Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without involution) (...) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t. (shrink)

In this paper we characterize, classify and axiomatize all axiomatic extensions of the Nilpotent Minimum Logic. Every axiomatic extension is complete with respect to a class of NM-chains. Given a family of NM-chains the number of elements of the largest odd finite NM-chain in the family and the number of elements of the largest even finite NM-chain in the family turns out to be a complete classifier.