Group-Like Uninorms

Sándor Jenei

doi:10.2991/ijcis.d.200706.001

Abstract

Uninorms play a prominent role both in the theory and the applications of aggregations and fuzzy logic. In this paper a class of uninorms, called group-like uninorms will be introduced and a complete structural description will be given for a large subclass of them. First, the four versions of a general construction—called partial lex product—will be recalled. Then two particular variants of them will be specified: the first variant constructs, starting from ℝ (the additive group of the reals) and modifying it in some way by ℤ’s (the additive group of the integers) what we will coin basic group-like uninorms, whereas the second variant can enlarge any group-like uninorm by a basic group-like uninorm resulting in another group-like uninorm. All grouplike uninorms obtained this way are “square” and have finitely many idempotent elements. On the other hand, we prove that any square group-like uninorm which has finitely many idempotent elements can be constructed by consecutive applications of the second variant (finitely many times) using only basic group-like uninorms as building blocks. Any basic group-like uninorm can be built by the first variant using only ℝ and ℤ, and any square group-like uninorm which has finitely many idempotent elements can be built using the second variant using only basic group-like uninorms: ultimately, all such uninorms can be built from ℝ and ℤ. In this way a complete characterization for square group-like uninorms which possess finitely many idempotent elements is given. The characterization provides, for potential applications in several fields of fuzzy theory or aggregation theory, the whole spectrum of choice of those square group-like uninorms which possess finitely many idempotent elements.

Highlights

Aggregation operations are crucial in numerous pure and applied fields of mathematics
Group-like uninorms, to be introduced below, form a subclass of involutive uninorms, and involutive uninorms play the same role among uninorms as the Łukasiewicz t-norm or in general, the class of rotation-invariant t-norms [14,15,16,17,18] do in the class of t-norms
Group-like uninorms will be defined as a particular subclass of involutive uninorms such that replacing their universe by an arbitrary linearly ordered set leads to the general notion of odd involutive FLe-chains: group-like uninorms are the monoidal operations of odd involutive FLe-chains over [0, 1]

Summary

INTRODUCTION

Aggregation operations are crucial in numerous pure and applied fields of mathematics. In order to narrow the gap between these two extremal classes, in [26,27] a deep knowledge has been gained about the structure of odd involutive FLe-chains, including a Hahn-type embedding theorem and a representation theorem by means of linearly ordered abelian groups and there-introduced constructions, called partial lex products [27] and the more general partial sublex products [26]: all odd involutive FLe-chains which have finitely many idempotent elements have a partial sublex product group-representation. We adapt the partial lex product construction to the narrower, more specific setting of group-like uninorms by introducing two particular variants of it These variants use only R and Z, the additive group of real numbers and integers, respectively. In our setting basic group-like uninorms have א0 prototypes, one for each natural number

PRELIMINARIES

Groupping Extensions Together

STRUCTURAL DESCRIPTION

Representation of Square Group-Like Uninorms by Basic Group-Like Uninorms

Open Problem