Boolean rings

Boolean rings

A boolean ring is a type of ring with a unit element, denoted as 1, and a zero element, denoted as 0. In a boolean ring, every element is idempotent, meaning that for any element A in the ring, A multiplied by itself equals A squared, which also equals A. It is important to note that any abstract boolean ring can be represented as a ring of subsets of some set, according to Stone's representation theorem.From: A First Course in Fuzzy Logic [2019]

Introduction

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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 Boolean algebraic system ⟨B,⋅,0,1⟩ is called a Boolean ring if it satisfies the conditions that there exist elements x,(x≠0) and y such that xy=yx=1. This is the condition of the field called Galois field, denoted by GF(2). The space of the functions f:C2n→GF(2),

Group of L-homeomorphisms and L _f-representability of Permutation Groups

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Published in Fuzzy Information and Engineering, 2020

Sini P

The classical mathematical theories have limitations for solving complicated problems that include uncertain data in many areas namely engineering, social science, medical science etc. One of the tools for dealing with these uncertainties is fuzzy sets. In 1965, Zadeh [1] introduced the theory of fuzzy sets. Later Gougen defined the concept of an -fuzzy set or an -set where is a semigroup, a partially ordered set, a lattice or a boolean ring [2]. We can extend most of the mathematical theories using the concept of an -set since -set is a generalisation of the fundamental mathematical concept of a set. Now many generalisations of fuzzy sets are presented and it gives a framework for generalising the classical mathematical theories [3–12].

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Introduction

Group of L-homeomorphisms and L f -representability of Permutation Groups

Group of L-homeomorphisms and L _f-representability of Permutation Groups