Dually Quasi

The variety DQD of semi-Heyting algebras with a weak negation, called dually quasi-De Morgan operation, and several of its subvarieties were investigated in the series [31], [32], [33], and [34]. In this paper we define and investigate a new subvariety JID of DQD, called “JI-distributive, dually quasi-De Morgan semi-Heyting algebras”, defined by the identity: x ∨ (y → z) ≈ (x ∨ y) → (x ∨ z), as well as the (closely related) variety DSt of dually Stone semi-Heyting algebras. We first prove that DSt and JID are discriminator varieties of level 1 and level 2 (introduced in [31]) respectively. Secondly, we give a characterization of subdirectly irreducible algebras of the subvariety JID1 of level 1 of JID. As a first application of it, we derive that the variety JID1 is the join of the variety DSt and the variety of De Morgan Boolean semi-Heyting algebras. As a second application, we give a concrete description of the subdirectly irreducible algebras in the subvariety JIDL1 of JID1 defined by the linear identity: (x → y) ∨ (y → x) ≈ 1, and deduce that the variety JIDL1 is the join of the variety DStHC generated by the dually Stone Heyting chains and the variety generated by the 4-element De Morgan Boolean Heyting algebra. Several applications of this result are also given, including a description of the lattice of subvarieties of JIDL1, equational bases of all subvarieties of JIDL1, and the amalgamation property of all subvarieties of DStHC. Date: September 27, 2017. 2000 Mathematics Subject Classification. Primary : 03G25, 06D20, 06D15; Secondary : 08B26, 08B15.