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Residuated structures with involution. (English) Zbl 1124.06011
Berichte aus der Mathematik. Aachen: Shaker; Darmstadt: Tech. Univ. Darmstadt, Fachbereich Mathematik (Dissertation) (ISBN 978-3-8322-5789-7/pbk). vi, 162 p. (2006).
A residuated lattice is involutive if it is endowed with an order involution such that \(xy \leq e'\) if and only if \(x \leq y'\). Equivalently, this can be defined in terms of the ‘dualizing’ constant \(e'\). Extending Mundici’s equivalence between MV-algebras and lattice-ordered groups, it is shown here that both can be captured, conveniently, in this framework. The same applies to the reduct of relation algebras with inverse of the complement as involution and to Girard quantales. Besides a detailed exposition of these relationships, the paper studies this class of structures from an universal algebraic point of view.
The most important results are: A cut-free Gentzen-system for equational calculus together with an algebraic completeness proof, whence a simplified proof of the decidability of the equational theory. Unsolvability of the word problem via an interpretation of groups. Construction of uncountably many minimal varieties. Undecidability of the equational theory in the modular case.

06F05 Ordered semigroups and monoids
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
03C05 Equational classes, universal algebra in model theory
06F07 Quantales