arXiv subjects
Publications and source records attributed to Hernán Javier San Martín.
In this manuscript, we provide an independent equational basis for the variety of reflexive Nelson algebras, a generalization of the variety of SNA-algebras. The proof of this result relies on a substantial number of technical arguments and computational examples. The result complements a broader study of reflexive Nelson algebras by showing that the variety admits an independent equational axiomatization.
The aim of this paper is to generalize the link between Heyting algebras and Nelson algebras, established independently by Fidel and Vakarelov at the end of the 1970s, in the framework of bounded distributive hemi-implicative lattices. For this purpose, we introduce the variety of hemi-Nelson algebras. Moreover, we characterize the lattice of congruences of a hemi-Nelson algebra in terms of certain implicative filters. We also esta\-blish an equivalence between the algebraic category of bounded distributive hemi-implicative lattices and the one of centered hemi-Nelson algebras.
Hemi-implicative semilattices (lattices), originally defined under the name of weak implicative semilattices (lattices), were introduced by the second author of the present paper. A hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,1)$ of type $(2,2,0)$ such that $(H,\wedge)$ is a meet semilattice, $1$ is the greatest element with respect to the order, $a\rightarrow a = 1$ for every $a\in H$ and for every $a$, $b$, $c\in H$, if $a\leq b\rightarrow c$ then $a\wedge b \leq c$. A bounded hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,0,1)$ of type $(2,2,0,0)$ such that $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice and $0$ is the first element with respect to the order. A hemi-implicative lattice is an algebra $(H,\wedge,\vee,\rightarrow,0,1)$ of type $(2,2,2,0,0)$ such that $(H,\wedge,\vee,0,1)$ is a bounded distributive lattice and the reduct algebra $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice. In this paper we introduce an equivalence for the categories of bounded hemi-implicative semilattices and hemi-implicative lattices, respectively, which is motivated by an old construction due J. Kalman that relates bounded distributive lattices and Kleene algebras.
A hemiimplicative semilattice is a bounded semilattice $(A, \wedge, 1)$ endowed with a binary operation $\to$, satisfying that for every $a, b, c \in A$, $a \leq b \to c$ implies $a \wedge b \leq c$ (that is to say, one of the conditionals satisfied by the residuum of the infimum) and the equation $a \to a = 1$. The class of hemiimplicative semilattices form a variety. These structures provide a general framework for the study of different structures of interest in algebraic logic. In any hemiimplicative semilattice it is possible to define a derived operation by $a \sim b := (a \to b) \wedge (b \to a)$. Endowing $(A, \wedge, 1)$ with the binary operation $\sim$ results again a hemiimplicative semilattice, which also satisfies the identity $a \sim b = b \sim a$. We call the elements of the subvariety of hemiimplicative semilattices satisfying $a \to b = b \to a$, a symmetric hemiimplicative semilattice. In this article, we study the correspondence assigning the symmetric hemiimplicative semilattice $(A, \wedge, \sim , 1)$ to the hemiimplicative semilattice $(A, \wedge, \to, 1)$. In particular, we characterize the image of this correspondence. We also provide many new examples of hemiimplicative semilattice structures on any bounded semillatice (possibly with bottom). Finally, we characterize congruences on the clases of hemiimplicative semilattices introduced as examples and we describe the principal congruences of hemiimplicative semilattices.