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Erschienen in:
08.02.2022 | Foundations
On a variety of hemi-implicative semilattices
Erschienen in: Soft Computing | Ausgabe 7/2022
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Abstract
A hemi-implicative semilattice is an algebra \((A,\wedge ,\rightarrow ,1)\) of type (2, 2, 0) such that \((A,\wedge ,1)\) is a bounded semilattice and the following conditions are satisfied: The class of hemi-implicative semilattices forms a variety. In this paper we introduce and study a proper subvariety of the variety of hemi-implicative semilattices, \(\mathsf {ShIS}\), which also properly contains some varieties of interest for algebraic logic. Our main goal is to show a representation theorem for \(\mathsf {ShIS}\). More precisely, we prove that every algebra of \(\mathsf {ShIS}\) is isomorphic to a subalgebra of a member of \(\mathsf {ShIS}\) whose underlying bounded semilattice is the bounded semilattice of upsets of a poset.
1.
for every \(a,b,c\in A\), if \(a\le b\rightarrow c\) then \(a\wedge b\le c\) and
2.
for every \(a\in A\), \(a\rightarrow a = 1\).