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Commutative residuated lattices with \((x\odot y)'=x'\vee y'\). (English) Zbl 1233.03062
Chajda, I. (ed.) et al., Proceedings of the 79th workshop on general algebra “79. Arbeitstagung Allgemeine Algebra”, 25th conference of young algebraists, Palack√Ĺ University Olomouc, Olomouc, Czech Republic, February 12–14, 2010. Klagenfurt: Verlag Johannes Heyn (ISBN 978-3-7084-0407-3/pbk). Contributions to General Algebra 19, 151-154 (2010).
A bounded commutative residuated lattice \({\mathcal L}=(L, \wedge, \vee, \odot, \rightarrow, 0,e,1)\) is called strict if it verifies the axiom \((x \odot y)'=x' \vee y'\), where \(x'=x \rightarrow 0\).
The author proves that for any strict bounded commutative residuated lattice \({\mathcal L}\), the set \(L^*=\{x' \mid x \in L\}\) is a Boolean algebra, and \({\mathcal L}\) is simple if and only if the unit \(e\) is an atom of \(L\).
For the entire collection see [Zbl 1201.08001].
03G25 Other algebras related to logic
03G05 Logical aspects of Boolean algebras
06F05 Ordered semigroups and monoids