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A polynomial representation of bounded lattices with an antitone involution. (English) Zbl 1243.06007
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, 23-24 (2010).
$$L=(L; \vee, \wedge, ', 0, 1)$$ be a bounded lattice (0 is the least and 1 the greatest element) with an antitone involution $$'$$, i.e. $$x\rightarrow x'$$ is a mapping satisfying $$x''=x$$ and $$x\leq y$$ implies $$y'\leq x'$$ for all $$x, y\in L$$.
From group theory it is known how a group $$G$$ may be considered as a group of permutations on $$G.$$ This is the so-called Cayley representation of a group. The authors show that a Cayley-like representation for bounded lattices with an antitone involution is possible. Instead of permutations they consider binary polynomials on $$L$$ of the form $$f_a(x,y)=(a\vee x)\wedge (a'\vee y),$$ for all $$a\in L$$.
For the entire collection see [Zbl 1201.08001].

MSC:
 06B15 Representation theory of lattices 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) 08A40 Operations and polynomials in algebraic structures, primal algebras