Varieties of De Morgan monoids: Minimality and irreducible algebras

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 ¬e or (ii) the union of an interval subalgebra [¬a,a] and two chains of idempotents, (¬a] and [a), where a=(¬e)2. In the latter case, the variety generated by [¬a,a] has no nontrivial idempotent member, and A/[¬a) is a Sugihara chain in which ¬e=e. It is also proved that there are just four minimal varieties of De Morgan monoids. These findings are then used to simplify the proof of a description (due to K. Świrydowicz) 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.