Varieties of De Morgan Monoids: Covers of Atoms

Abstract

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids may be mapped by noninjective homomorphisms. The homomorphic preimages of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety (C4) within U are revealed here. There are just ten of them (all finitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety—in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of (C4) [or of (D4)] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.

Singly generated quasivarieties and residuated structures

“…Indeed, Slaney [65] proved that the free 0‐generated De Morgan monoid has exactly 3088 elements. Its congruence lattice has just 68 elements, no two of which produce isomorphic factor algebras [53, Corollary 3.6]. Let A 1 , …, A 68 denote the factor algebras, where A 1 is trivial.…”

“…By the Homomorphism Theorem, these are all of the 0‐generated De Morgan monoids, up to isomorphism. The minimal quasivarieties of De Morgan monoids are just and , [53, Theorem 3.4]. As passive structural completeness persists in subquasivarieties, the next result is a characterization of the PSC quasivarieties of De Morgan monoids.…”

“…Theorem 8.4(2) suggests that C 4 has more interesting homomorphic pre‐images than the other simple 0‐generated De Morgan monoids. We therefore make the abbreviation The quasivariety is not a variety [53, § 4]. It is therefore not obvious that possesses a largest subvariety, but in fact it does.…”

“…(We conjecture that and are not SC.) In the lattice of subvarieties of , the unique atom has just six covers, identified in [53, Theorem 8.10]. The varietal join of those six covers is HSC [53, Theorem 8.13].…”

“…Proof The join‐irreducible covers of the four atoms are described in [53, Theorem 7.2]. With four exceptions, each has the JEP, as it is either a subvariety of or of (and is thus PSC) or has the form for a simple algebra A .…”