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Hilbert algebras as implicative partial semilattices. (English) Zbl 1125.03047
The infimum of two elements \(a\) and \(b\) of a Hilbert algebra is called the compatible meet of \(a\) and \(b\) if these elements are compatible in a certain sense. The main goal of the present paper is to study Hilbert algebras equipped with the compatible meet operation, which normally is partial. It is proved that a partial lower semilattice is a reduct of such an expanded Hilbert algebra if and only if both algebras have the same filters (see Theorem 3.9). Thus, an expanded Hilbert algebra is an implicative partial semilattice and conversely. Another important contribution of the paper is the characterization of the implication in an implicative partial semilattice in terms of filters of the underlying semilattice (see Theorem 4.5).

MSC:
03G25 Other algebras related to logic
06A12 Semilattices
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