Morgan-Stone Lattices versus De Morgan Lattices

EasyChair Preprint no. 11668

Abstract

Morgan-Stone (MS) lattices are axiomatized by the
constant-free identities of those axiomatizing
Morgan-Stone (MS) algebras, in which case double negation
is an endomorphism of any MS lattice onto its De Morgan lattice
subalgebra, and so this point has interesting consequences
concerning the issues of lattices of [quasi-]varieties of
MS lattices facilitating finding these much.
First, we prove that the variety of MS lattices is the
quasi-variety generated by a six-element one
with lattice reduct being the direct product of the three- and two-element
chain lattices, in which case subdirectly-irreducible
MS lattices are exactly isomorphic copies of non-one-element
subalgebras of the six-element generating MS lattice with
the double negation endomorphism kernel being
the only non-trivial congruence of any non-simple one, and
so, by a universal tool elaborated here,
we get a 29-element non-chain distributive lattice
of varieties of MS lattices, isomorphic to the one
of sets of such subalgebras containing
embedable ones.
And what is more, we prove that any sub-quasi-variety of
the quasi-equational join (viz., the quasi-variety generated by the union)
of a finitely-generated quasi-variety of MS lattices and the variety of De Morgan
lattices, including the former, is the quasi-equational join
of its intersection with the latter and the former.
As a consequence, using the eight-element non-chain distributive
lattice L of quasi-varieties of De Morgan lattices, found earlier,
we prove that the lattice of quasi-varieties of
strong/quasi-strong MS lattices, being a non-chain distributive (15/29)-element
one embedable into the direct product of L and
a (2/5)-element chain.

Keyphrases: algebra, homomorphism, lattice

@Booklet{EasyChair:11668,
  author = {Alexej Pynko},
  title = {Morgan-Stone Lattices versus De Morgan Lattices},
  howpublished = {EasyChair Preprint no. 11668},

  year = {EasyChair, 2024}}