States on BE-algebras

Abstract

Many new fields of science require a probability theory based on non-classical logics. We know multiple-valued logics are non-classical logics and became popular in computer science since it was understood that they play a fundamental role in fuzzy logics. In analogous to probability measure, the states on multiple-valued algebras proved to be the most suitable models for averaging the truth-value in their corresponding logics. Mundici introduced states (an analog of probability measures) on MV-algebras in 1995, as averaging of the truth-value in Mundici (Studia Logica, 55:113–127, 1995) [184]. Since middle 1990s, mainly after Mundici’s paper (Mundici in Studia Logica, 55:113–127, 1995, [184]), on probability theory on MV-algebras, there has been an increasing amount of study on generalizations of probability theoretical concepts, most notable states, on various logic origin algebraic structures. In (Borzooei et al., in Kochi Math, 9:27–42, 2014, [24]), R. Borzooei et al. studied the states on BE-algebras. Bosbach state was introduced by R. Bosbach in (Axiomatic und Arithmetic, Fundamenta Mathe maticae 64:257–287, 1969), [25] and (Kongruenzen and Quotiente, Fundamenta Math ematicae 69:1-14, 1970, [26]). The notion of a Bosbach state has been studied for other algebras of fuzzy structures such as pseudo BL-algebras (Georgescu in Boshbatch states on fuzzy structures, Soft Comput, 8:217–230, 2004, [99]), bounded non-commutative $R\mathfrak {l}$-monoids (Dvurecenskij and Rachunek in Discrete Math 306:1317–1326, 2006, [90]), (Dvurecenskij and Rachunek in Semigroup Forum 56:487–500, 2006, [91]), residuated lattices (Ciungu in Appl Funct Anal 2:175–188, 2002 [55]), pseudo BCK-semilattices, and pseudo BCK-algebras (Kuhr in Pseudo-BCK-algebras and related structures, Univerzite Palackeho Olomouci, 2007, [163]). In (Busneag in Math Comp Sci Ser, 37:58–64, 2010, [31]), C. Busneag developed the theory of state-morphisms on Hilbert algebras and got some results relative to the theory of Bosbach states on bounded and non-bounded Hilbert algebras.