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

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