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Fuzzy Epistemic Logic is an important formalism for approximate reasoning. It extends the well known basic propositional logic BL, introduced by Hajek, by offering the ability to reason about possibility and necessity of fuzzy propositions. We consider an algebraic approach to study this logic, introducing Epistemic BL-algebras. These algebras turn to be a generalization of both, Pseudomonadic Algebras introduced by cite{Bez2002} and serial, euclidean and transitive Bi-modal Godel Algebras proposed by cite{CaiRod2015}. We present the connection between this class of algebras and fuzzy possibilistic frames, as a first step to solve an open problem proposed by Hajek cite[chap. ~8]{HajekBook98}.
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A mistake concerning the ultra textit{LI}-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an textit{LI}-ideal to be an ultra textit{LI}-ideal are given. Moreover, the notion of an textit{LI}-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) textit{LI}-ideal and an textit{ILI}-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper textit{LI}-ideal and Boolean textit{LI}-ideal, (2) prime proper textit{LI}-ideal and textit{ILI}-ideal, (3) proper obstinate textit{LI}-ideal, (4) ultra textit{LI}-ideal.
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stones representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stones theorem from Boolean algebra.
For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise L ukasiewicz negation $ eg x={ eg alphamid alphain x}$, (truncated) Minkowski sum, $xoplus y={alphaoplus betamid alpha in x,,,betain y}$, pointwise L ukasiewicz conjunction $xodot y= eg( eg xoplus eg y)$, the operators $Delta x=[min x,min x]$, $ abla x=[max x,max x]$, and distinguished constants $0=[0,0],,, 1=[1,1],,,, mathsf{i} = A$. We list a few equations satisfied by the algebra $mathcal I(A)=(I(A),0,1,mathsf{i}, eg,Delta, abla,oplus,odot)$, call IMV-algebra every model ofthese equations, and show that, conversely, every IMV-algebra is isomorphic to the IMV-algebra $mathcal I(B)$ of all intervals in some MV-algebra $B$. We show that IMV-algebras are categorically equivalent to MV-algebras, and give a representation of free IMV-algebras. We construct L ukasiewicz interval logic, with its coNP-complete consequence relation, which we prove to be complete for $mathcal I([0,1])$-valuations. For any class $mathsf{Q}$ of partially ordered algebras with operations that are monotone or antimonotone in each variable, we consider the generalization $mathcal I_{mathsf{Q}}$ of the MV-algebraic functor $mathcal I$, and give necessary and sufficient conditions for $mathcal I_{mathsf{Q}}$ to be a categorical equivalence. These conditions are satisfied, e.g., by all subquasivarieties of residuated lattices.
We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fraisse limits of suitable classes of structures. Moreover by means of Fraisse theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.
The only C*-algebras that admit elimination of quantifiers in continuous logic are $mathbb{C}, mathbb{C}^2$, $C($Cantor space$)$ and $M_2(mathbb{C})$. We also prove that the theory of C*-algebras does not have model companion and show that the theory of $M_n(mathcal {O_{n+1}})$ is not $forallexists$-axiomatizable for any $ngeq 2$.
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