A note on regular De Morgan semi-Heyting algebras

The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of RDMSH1. Secondly, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras that contains RDMSH1 . Furthermore, we prove that every subvariety of RDQDStSH1, and hence of RDMSH1, has Amalgamation Property. The note concludes with some open problems for further investigation.