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:
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1.
for every \(a,b,c\in A\), if \(a\le b\rightarrow c\) then \(a\wedge b\le c\) and
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2.
for every \(a\in A\), \(a\rightarrow a = 1\).
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.
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Notes
Let \((A,\le )\) be a poset. If any two elements \(a, b \in A\) have a greatest lower bound (i.e., an infimum), which is denoted by \(a\wedge b\), then the algebra \((A,\wedge )\) is called a meet semilattice. Throughout this article we write semilattice in place of meet semilattice. A semilattice \((A,\wedge )\) is said to be bounded if it has a greatest element; in this case we write \((A,\wedge ,1)\), where 1 is the greatest element of \((A,\le )\).
Hemi-implicative lattices were originally introduced in [13] under the name of weak implicative lattices.
It is interesting to note that \(\mathsf {RWH}\cap \mathsf {Hil}^{l}= \mathsf {Hey}\), where \(\mathsf {Hey}\) is the variety of Heyting algebras. Indeed, it is immediate that \(\mathsf {Hey}\subseteq \mathsf {RWH}\cap \mathsf {Hil}^{l}\). Conversely, let \(A\in \mathsf {RWH}\cap \mathsf {Hil}^{l}\) and \(a\in A\). Since \(1\rightarrow a = a\) then it follows from [5, Proposition 4.20] and [5, Proposition 4.22] that \(A\in \mathsf {Hey}\). Therefore, \(\mathsf {RWH}\cap \mathsf {Hil}^{l}= \mathsf {Hey}\).
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Acknowledgements
This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina), Universidad Nacional de La Plata [PPID/X047] and a subvention of Cicitca (Universidad Nacional de San Juan, Argentina).
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Castiglioni, J.L., Fernández, V., Mallea, H.F. et al. On a variety of hemi-implicative semilattices. Soft Comput 26, 3187–3195 (2022). https://doi.org/10.1007/s00500-022-06807-4
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DOI: https://doi.org/10.1007/s00500-022-06807-4