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

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.


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
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