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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 \(t\).

MSC:

03G10 Logical aspects of lattices and related structures
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B25 Decidability of theories and sets of sentences
06F05 Ordered semigroups and monoids
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