Morgan-Stone lattices

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 nine non-one-element subalgebras of the six-element generating MS lattice, and so we get a 29-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. In this connection, we also prove that any relatively simple relatively subdirectly-representable abstract hereditary multiplicative subclass of the equational join of the varieties of De Morgan lattices/algebras and Stone ones is a sub-variety of the former.