple, is a restricted implicative variety, whenever it is locally finite, its simple/(finitely-)subdirectly-irreducible members being exactly isomorphic copies of non-one-element subalgebras of ultraproducts of members of the class. Likewise, we prove that the quasivariety generated by a [finite ]class of [finite ]lattice expansions, non-one-element finite subalgebras of which are all subdirectly-irreducible, is restricted finitely disjunctive, whenever it is a locally finite variety, its (finitely-)subdirectly-irreducibles being (exactly )/[exactly ] isomorphic copies of non-one-element subalgebras of ultraproducts of members of the class.

Keyphrases: disjunctive, filtral, implicative, quasivariety

@booklet{EasyChair:3096,
  author    = {Alexej Pynko},
  title     = {Implicativity Versus Filtrality, Disjunctivity and Equality Determinants},
  howpublished = {EasyChair Preprint 3096},
  year      = {EasyChair, 2023}}