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

EasyChair Preprint no. 3942

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

@Booklet{EasyChair:3942,
  author = {Alexej Pynko},
  title = {Subprevarieties of Algebraic Systems Versus Extensions of Logics: Application to Some Many-Valued Logics},
  howpublished = {EasyChair Preprint no. 3942},

  year = {EasyChair, 2020}}