The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a (...) Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. (shrink)

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras . Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is locally finite. Furthermore we prove that the lattice of subvarieties of QSA's is an -chain. MSC: 03G25, 06D16, 06E15.

We determine the number of non‐isomorphic semi‐Heyting algebras on an n‐element chain, where n is a positive integer, using a recursive method. We then prove that the numbers obtained agree with those determined in [1]. We apply the formula to calculate the number of non‐isomorphic semi‐Heyting chains of a given size in some important subvarieties of the variety of semi‐Heyting algebras that were introduced in [5]. We further exploit this recursive method to calculate the numbers of non‐isomorphic semi‐Heyting chains with (...) n elements such that removing the mth element () we are left with a subalgebra. We also solve a related problem posed in [1] of determining the number of ways a semi‐Heyting chain with elements can be extended to a n element semi‐Heyting chain by adding a new element in the mth place. Finally we combine these results by finding a second way to calculate the numbers that provides some extra information. (shrink)

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ ≈ x ∧ y, x ∧ ≈ x ∧ [ → ], and x → x ≈ 1.

The variety \ of implication zroupoids and a constant 0) was defined and investigated by Sankappanavar :21–50, 2012), as a generalization of De Morgan algebras. Also, in Sankappanavar :21–50, 2012), several subvarieties of \ were introduced, including the subvariety \, defined by the identity: \, which plays a crucial role in this paper. Some more new subvarieties of \ are studied in Cornejo and Sankappanavar that includes the subvariety \ of semilattices with a least element 0. An explicit description of (...) semisimple subvarieties of \ is given in Cornejo and Sankappanavar. It is a well known fact that there is a partial order ) induced by the operation ∧, both in the variety \ of semilattices with a least element and in the variety \ of De Morgan algebras. As both \ and \ are subvarieties of \ and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation \ on \ is actually a partial order in some subvariety of \ that includes both \ and \. The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety \ is a maximal subvariety of \ with respect to the property that the relation \ is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in \ that can be defined on an n-element chain -chains), n being a natural number. Secondly, we answer this problem in our second main theorem which says that, for each \, there are exactly n nonisomorphic \-chains of size n. (shrink)