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Introduction
Published in Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski, Computer Arithmetics for Nanoelectronics, 2018
Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski
A universal algebra consists of a set of elements and operations on this set. Boolean algebras are a particular case of universal algebra. There is an infinite number of different Boolean algebras. The basic requirements of any algebra employed to describe and manipulate Boolean functions include Functional completeness - The algebra must be capable of describing and manipulating an arbitrary Boolean function.Flexibility - The algebra must be amenable to the manipulation of Boolean functions so that design algorithms can be set up and employed with reasonable ease.Implementability - The basic connectives and any high-level functional operations should have simple and reliable physical logic circuit counterparts.
On the representations of L-equivalence relations on L-fuzzy sets with applications to locally vague environments
Published in International Journal of General Systems, 2020
Since the early stage of the fuzzy set theory, fuzzy equivalence relations play a key role in the development of several areas such as approximate reasoning (Klawonn and Luis Castro 1995; Demirci 2000; Klawonn 2000), fuzzy control (Klawonn and Kruse 1993; Klawonn, Gebhardt, and Kruse 1995; Klawonn and Kruse 1997), fuzzy universal algebra (Belohlávek 2002; Ignjatović, Ćirić, and Bogdanović 2009), vague group (Demirci 2003; Recasens 2010), vague lattice (Demirci 2005), fuzzy topology (Höhle and Šostak 1999) and fuzzy transform (Moko and Hurtik 2018). The concept of fuzzy equivalence relation has appeared under different names, e.g. similarity relation (Zadeh 1971), likeness relation (Trillas and Valverde 1984), probabilistic relation (Trillas and Valverde 1984), T-indistinguishability operator (Valverde 1985; Recasens 2010), T-equivalence (De Baets and Mesiar 1997, 2002), -fuzzy equivalence relation (Klawonn and Luis Castro 1995), equality relation (Klawonn and Kruse 1993; Klawonn 2000), -valued similarity relation (Höhle 1988) and -equivalence relation (Demirci 2003), depending on the context and the truth-value structure it is based on.