Strong K ‐Modal BL ‐Algebras

Abstract

ABSTRACT In this paper, we present the corresponding algebraic structure for ‐modal ‐logic (abbreviated as ) proposed by Eslami et al. [ Journal of Intelligent and Fuzzy Systems ] 29 (2015): 337–388, which we call a strong ‐modal ‐algebra. We demonstrate that the equivalent algebraic semantics for is the variety of strong ‐modal ‐algebras, and is algebraizable in the sense of Blok and Pigozzi. We prove that is strongly sound and complete with respect to strong ‐modal ‐algebras. The local deduction theorem is also established for . Furthermore, we show that is strongly sound and complete with respect to linearly ordered strong ‐modal ‐algebras. We show that is a conservative expansion of Hájek's basic logic. Finally, using algebraizability, we analyze congruences in strong ‐modal ‐algebras.