Download PDFOpen PDF in browserImplicativity Versus Filtrality, Disjunctivity and Finite SemiSimplicityEasyChair Preprint no. 3096, version 425 pages•Date: July 10, 2023AbstractExtending the notion of an implicative system for a class of algebras by admitting existential parameters, we come to that of [ restricted}, viz., parameterless] one, {quasi}varieties with {relatively} subdirectlyirreducibles {[i.e., those generated by subclasses]} with [restricted] implicative system being called [restricted] implicative. Likewise, a {quasi}variety is said to be {relatively} [sub]directly filtral/congruencedistributive, if {relative} congruences /lattice of any [sub]direct product of its {relatively} subdirectlyirreducibles are/is filtral/distributive, prevarieties (viz., abstract hereditary multiplicative classes) generated by subclasses with <finite> disjunctive system being called <finitely> disjunctive. The main general results of the work are that any /{quasi}equational {pre}variety is /<finitely> disjunctive iff it is {relatively} congruencedistributive with {its members isomorphic to subdirect products of relatively finitelysubdirectlyirreducible ones}/ and the class of its {relatively} finitelysubdirectlyirreducible members being ``a universal /<firstorder> model class''``hereditary /<and closed under ultraproducts$>'', while any {quasi}variety is [restricted] implicative it is {relatively} [sub]directly filtral iff it is {relatively} [<finitely>semisimple (i.e., its {relatively} [<finitely>]subdirectlyirreducibles are {relatively} simple) and [sub]directly congruencedistributive with the class of {relatively} simple members being ``a [universal] firstorder model one''``[hereditary and] closed under ultraproducts'' [iff it is disjunctive and {relatively} finitelysemisimple] if[f] it is {relatively} semisimple and has [R]EDP{R}C. Keyphrases: disjunctive, filtral, implicative, quasivariety Download PDFOpen PDF in browser 
