The Construction of an Abelian Group from The Set of Triangular Fuzzy Numbers

Abstract

A group is a non-empty set with a binary operation that satisfies some axioms, and is called an abelian group if the binary operation is commutative. A triangular fuzzy numbers (TFNs), denoted ˜A = (a, δ, γ), is a fuzzy subset of the real numbers R characterized by a center a ∈ R and spreads left and right δ, γ ∈ R+, representing the uncertainty around a. In this research, we will construct an abelian group from the set of TFNs over the binary operation ⊕ is given by (a1, δ1, γ1) ⊕ (a2, δ2, γ2) = (a1 + a2, δ1 + δ2, γ1 + γ2) for each (a1,δ1, γ1), (a2, δ2, γ2) ∈ ˜A. However, the tuple ( ˜A, ⊕) is not constructible as an abelian group. We define another binary operation to construct an abelian group from ˜A, i.e. ’∗^∼’ where (a1, δ1, γ1) ∗^∼ (a2, δ2,γ2) = (a1 + a2, δ1 · δ2, γ1 · γ2) for all (a1, δ1, γ1), (a2, δ2, γ2) ∈ ˜A. We proved that ( ˜A, ∗^∼) is an abelian group by verifying the associativity of ∗^∼, the existence of an identity element, the existence of inverses and the commutativity of the operation. This research provides the foundation for further  TFNs under the proposed binary operation.