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Adding Involution to Residuated Structures

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Abstract

Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2≤x but fails to preserve the contraction property x≤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.

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Authors and Affiliations

  1. School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Tatsunokuchi, Ishikawa, 923-1292, Japan

    Nikolaos Galatos

  2. School of Mathematical and Statistical Sciences, University of Natal, King George V Ave, Durban, 4001, South Africa

    James G. Raftery

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  1. Nikolaos Galatos
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Galatos, N., Raftery, J.G. Adding Involution to Residuated Structures. Studia Logica 77, 181–207 (2004). https://doi.org/10.1023/B:STUD.0000037126.29193.09

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