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**Adding involution to residuated structures.**
*(English)*
Zbl 1062.03059

Summary: Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom \(x^{2} \leqslant x\) but fails to preserve the contraction property \(x\leqslant x^2\). 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 \(t\).