Lattice of closure endomorphisms of a Hilbert algebra. (English) Zbl 1437.03183

Summary: A closure endomorphism of a Hilbert algebra \(A\) is a mapping that is simultaneously an endomorphism of and a closure operator on \(A\). It is known that the set \(C E\) of all closure endomorphisms of \(A\) is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of \(A\), anti-isomorphic to the lattice of certain closure retracts of \(A\), and compactly generated. The set of compact elements of \(C E\) coincides with the adjoint semilattice of \(A\); conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.


03G25 Other algebras related to logic
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06F35 BCK-algebras, BCI-algebras
08A35 Automorphisms and endomorphisms of algebraic structures
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