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Endomorphisms of implication algebras. (English) Zbl 1309.03025

In an earlier paper [Semigroup Forum 84, No. 1, 25–32 (2012; Zbl 1243.03077)], the author has proved that two finite implication algebras are isomorphic if their endomorphism monoids are so. Now, this result is extended to arbitrary implication algebras. The main tool is the topological duality for implication, or Tarski, algebras; see [S. A. Celani and L. M. Cabrer, Algebra Univers. 58, No. 1, 73–94 (2008; Zbl 1136.03046)].
Reviewer’s remark: It seems that the author’s theorem could be proved also by completely algebraic methods. In such a manner, C. Tsinakis has shown a similar result for the so called principal Brouwerian semilattices in [Houston J. Math. 5, 427–436 (1979; Zbl 0431.06003)]. He also notes there that any Boolean lattice, viewed as a member of the class of Brouwerian semilattices, is principal. This observation pertains equally to implication algebras, which are, in fact, reducts of such Boolean lattices.

MSC:

03G25 Other algebras related to logic
06E75 Generalizations of Boolean algebras
08A35 Automorphisms and endomorphisms of algebraic structures
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References:

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[2] S. A. Celani, L. M. Cabrer, Topological duality for Tarski algebras, Algebra Universalis 58 (2008), 73-94. · Zbl 1136.03046
[3] H. Gaitán, Finite Tarski algebras are determined by their endomorphisms, Semigroup Forum 84 (2012), 25-32. · Zbl 1243.03077
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