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

JUAN MANUEL CORNEJO; HANAMANTAGOUDA P. SANKAPPANAVAR

BULLETIN OF THE SECTION OF LOGIC

Lodz University Press

Año: 2022 vol. 51 p. 555 - 645

0138-0680

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. Firstly, we present a Hilbert-style axiomatization of a new %implicative 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 $mathbb{DHMSH}$. algebras. 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 %for $mathcal{DHMSH}$,and that the lattice of axiomatic extensions of $mathcal{DHMSH}$ is dually isomorphic to the lattice of subvarieties of $mathbb{DHMSH}$. A newaxiomatization 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 L ukasiewicz logic.Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan-G"{o}del logics and dually pseudocomplemented G"{o}del 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.