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EQ-algebras

机译：情商代数

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摘要

We introduce a new class of algebras called EQ-algebras. An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. These algebras are intended to become algebras of truth values for a higher-order fuzzy logic (a fuzzy type theory, FTT). The motivation stems from the fact that until now, the truth values in FTT were assumed to form either an IMTL-, BL-, or MV-algebra, all of them being special kinds of residuated lattices in which the basic operations are the nionoidal operation (multiplication) and its residuum. The latter is a natural interpretation of implication in fuzzy logic; the equivalence is then interpreted by the biresiduum, a derived operation. The basic connective in FTT, however, is a fuzzy equality and, therefore, it is not natural to interpret it by a derived operation. This defect is expected to be removed by the class of EQ-algebras introduced and studied in this paper. From the algebraic point of view, the class of EQ-algebras generalizes, in a certain sense, the class of residuated lattices and so, they may become an interesting class of algebraic structures as such.
机译：我们介绍一种称为EQ代数的新型代数。一个EQ代数具有三个基本的二进制运算（满足，乘法和模糊等式）和一个顶级元素。这些代数旨在成为高阶模糊逻辑（模糊类型理论，FTT）的真值代数。动机源于以下事实：到目前为止，假设FTT中的真值形成了IMTL，BL或MV代数，它们都是特殊类型的剩余格，其中基本运算是正弦运算（乘法）及其残差。后者是对模糊逻辑含义的自然解释。等价性由biresiduum（派生的运算）解释。但是，FTT中的基本连接词是模糊等式，因此，通过派生运算来解释它是不自然的。可以通过本文介绍和研究的EQ代数类来消除此缺陷。从代数的角度来看，EQ代数的类在某种意义上概括了剩余格的类，因此，它们可能成为有趣的代数结构类。

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