Morgan-Stone Lattices

EasyChair Preprint 10296, version 6

Abstract

Morgan-Stone (MS) lattices are axiomatized by the
constant-free identities of those axiomatizing
Morgan-Stone (MS) algebras.
Applying the technique of characteristic functions of prime
filters as homomorphisms from lattices onto the two-element chain
one and their products, we prove that the variety of MS lattices is the abstract
hereditary multiplicative class generated by a six-element one
with an equational disjunctive system
expanding the direct product of the three- and two-element
chain distributive lattices, in which case subdirectly-irreducible
MS lattices are exactly isomorphic copies of non-one-element
subalgebras of the six-element generating MS lattice, and so
we get a 26-element non-chain distributive lattice
of varieties of MS lattices subsuming the four-/three-element chain one
of ``De Morgan''/Stone lattices/algebras
(viz., constant-free versions
of De Morgan algebras)/(more precisely, their term-wise definitionally equivalent
constant-free versions, called Stone lattices).
Among other things, we provide an REDPC scheme for MS lattices.
Laying a special emphasis onto
the [quasi-]equational join (viz., the [quasi-]variety generated by the union)
of De Morgan and Stone lattices, we find
a fifteen-element non-chain distributive lattice of its sub-quasi-varieties
subsuming the eight-element one
of those of the variety of De Morgan lattices found earlier,
each of the rest being the quasi-equational join of its
intersection with the variety of De Morgan lattices
and the variety of Stone lattices.

Keyphrases: De Morgan lattice, REDPC, Stone algebra, quasi-variety

@booklet{EasyChair:10296,
  author    = {Alexej Pynko},
  title     = {Morgan-Stone Lattices},
  howpublished = {EasyChair Preprint 10296},
  year      = {EasyChair, 2023}}