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)].


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
Full Text: DOI


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