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. This theorem is 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.