On the Composition of Overlap and Grouping Functions

: Obtaining overlap/grouping functions from a given pair of overlap/grouping functions is an important method of generating overlap/grouping functions, which can be viewed as a binary operation on the set of overlap/grouping functions. In this paper, ﬁrstly, we studied closures of over-lap/grouping functions w.r.t. (cid:126) -composition. In addition, then, we show that these compositions are order preserving. Finally, we investigate the preservation of properties like idempotency, migrativity,

The construction of the following overlap/grouping functions was developed in many literature works [1,4,13,15,16,21,27,28]. Obtaining overlap/grouping functions from given overlap/grouping functions is one of the methods to generate overlap/grouping functions. We consider this work as a composition of two or more overlap/grouping functions. As mentioned above, some properties are important for overlap/grouping functions. Thus, it raises the question of whether the new generated overlap/grouping function still satisfies the properties of overlap/grouping functions. In this paper, we consider properties preservation of four compositions such as meet operation, join operation, convex combination, and -composition of overlap/grouping functions. These results might serve as a certain criteria for choices of generation methods of overlap/grouping functions from given overlap/grouping functions.
The paper is organized as follows: In Section 2, we recall the concepts of overlap/grouping functions and their properties. In Section 3, we studied the closures of overlap/grouping functions w.r.t. -composition. In Section 4, we study the order preservation of compositions. In Section 5, we study properties' preservation of compositions. In Section 6, conclusions are briefly summed up.

Properties of Overlap and Grouping Functions
(PS) Power stable [29]:

Compositions of Overlap and Grouping Functions and Their Closures
In the following, we list four compositions of overlap/grouping functions including meet, join, convex combination, and -composition. In addition, we then studied their closures.

Compositions of Overlap and Grouping Functions
For any two overlap (or grouping) functions O 1 and O 2 , a convex combination of O 1 and O 2 is defined as for all (η, ξ) ∈ [0, 1] 2 and λ ∈ [0, 1]. For any two overlap (or grouping) functions O 1 and O 2 , the -composition of O 1 and O 2 is defined as for all (η, ξ) ∈ [0, 1] 2 .
Unfortunately, -composition of two bivariate functions does not preserve (O1). For This means -composition of two overlap/grouping functions is not closed.
However, it is possible to find an example that -composition of two overlap/grouping functions is also an overlap/grouping function. For example, for two given overlap func- The summary of the closures of two bivariate functions w.r.t. these compositions is shown in Table 1.

Order Preservation
In the following we show that the meet operation, join operation, convex combination, and -composition of overlap/grouping functions are order preserving.

Theorem 2. Suppose that four overlap functions have O
Proof. The case for meet operation, join operation, and convex combination are straightforward. We show only that -composition preserves order. For any η, ξ ∈ [0, 1], from .

Properties Preservation
In the following, we study properties preserved by meet operation, join operation, convex combination, and -composition of overlap/grouping functions.

Summary
Thus far, we have studied the basic properties of overlap/grouping functions w.r.t. the meet operation, join operation, convex combination, and -composition. The summary of the properties of overlap/grouping functions w.r.t. the meet operation, join operation, convex combination, and -composition is shown in Table 2.

Conclusions
This paper studies the properties preservation of overlap/grouping functions w.r.t. meet operation, join operation, convex combination, and -composition. The main conclusions are listed as follows.
(1) Closures of two bivariate functions w.r.t. meet operation, join operation, convex combination, and -composition have been obtained in Table 1. Note that -composition does not preserve (O1), and -composition of overlap/grouping functions is not closed. In other words, -composition can not be used to generate new overlap/grouping functions. (2) We show that meet operation, join operation, convex combination, and -composition of overlap/grouping functions are order preserving, see Theorems 2 and 3. (3) We have investigated the preservation of the law of (ID), (MI), (HO-k), (k-LI), and (PS) w.r.t. meet operation, join operation, convex combination, and -composition, which can be summarized in Table 2.
These results can be served as a certain criteria for choices of generation methods of overlap/grouping functions from given overlap/grouping functions. For example, convex combination does not preserve (PS). Thus, we can not generate a power stable overlap function from two power stable overlap functions by their convex combination.
As we know, overlap/grouping functions have been extended to interval-valued and complex-valued overlap/grouping functions. Could similar results be carried over to the interval-valued and complex-valued settings? Moreover, special overlap/grouping functions such as Archimedean and multiplicatively generated overlap/grouping functions have been studied. In these cases, many restrictions have been added. For further works, it follows that we intend to consider properties preservation of these overlap/grouping functions w.r.t. different composition methods.