Products of elements in vague semigroups and their implementations in vague arithmetic


Demirci M.

FUZZY SETS AND SYSTEMS, vol.156, no.1, pp.93-123, 2005 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 156 Issue: 1
  • Publication Date: 2005
  • Doi Number: 10.1016/j.fss.2005.04.001
  • Journal Name: FUZZY SETS AND SYSTEMS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.93-123
  • Keywords: semigroup, fuzzy semigroup, vague semigroup, fuzzy arithmetic, vague arithmetic, fuzzy equivalence relation, VALUED EQUIVALENCE-RELATIONS, FUZZY FUNCTIONS, INDISTINGUISHABILITY OPERATORS, PART II, FOUNDATIONS, ALGEBRA, CONSTRUCTIONS, FUNDAMENTALS, RESPECT
  • Akdeniz University Affiliated: Yes

Abstract

Vague arithmetic different from the present literature of fuzzy arithmetic has been proposed in [Demirci (Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 10(1) (2002) 25; Internat. J. General Systems 32(2) (2003) 157, 177)] to model vaguely defined arithmetic operations resulting from the indistinguishability of real numbers. The main motivating problem of this paper is to introduce the notion of vague product (sum) of a finite number of real numbers in vague arithmetic, and to point out their fundamental properties. From a more abstract mathematical point of view, the vague product (sum) of a finite number of real numbers in vague arithmetic and their properties can be considered as the vague product of a finite number of elements in vague semigroups and their relevant properties. For this reason, a large part of this paper is devoted to the vague product of a finite number of elements in vague semigroups and their elementary properties. As a direct implementation of the present results, it is shown that the vague product (sum) of a finite number of real numbers in vague arithmetic can be easily evaluated in terms of the underlying many-valued equivalence relations. Furthermore, various non-trivial examples for the vague product (sum) of a finite number of real numbers in vague arithmetic are designed, and a simple technique for the construction of such non-trivial examples is stated. (c) 2005 Elsevier B.V. All rights reserved.