Download PDFOpen PDF in browserImplicativity Versus Filtrality, Disjunctivity and Equality DeterminantsEasyChair Preprint no. 3096, version 227 pages•Date: April 12, 2020AbstractThe 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 ]semisimple and [relatively ](sub)directly congruencedistributive, while the class of all its [relatively ]simple and oneelement members is either a (universal )firstorder model class or (both hereditary and )ultraclosed, if(f) it is [relatively ]semisimple and has (R)EDP[R]C. Likewise, a [quasi]variety is (finitely )restricted disjunctive iff it is [relatively ]congruencedistributive, while the class of all its [relatively ]finitelysubdirectlyirreducible and oneelement members is a universal (firstorder )model class, that is, (both ultraclosed and )hereditary. In addition, we prove that a locally finite [quasi]variety is restricted implicative iff it is both (finitely )restricted disjunctive and [relatively ]semisimple. Moreover, we prove that the quasivariety generated by a class of lattice expansions, nononeelement finite subalgebras of which are all simple, is a restricted implicative variety, whenever it is locally finite, its simple/(finitely)subdirectlyirreducible members being exactly isomorphic copies of nononeelement subalgebras of ultraproducts of members of the class. Likewise, we prove that the quasivariety generated by a [finite ]class of [finite ]lattice expansions, nononeelement finite subalgebras of which are all subdirectlyirreducible, is restricted finitely disjunctive, whenever it is a locally finite variety, its (finitely)subdirectlyirreducibles being (exactly )/[exactly ] isomorphic copies of nononeelement subalgebras of ultraproducts of members of the class. Keyphrases: disjunctive, filtral, implicative, quasivariety
