## 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

### Citations:

Zbl 1082.03056; Zbl 1280.03063
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### References:

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