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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, Palacký 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].
##### MSC:
 03G25 Other algebras related to logic 03G05 Logical aspects of Boolean algebras 06F05 Ordered semigroups and monoids