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Hybrid generalized Bosbach and Rie c̆ an states on non-commutative residuated lattices

Generalized Bosbach and Rie c̆ an states, which are useful for the development of an algebraic theory of probabilistic models for commutative or non-commutative fuzzy logics, have been investigated in the literature. In this paper, a new way arising from generalizing residuated lattice-based filters from commutative case to non-commutative one is applied to introduce new notions of generalized Bosbach and Rie c̆ an states, which are called hybrid ones, on non-commutative residuated lattices is provided, and the relationships between hybrid generalized states and those existing ones are studied, examples show that they are different. In particular, two problems from L.C. Ciungu, G. Georgescu, and C. Mure, “Generalized Bosbach States: Part I” (Archive for Mathematical Logic 52 (2013):335–376) are solved, and properties of hybrid generalized states, which are similar to those on commutative residuated lattices, are obtained without the condition “strong”.