Generalized Bosbach states. II (Q377478)

In the last period, we are witnesses of many attempts how to introduce a state, an analogue of a probability measure, for some interesting algebraic structures connected with logic. The first one was a state introduced for MV-algebras, later for BL-algebras, called Bosbach states and Riečan states, respectively. These notions were generalized for pseudo MV-algebras, pseudo BL-algebras, MTL-algebras, and residuated lattice. In the considered cases, a state has values in the real interval \([0,1]\). The paper under review is a continuation of the research from the paper by the same authors [Arch. Math. Logic 52, No. 3--4, 335--376 (2013; Zbl 1296.03040)]. Also in this paper, the authors define Bosbach states as mappings from a (commutative) residuated lattice with values into a (commutative) residuated lattice; the case of the real interval \([0,1]\) is not excluded. The main results consist of exhibiting some convergence properties implied by states are introduced. Using the notion of similarity convergence, three notions of continuity of a generalized Bosbach state are introduced. Consequently, some completions of residuated lattices (\(s\)-completion, metric completion) are investigated.