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Congruences and closure endomorphisms of Hilbert algebras. (English) Zbl 1320.03090

A closure endomorphism on a Hilbert algebra \(H\) is its endomorphism which happens to be a closure operator w.r.t. the natural order on \(H\). The author describes several elementary properties of closure endomorphisms, characterises closure endomorphisms of finite Hilbert algebras and shows that all congruences of a finite Hilbert algebra are kernels of closure endomorphisms. Moreover, all classes of every congruence have the greatest element. Let \(H_{\mathrm{irr}}\) stand for the subalgebra of a finite Hilbert algebra \(H\) which consists of the unit and irreducible elements of \(H\). The main result: two finite Hilbert algebras \(H_1\) and \(H_2\) have isomorphic endomorphism monoids iff their subalgebras \({H_1}_{\mathrm{irr}}\) and \({H_2}_{\mathrm{irr}}\) are isomorphic.
Reviewer’s remark: Closure endomorphisms of Hilbert algebras have already been studied by the reviewer in [Contrib. Gen. Algebra 16, 25–34 (2005; Zbl 1082.03056); Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 51, No. 2, 41–51 (2012; Zbl 1280.03063)].

MSC:

03G25 Other algebras related to logic
06A12 Semilattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
08A30 Subalgebras, congruence relations
08A35 Automorphisms and endomorphisms of algebraic structures
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