Implicativity Versus Filtrality, Disjunctivity and Finite Semi-Simplicity

EasyChair Preprint no. 3096, version 4

Abstract

Extending the notion of an implicative system for a class of algebras by
admitting existential parameters, we come to that of [
restricted}, viz., parameter-less] one,
{quasi-}varieties with {relatively}
subdirectly-irreducibles
{[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/congruence-distributive, if
{relative} congruences /lattice of any [sub]direct product of its
{relatively} subdirectly-irreducibles are/is
filtral/distributive, pre-varieties
(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} congruence-distributive
with {its members isomorphic to subdirect products
of relatively finitely-subdirectly-irreducible ones}/
and the class of its {relatively}
finitely-subdirectly-irreducible members being
``a universal /<first-order> model
class''|``hereditary /<and closed under
ultra-products$>'', while
any {quasi-}variety is [restricted] implicative
it is {relatively} [sub]directly filtral iff
it is {relatively} [<finitely->semi-simple
(i.e., its {relatively} [<finitely->]subdirectly-irreducibles are
{relatively} simple) and [sub]directly congruence-distributive
with the class of {relatively} simple members being
``a [universal] first-order model one''|``[hereditary and]
closed under ultra-products''
[iff it is disjunctive and {relatively} finitely-semi-simple]
if[f] it is {relatively} semi-simple and has [R]EDP{R}C.

Keyphrases: disjunctive, filtral, implicative, quasivariety

@Booklet{EasyChair:3096,
  author = {Alexej Pynko},
  title = {Implicativity Versus Filtrality, Disjunctivity and Finite Semi-Simplicity},
  howpublished = {EasyChair Preprint no. 3096},

  year = {EasyChair, 2023}}