On commutative weak Heyting algebras and some of its subreducts

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 Godel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety \(\mathcal L^{\rm Com}\) of commutative semi-Heyting algebras and the variety \(\mathcal L^{\vee}\) generated by the chains in which a < b implies a → b = b. We also study the variety \(\mathcal C\) generated within \(\mathcal{SH}\) by \(\mathcal L_{\mathcal H}\), \(\mathcal L_\vee\) and \(\mathcal L_{\rm Com}\). In particular we prove that \(\mathcal C\) is locally finite and we obtain a construction of the finitely generated free algebra in \(\mathcal C\).