Varieties of De Morgan monoids: minimality and irreducible algebras

TL;DR

This paper classifies finitely subdirectly irreducible De Morgan monoids, identifies four minimal varieties, and simplifies the understanding of their subvariety lattice, shedding light on models and axiomatic extensions of relevant logic.

Contribution

It provides a detailed classification of certain De Morgan monoids and identifies the four minimal varieties, advancing the structural understanding of these algebras.

Findings

Finitely subdirectly irreducible De Morgan monoids are either Sugihara chains or unions of specific subalgebras.

There are exactly four minimal varieties of De Morgan monoids.

The results simplify the description of the subvariety lattice of relevant algebras.

Abstract

It is proved that every finitely subdirectly irreducible De Morgan monoid A (with neutral element e) is either (i) a Sugihara chain in which e covers not(e) or (ii) the union of an interval subalgebra [not(a), a] and two chains of idempotents, (not(a)] and [a), where a = (not(e))^2. In the latter case, the variety generated by [not(a), a] has no nontrivial idempotent member, and A/[not(a)) is a Sugihara chain in which not(e) = e. It is also proved that there are just four minimal varieties of De Morgan monoids. This theorem is then used to simplify the proof of a description (due to K. Swirydowicz) of the lower part of the subvariety lattice of relevant algebras. The results throw light on the models and the axiomatic extensions of fundamental relevance logics.