On good EQ-algebras

A special algebra called EQ-algebra has been recently introduced by Vilém Novák. Its original motivation comes from fuzzy type theory, in which the main connective is fuzzy equality. EQ-algebras have three binary operations – meet, multiplication, and fuzzy equality – and a unit element. They open t...

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Bibliographic Details
Published in:Fuzzy sets and systems 2011-09, Vol.178 (1), p.1-23
Main Authors: El-Zekey, Moataz, Novák, Vilém, Mesiar, Radko
Format: Article
Language:English
Subjects:
Applied sciences
BCK-algebras
Circuit properties
Computer science
control theory
systems
Digital circuits
Electric, optical and optoelectronic circuits
Electronic circuits
Electronics
EQ-algebras
Exact sciences and technology
Fuzzy equality
Fuzzy logic
General logic
Information, signal and communications theory
Logic and foundations
Mathematical logic, foundations, set theory
Mathematical methods
Mathematics
Miscellaneous
Representable algebras
Residuated lattices
Sciences and techniques of general use
Telecommunications and information theory
Theoretical computing
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Summary:A special algebra called EQ-algebra has been recently introduced by Vilém Novák. Its original motivation comes from fuzzy type theory, in which the main connective is fuzzy equality. EQ-algebras have three binary operations – meet, multiplication, and fuzzy equality – and a unit element. They open the door to an alternative development of fuzzy (many-valued) logic with the basic connective being a fuzzy equality instead of an implication. This direction is justified by the idea presented by G.W. Leibniz that “a fully satisfactory logical calculus must be an equational one.” In this paper, we continue the study of EQ-algebras and their special cases. We introduce and study the prefilters and filters of separated EQ-algebras. We give great importance to the study of good EQ-algebras. As we shall see in this paper, the “goodness” property (and thus separateness) is necessary for reasonably behaving algebras. We enrich good EQ-algebras with a unary operation Δ (the so-called Baaz delta), fulfilling some additional assumptions that are heavily used in fuzzy logic literature. We show that the characterization theorem obtained until now for representable good EQ-algebras also hold for the enriched algebra.
ISSN:0165-0114
1872-6801
DOI:10.1016/j.fss.2011.05.011