A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions

Semi-Heyting algebras were introduced by the second-named author during 1983-85 as an abstraction of Heyting algebras. The first results on these algebras, however, were published only in 2008 (see [San08]). Three years later, in [San11], he initiated the investigations into the variety DHMSH of dually hemimorphic semi-Heyting algebras obtained by expanding semi-Heyting algebras with a dually hemimorphic operation. His investigations were continued in a series of papers thereafter. He also had raised the problem of finding logics corresponding to subvarieties of DHMSH, such as the variety DMSH of De Morgan semi-Heyting algebras, and DPCSH of dually pseudocomplemented semi-Heyting algebras, as well as logics to 2, 3, and 4-valued DHMSH-matrices. In this paper, we first present a Hilbert-style axiomatization of a new implicative logic called--Dually hemimorphic semi-Heyting logic, (DHMSH, for short)-- as an expansion of semi-intuitionistic logic by the dual hemimorphism as the negation and prove that it is complete with respect to the variety DHMSH of dually hemimorphic semi-Heyting algebras as its equivalent algebraic semantics (in the sense of Abstract Algebraic Logic). Secondly, we characterize the (axiomatic) extensions of DHMSH in which the Deduction Theorem holds. Thirdly, we present several logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH, thus solving the problem mentioned earlier. We also provide new axiomatizations for Moisil's logic and the 3-valued Lukasiewicz logic.