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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.

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
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