Semi-Nelson algebras

IGNACIO DARIO VIGLIZZO; JUAN MANUEL CORNEJO

Bahia Blanca

Congreso; Reunión Anual de la UMA; 2016

Instituto de Matemática, Universidad Nacional del Sur

We apply the well known construction of Nelson algebras from Heyting algebras due to Vakarelov [{\em D.~Vakarelov. Notes on {${\cal N}$}-lattices and constructive logic with strong negation. Studia Logica, 36(1--2):109--125, 1977.}] to semi-Heyting algebras [{\em Hanamantagouda~P. Sankappanavar. Semi-{H}eyting algebras: an abstraction from {H}eyting algebras. In Proceedings of the 9th ``{D}r. {A}ntonio {A}. {R}. {M}onteiro'' {C}ongress, pages 33--66, Bah\'\i a Blanca, 2008. Univ. Nac. del Sur.}], and describe a new variety of what we naturally denominated semi Nelson algebras. Given a semi Heyting algebra $A$, the construction consists of defining over the set $V(A) = \{(a,b) \in A^2:\ a \wedge b = 0\}$ the following operations: \begin{itemize}\item[] $(a,b) \sqcap (c,d) = (a \wedge c, b \vee d)$ \item[] $(a,b) \sqcup (c,d) = (a \vee c, b \wedge d)$,\item[] $(a,b) \to (c,d) = (a \Rightarrow c, a \wedge d)$,\item[] $\sim (a,b) = (b,a)$,\item[] $\top = (1,0)$. \end{itemize} We characterize the lattice of congruences of a semi-Nelson algebra through some of its deductive systems, use this to find the subdirectly irreducible algebras, prove that the variety is arithmetical, has equationally definable principal congruences, has the congruence extension property and describe the semisimple subvarieties.