Algebraic structures for fuzzy numbers from categorial point of view

Abstract

Using a classic algebraic construction additive and multiplicative structures (as commutative monoids) for fuzzy numbers are obtained. Moreover, we realize here an isomorphism between these structures.

Quotient algebraic structures on the set of fuzzy numbers

“…For the convenience of calculus, the fuzzy numbers are usually represented by their level sets, obtaining the parametric representation (see [18,20,53]), or by its two sides, considered as a pair of functions x − A and x + A , defined on the interval [0, 1] (see [13,35]). In this paper, we provide a completion of the results obtained by A. M. Bica in [6], indicating the nature of the quotient set obtained in this mentioned paper, for the additive and multiplicative structures of the set of fuzzy numbers and extending these results from unimodal fuzzy numbers to flat fuzzy numbers. More precisely, we will characterize the factor groups by using the set of the continuous functions with bounded variation on [0,1].…”

“…Firstly, using the extension principle, are defined and studied the arithmetic operations with fuzzy numbers and their properties (see [4,15,16,17,18,19,21,25,28,34,36,38,40,42,46,47,48,55,56]). Since in fuzzy arithmetic some of the usual properties DOI: 10.14736/kyb-2015-2-0255 of operations are missing, such as the nonexistence of the opposite of a (noncrisp) fuzzy number and the absence of the distributivity law of the scalar product for the sum of crisp numbers, several equivalence relations were proposed in order to avoid these defects (see [2,6,7,38,39,40,41,43,44,45,46,48,51]) and obtaining group properties for the quotient set. Since the set of fuzzy numbers is not a group with the addition, the difference of two fuzzy numbers is only a partial operation being defined as a substraction (see [48]) or by using the Hukuhara and generalized Hukuhara difference (see [54]).…”

“…In [6,Remark 25], it is mentioned that the quotient set F V / ∼ ⊕ of the fuzzy numbers set F V, has the properties R ⊂ F V / ∼ ⊕ and F V / ∼ ⊕ = R, but the nature of F V / ∼ ⊕ is not specified. As a main contribution of this paper we determine this set F V / ∼ ⊕ (denoted here by F), showing that it is topologically isomorphic with the set BVC [0, 1] of all continuous functions with bounded variation on [0, 1].…”

On Isomorphism Theorems for MI-groups

“…As we have mentioned above the main idea of our generalization of groups consists in the introduction of a set of pseudoidentities which possess similar properties like the identity (neutral) element. As a consequence of the definition we obtain that groups as well as commutative monoids equipped with an involutive operation representing inversion proposed by Bica in [1] (see [2]) form special subclasses of MI-groups.…”

“…In [2], the inverse elements are defined using an involutive monoidal automorphism. Since we deal here with MI-monoids being not abelian in general, we consider an MI-monoidal isomorphism of MI-monoid onto its dual which only reverses the order of operands.…”