The Variety of Semi-Heyting Algebras Satisfying the Equation

M. ABAD, J.M. CORNEJO, J.P. DÍAZ VARELA

Reports on mathematical logic

Jagiellonian University Press

Lugar: KRAKOW ; Año: 2011 vol. 46 p. 75 - 90

0137-2904

In cite{SanSH}, Sankappanavar introduced a new equational class$SH$ of algebras, which he called ``{it semi-Heyting Algebras}´´,as an abstraction of Heyting algebras. This variety includes Heytingalgebras and share with them some rather strong properties. Forexample, the variety of semi-Heyting algebras is arithmetical,semi-Heyting algebras are pseudocomplemented distributive latticesand their congruences are determined by filters. Sankappanavarintroduced in his work several subvarieties of $SH$, for instance,the variety $SH^S$ of Stone semi-Heyting algebras, the variety$SH^B$ of Boolean semi-Heyting algebras, the variety $mathcal{QH}$of quasi-Heyting algebras, the variety $SH^C$ generated bysemi-Heyting chains, investigated in cite{AbCorDVarCSH}, thevariety $mathcal{FTT}$ in which $0 o 1 approx 1$, the variety$mathcal{FTF}$ in which $0 o 1 approx 0$, and so on. These newvarieties seem to be of interest from the point of view ofnon-classical logic, since they can provide a new interpretation forthe implication connective.