Semi-Heyting Algebras Term Equivalent to Gödel Algebras

MANUEL ABAD; JOSÉ PATRICIO DÍAZ VARELA; JUAN MANUEL CORNEJO

ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS

SPRINGER

Lugar: Berlin; Año: 2013 p. 1 - 18

0167-8094

In this paper we investigate those subvarieties of the variety $mathcal {SH}$ of semi-Heyting algebras which are term-equivalent to the variety $mathcal L_{mathcal H}$ of G"{o}del algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety $mathcal L^{Com}$ of commutative semi-Heyting algebras and the variety $mathcal L^{ee}$ generated by the chains in which $a < b$ implies $a o b = b$. We also study the variety $mathcal C$ generated within $mathcal {SH}$ by $mathcal L_{mathcal H}$, $mathcal L_ee$ and $mathcal L_{Com}$. In particular we prove that $mathcal C$ is locally finite and we obtain a construction of the finitely generated free algebra in $mathcal C$.