Archive for Mathematical Logic 52 (7-8):689-706 (2013)
AbstractBosbach and Riečan states on residuated lattices both are generalizations of probability measures on Boolean algebras. Just from the observation that both of them can be defined by using the canonical structure of the standard MV-algebra on the unit interval [0, 1], generalized Riečan states and two types of generalized Bosbach states on residuated lattices were recently introduced by Georgescu and Mureşan through replacing the standard MV-algebra with arbitrary residuated lattices as codomains. In the present paper, the Glivenko theorem is first extended to residuated lattices with a nucleus, which gives several necessary and sufficient conditions for the underlying nucleus to be a residuated lattice homomorphism. Then it is proved that every generalized Bosbach state (of type I, or of type II) compatible with the nucleus on a nucleus-based-Glivenko residuated lattice is uniquely determined by its restriction on the nucleus image of the underlying residuated lattice, and every relatively generalized Riečan state compatible with the double relative negation on an arbitrary residuated lattice is uniquely determined by its restriction on the double relative negation image of the residuated lattice. Our results indicate that many-valued probability theory compatible with nuclei on residuated lattices reduces in essence to probability theory on algebras of fixpoints of the underlying nuclei
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The Existence of States Based on Glivenko Semihoops.Pengfei He, Juntao Wang & Jiang Yang - 2022 - Archive for Mathematical Logic 61 (7):1145-1170.
References found in this work
Residuated Lattices: An Algebraic Glimpse at Substructural Logics.Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski & Hiroakira Ono - 2007 - Elsevier.
Averaging the Truth-Value in Łukasiewicz Logic.Daniele Mundici - 1995 - Studia Logica 55 (1):113 - 127.
Free Algebras in Varieties of Glivenko MTL-Algebras Satisfying the Equation 2(X2) = (2x)2.Roberto Cignoli & Antoni Torrens Torrell - 2006 - Studia Logica 83 (1-3):157-181.
Metamathematics of Fuzzy Logic.Petr Hájek - 1998 - Dordrecht, Boston and London: Kluwer Academic Publishers.
Glivenko Like Theorems in Natural Expansions of BCK‐Logic.Roberto Cignoli & Antoni Torrens Torrell - 2004 - Mathematical Logic Quarterly 50 (2):111-125.