|
Download PDFOpen PDF in browserMorgan-Stone Lattices versus De Morgan LatticesEasyChair Preprint 1166815 pages•Date: January 3, 2024AbstractMorgan-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 Download PDFOpen PDF in browser |
|
|