# l-Hemi-Implicative Semilattices

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## Abstract

An l-hemi-implicative semilattice is an algebra $$\mathbf {A} = (A,\wedge ,\rightarrow ,1)$$ such that $$(A,\wedge ,1)$$ is a semilattice with a greatest element 1 and satisfies: (1) for every $$a,b,c\in A$$, $$a\le b\rightarrow c$$ implies $$a\wedge b \le c$$ and (2) $$a\rightarrow a = 1$$. An l-hemi-implicative semilattice is commutative if if it satisfies that $$a\rightarrow b = b\rightarrow a$$ for every $$a,b\in A$$. It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an derived operation by $$a \sim b := (a \rightarrow b) \wedge (b \rightarrow a)$$. Endowing $$(A,\wedge ,1)$$ with the binary operation $$\sim$$ the algebra $$(A,\wedge ,\sim ,1)$$ results an l-hemi-implicative semilattice, which also satisfies the identity $$a \sim b = b \sim a$$. In this article, we characterize the (derived) commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element (possibly with bottom). Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices.

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## Author information

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Correspondence to Hernán Javier San Martín.

Presented by Jacek Malinowski

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Castiglioni, J.L., San Martín, H.J. l-Hemi-Implicative Semilattices. Stud Logica 106, 675–690 (2018). https://doi.org/10.1007/s11225-017-9759-3