Gaitán, Hernando Endomorphisms of implication algebras. (English) Zbl 1309.03025 Demonstr. Math. 47, No. 2, 284-288 (2014). 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. Reviewer: Jānis Cīrulis (Riga) Cited in 1 Document MSC: 03G25 Other algebras related to logic 06E75 Generalizations of Boolean algebras 08A35 Automorphisms and endomorphisms of algebraic structures Keywords:endomorphism monoid; implication algebra; Tarski space; topological duality Citations:Zbl 1243.03077; Zbl 1136.03046; Zbl 0431.06003 PDF BibTeX XML Cite \textit{H. Gaitán}, Demonstr. Math. 47, No. 2, 284--288 (2014; Zbl 1309.03025) Full Text: DOI OpenURL References: [1] J. C. Abbott, Semi-Boolean algebras, Mat. Vesnik 19 (1967), 177-198. · Zbl 0153.02704 [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 [4] C. J. Maxon, On semigroups of Boolean ring endomorphism, Semigroup Forum 4 (1972), 78-82. [5] K. D. Magill, The semigroup of endomorphisms of a Boolean ring, J. Aust. Math. Soc. 11 (1970), 411-416. [5] H. Rasiowa, An Algebraic Approach to Non-Classical Logics, North Holland, Studies in Logic and the Foundations of Mathematics, Vol 78, Amsterdam, 1974. · Zbl 0299.02069 [6] B. M. Schein, Ordered sets, semilattices, distributive lattices and Boolean algebras with homomorphic endomorphism semigroups, Fund. Math. 68 (1970), 31-50. · Zbl 0197.28902 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.