Adjoint semilattice and minimal Brouwerian extensions of a Hilbert algebra. (English) Zbl 1280.03063

Cīrulis, Jānis

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 51, No. 2, 41-51 (2012).

Let \(A:=(A,\rightarrow,1)\) be a Hilbert algebra. A closure endomorphism is a mapping \(\varphi :A\rightarrow A\) which is a closure operator and an endomorphism. For example, if \(p\in A\), the mapping \(\alpha_p:A\rightarrow A\) defined by \(\alpha_px:=p\rightarrow x\) is a closure endomorphism. For every finite subset \(P:=\{p_1,...,p_n\}\) of \(A\), the mapping \(\alpha_P:=\alpha_{p_n}\circ\dots\circ\alpha_{p_1}\) is a closure endomorphism, called finitely generated (if \(P=\emptyset\), then \(\alpha_P=\varepsilon\) is the identity mapping). The set \(\mathrm{CE}^f\) of all such mappings is closed under composition and the algebra \((\mathrm{CE}^f,\circ,\varepsilon)\) is a lower bounded join-semilattice, called the adjoint semilattice of the Hilbert algebra \(A\).
In this paper it is shown that the adjoint semilattice \(\mathrm{CE}^f\) is isomorphic to the semilattice of finitely generated filters of \(A\) and subtractive (dually implicative) and its generating set turns out to be closed under subtraction and is an order dual of \(A\). The lattice of ideals of \(\mathrm{CE}^f\) is isomorphic to the lattice of filters of \(A\). A minimal Brouwerian extension of \(A\) is shown to be dually isomorphic to the adjoint semilattice of \(A\). Embedding of \(A\) into its minimal Brouwerian extension preserves all existing joins, but in this paper the author characterizes also the preserved meets.

Reviewer: Florentina Chirteş (Craiova)

MSC:

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

Keywords:

adjoint semilattice; Brouwerian extension; closure endomorphism; compatible meet; Hilbert algebra; implicative semilattice; subtraction; finitely generated filters; lattice of ideals; lattice of filters

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