State BL-algebras
The concept of a state MV-algebra was firstly introduced by Flaminio and Montagna (An algebraic approach to states on MV-algebras. In: Novk V (ed) Fuzzy logic 2, proceedings of the 5th EUSFLAT conference, September 1114, Ostrava, vol II, pp 201206, 2007; Int J Approx Reason 50:138152, 2009) as an MV-algebra with internal state as a unary operation. Di Nola and Dvure?enskij (Ann Pure Appl Logic 161:161173, 2009a; Math Slovaca 59:517534, 2009b) gave a stronger version of a state MV-algebra. In the present paper, we introduce the notion of a state BL-algebra, or more precisely, a BL-algebra with internal state. We present different types of state BL-algebras, like strong state BL-algebras and state-morphism BL-algebras, and we study some classes of state BL-algebras. In addition, we give a sample of important examples of state BL-algebras and present some open problems.
state bl-algebra mv-algebra state bl-operator state-operator strong state-operator state-morphism-operator