Abstract
Uninorms play a prominent role both in the theory and applications of Aggregations, Fuzzy Theory, and of Mathematical Fuzzy Logic. In this paper the class of group-like uninorms is introduced. First, two variants of a construction method – called partial-lexicographic product – will be recalled; it constructs a large subclass of group-like FL\(_e\)-algebras. Then two specific ways of applying the partial-lexicographic product construction to construct uninorms will be presented. The first one constructs starting from \(\mathbb R\) and modifying it in some way by \(\mathbb Z\)’s, what we call basic group-like uninorms, whereas with the second one may extend group-like uninorms by using \(\mathbb Z\) and a basic uninorm to obtain further group-like uninorms. All group-like uninorms obtained this way have finitely many idempotents. On the other hand, we assert that the only way to construct group-like uninorms which have finitely many idempotents is to apply this extension (by a basic group-like uninorm) consecutively, starting from a basic group-like uninorm. In this way a complete characterization for group-like uninorms which possess finitely many idempotents is given. The obtained uninorm class can be candidate for the aggregation operation of several applications. The paper is illustrated with several 3D plots.
This work was supported by the GINOP 2.3.2-15-2016-00022 grant.
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Notes
- 1.
Other terminologies for FL\(_e\)-algebras are: pointed commutative residuated lattices or pointed commutative residuated lattice-ordered monoids.
- 2.
That is, there exists a binary operation
such that 
if and only if 
; this equivalence is called residuation condition or adjointness condition, (
) is called an adjoint pair. Equivalently, for any x, z, the set 
has its greatest element, and 
is defined as this element: 
. - 3.
Note that intuitively it would make up for a coordinatewise definition, too, in the second line of (1) to define it as
. But \(\bot \) is not amongst the set of possible second coordinates. However, since \(X_{gr}\) is discretely embedded into X, if 
would be an element of the algebra then it would be equal to 
. - 4.
The rank of an involutive FL\(_e\)-algebra is positive if \(t>f\), negative if \(t<f\), and 0 if \(t=f\).
- 5.
The negative cone consists of the elements which are smaller or equal to t.
such that 
if and only if 
; this equivalence is called residuation condition or adjointness condition, (
) is called an adjoint pair. Equivalently, for any x, z, the set 
has its greatest element, and 
is defined as this element: 
.
. But 
would be an element of the algebra then it would be equal to 
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Jenei, S. (2019). A New Class of Uninorm Aggregation Operations for Fuzzy Theory. In: Rutkowski, L., Scherer, R., Korytkowski, M., Pedrycz, W., Tadeusiewicz, R., Zurada, J. (eds) Artificial Intelligence and Soft Computing. ICAISC 2019. Lecture Notes in Computer Science(), vol 11508. Springer, Cham. https://doi.org/10.1007/978-3-030-20912-4_28
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