# Subprevarieties of Algebraic Systems Versus Extensions of Logics: Application to Some Many-Valued Logics

### EasyChair Preprint no. 3942

22 pages•Date: July 25, 2020### Abstract

Here, we study applications of the factual interpretability of [equ\-ivalence between]

the equality-free infinitary universal Horn theory

(in particular, the sentential logic) of a class of algebraic systems

(in particular, logical matrices)

[with equality uniformly definable by

a set of atomic equality-free formulas] in [and]

the prevariety generated by the class, in which case

the lattice of extensions of the former is

a Galois retract of [dual to]

that of all subprevarieties of the prevariety,

the retraction [duality] retaining relative equality-free

infinitary universal Horn axiomatizations.

As representative instances,

we explore:

(1) the classical (viz., Boolean) expansion of Belnap's four-valued

logic that is not equivalent to any class of pure algebras

but is equivalent

to the quasivariety of filtered De Morgan Boolean algebras that

are matrices with underlying algebra being a De Morgan

Boolean algebra,

truth predicate being a filter of it and equality being

definable by a strong equivalence connective,

proving that prevarieties of such structures

form an eight-element non-chain distributive lattice,

and so do extensions of the expansion involved;

(2) Kleene's three-valued logic that is neither interpretable

in pure algebras nor equivalent to a prevariety

of algebraic systems, but is interpretable into

the quasivariety of resolutional filtered

Kleene lattices that are matrices

with underlying algebra being a Kleene lattice

and truth predicate being a filter of it,

satisfying the Resolution rule,

proving that proper extensions of the logic

form a four-element diamond.

**Keyphrases**: algebra, logic, model