Dynamic Ordered Weighted Averaging Functions for Complete Lattices^{1}^{1}1Preprint submited to Soft Computing
Abstract
In this paper we introduce a class of operators on complete lattices called Dynamic Ordered Weighted Averaging (DYOWA) functions. These functions provide a generalized form of an important class of aggregation functions: The Ordered Weighted Averaging (OWA) functions, whose applications can be found in several areas like: Image Processing and Decision Making. The wide range of applications of OWAs motivated many researchers to study their variations. One of them was proposed by Lizassoaim and Moreno in 2013, which extends those functions to complete lattices. Here, we propose a new generalization of OWAs that also generalizes the operators proposed by Lizassoaim and Moreno.
Keywords:
Aggregations functions tnorms tconorms OWA functionsDYOWA functions Complete lattices.∎
1 Introduction
After the contributions of (Zadeh, 1965) in the field of Fuzzy Sets, many extensions of classical mathematical theories have been developed, with several possibilities of application. Applications in areas like: Image Processing and Decision Making require some special functions capable of encoding a set of multiple values in a single value; these functions are called: Aggregation functions (Beliakov et al, 2016; Bustince et al, 2013; Chen and Hwang, 1992; Dubois and Prade, 2004; Liang and Xu, 2014; Paternain et al, 2015, 2012; Zhou et al, 2008).
Aggregation functions can be classified into four classes: Averaging, conjunctive, disjunctive and mixed. The disjunctive and conjunctive aggregations functions provide, respectively, models for disjunctions and conjunctions in Fuzzy Logic (Beliakov et al, 2016; Bustince et al, 2010, 2012; Dimuro and Bedregal, 2014; Dubois and Prade, 1985; Klement et al, 2000; Farias et al, 2016b). On the other hand, averaging aggregation functions can be applied, for example, in fields like image processing and decision making (Paternain et al, 2015; Bustince et al, 2011; Yager, 1988; Zadrozny and Kacprzyk, 2006).
A special type of averaging aggregation is called: Ordered Weighted Averaging function, or simply OWA, function. It was developed by Yager (Yager, 1988) with the intention to study the problem of multiple decision making, however many other applications for such operators have arisen since then (Paternain et al, 2015; Llamazares, 2015; Torra and Godo, 2002; Lin and Jiang, 2014).
Some variations of OWAs can be found in literature; e.g. see (Chen and Hwang, 1992; Cheng and Chang, 2006; Merigó and GilLafuente, 2009; Merigó, 2012; Yager, 2006). All of them are defined on the set . In 2013, Lizasoain and Moreno (Lizasoain and Moreno, 2013) generalized those operators to any complete lattice .
All of such different approaches of OWAs have an essential common factor: They use a fixed vector of weights for the final calculation. In this paper, we propose a new way of generalization of OWAs on complete lattices; the vector of weights is determined from the input arguments providing a “dynamic flavour”. More precisely, the weights are variables defined from the input vector.
We start this paper by exposing some basic concepts such as: Aggregation functions, OWAs, Tnorms and Tconorms. Sections 3 and 4 provide the extension of some concepts previously listed to complete lattices; they also expose the generalization proposed by Lizasoain and Moreno. In section 5, we introduce our proposal of generalizing OWA for complete lattices, we study some of its properties and present some examples in different environments. We will show in this part of the paper that the OWA functions proposed here, as well as those proposed by Yager, are averaging functions. We will also prove that the Yager operators can be obtained as a particular case of our OWA operators. To conclude, we bring the section of conclusions and future works.
2 Aggregation Functions
The aggregation functions are mathematical tools that allow you to perform grouping complex information into a more simple information. More precisely, these functions are rules that associate each dimensional input to a unique value, the output. The formal definition is presented below:
Definition 1 (Aggregation Function)
A function which satisfies the following properties:

and ;

whenever for all .
is called of nary aggregation function.
Applications of aggregation functions can be found, for example, in decisionmaking problems and in the formulation of some fuzzy logic connectives (see (Beliakov et al, 2016)). In the following, we introduce some notations that will be used in this paper.
Remark 1

We use to denote the dimentional vector whose coordinates belong to the set .

Functions as and are classical example of nary aggregations function.

In order to simplify the terminology in some points of the text we use the term aggregate function instead of nany aggregation function.
Aggregation functions can be classified into four different types:
Definition 2
Let be an aggregation function. We say that is a:

Averaging aggregation function if for any ;

Conjuntive aggregation function if for all ;

Disjuntive aggregation function if for all ;

Mixed aggregation function if it does not belong to any of the previous classes.
Table 1 presents some examples of aggregation functions.
Function  Averaging  Conjunctive  Disjunctive  Mixed 

X  X  
X  X  
X  
X  
X  
X 
Definition 3
A function satisfies the properties of:

Idempotency if for all ;

Symmetry if for any permutation of the set we have ;

Neutral element if there is a element such that for all allocated in any coordinate we have to ;

Absorption if has an absorption element , i.e., if for all we have to ;

Homogeneity if for any we have to ;

Zero divizor if there is such that ;

One divizor if there is such that ;

Associativity if and for any .
Table 2 presents some examples of aggregations functions which satisfy such properties.
Aggregation function  Properties 

(IP), (SP), (NP), (AP), (HP) and (ASP)  
(IP), (SP), (NP), (AP), (HP) and (ASP)  
(IP), (SP), (HP) and (ASP)  
(SP), (NP), (AP) and (ASP)  
(SP), (NP), (AP) and (ASP)  
(SP) and (AP) 
2.1 OWA Functions
The Ordered Weighted Averaging – OWA function, defined by Yager in (Yager, 1988), constitute an important family of averaging aggregation functions, which have been widely studied by many researchers around the world, motivated by its wide range of applications. Applications of OWA can be found, for example, in image processing (Paternain et al, 2015; Bustince et al, 2011; Zadrozny and Kacprzyk, 2006), in neural networks (Amin and Emrouznejad, 2011a, b; Emrouznejad, 2008) and in decision making (Cheng and Chang, 2006; Ahn, 2008; Miguel et al, 2016). The definition of this important class of functions is presented below:
Definition 4 (OWA Funtion)
Given a dimentional vector of weights ^{2}^{2}2A dimentional vector of weights is such that . , the function
where is the descending ordernation of the vector , is called of Ordered Weighted Averaging function or simply OWA function.
It is not difficult to show that for any vector of weights , the function is an averaging aggregation function. Furthermore, are continuous functions which satisfy: (IP), (SP) and (HP), but do not: (ZD) and (OD). They are parametric functions; namely: Depending on the vector of weights it will simulate an average aggregation function. Below, we present some examples:
Example 1

is obteined by vector of weights ;

can be obteined by vector of weights ;

is the OWA function with ;

The median,
can be found from:

If is odd, then for all and .

If is even, then for all and , and .

2.2 tnorms and tconorms
Some aggregation functions provide models for conjunctions and disjunctions in fuzzy logic. These operators are called respectively of tnorms and tconorms:
Definition 5 (tnorms)
A tnorm is a function which satisfies:

for all ;

for any ;

for any ;

whenever .
Definition 6 (tconorms)
A tconorm is a function such that:

for any ;

for all ;

for all ;

whenever .
Tnorms are conjunctive and tconorms are disjunctive aggregation functions. Table 3 contains some examples of tnorms and tconorms. The reader can find in (Klement et al, 2000) a deeper insight about tnorms and tconorms on and in (Bedregal et al, 2006; Baets and Mesiar, 1999; Cooman and Kerre, 1994; Palmeira et al, 2014) on some class of the lattices.
tnorms  tconorms 

3 Aggregations, tnorms and tconorms for complete lattices
A complete lattice is a partial order, , in which any subset has supremum and infimum elements, denoted respectively by and (Birkhoff, 1961; Gierz et al, 1980). Complete lattices are bounded; i.e. they have top, , and bottom elements, .
The properties (IP), (SP), (NP), (AP), (HP), (ZD), (OD) and (ASP) can be extended to lattices, as well as: aggregations, tnorms and tconorms.
Definition 7
An isotonic function^{3}^{3}3A function is isotonic if, , whenever , for all . such that:

;

is called of aggregation function on .
Definition 8 ((Baets and Mesiar, 1999))

An isotonic binary operator which satisfies (SP) and (ASP), and have as neutral element is a tnorm on .

An isotonic binary operator that satisfies (SP) and (ASP), and have as neutral element is a tconorm on .
The associativity of and allows us to define nany operators, as follow:
and
Proposition 1
Let be a tnorm and a tconorm on a complete lattice . Then, for any are valid:

and ;

and ;

and ;

.
4 OWA operators for complete lattices
Several variations of OWA functions defined on the interval can be found in literature; e.g. IGOWA, IGCOWA, POWA and cOWA (Merigó and GilLafuente, 2009; Merigó, 2012; Yager, 2006; Chen and Zhou, 2011). Another approach is due to Lizasoain and Moreno (Lizasoain and Moreno, 2013) which generalized OWAs to complete lattices.
Definition 9 (Definition 3.3 of (Lizasoain and Moreno, 2013))
Let be a complete lattice and be a tnorm and a tconorm. We say that is a:

vector of weights on whenever ;

distributive vector of weights on whenever it satisfies the (i) and
Remark 2
If is a complete lattice and and , then is a:

vector of weights if, and only if, ;

distributive vector of weights if, and only if, satisfies and , for all .
To calculate the output of an OWA (in the sense of Yager), we need to sort the ndimensional input vector in a decreasing way. This process is always possible when the underlying complete lattice is the linear order, but there are complete lattices with pairs of noncomparable elements. For this reason, we need to define an auxiliary vector from the input vector. This is done using the following Lemma:
Lemma 1 (Lemma 3.1 of (Lizasoain and Moreno, 2013))
Let be a complete lattice. For any , consider the following values:







.
Then, . If is totally ordered, then there is a permutation, , for the set , such that .
A proof of this Lemma can be found in (Lizasoain and Moreno, 2013). To simplify the notations we use the following definition:
Definition 10
If is a complete lattice, then the function defined by:
where is the dimentional vector obtained according to Lemma 1, is called of LizassoainMoreno function.
Example 2
If is a complete lattice and , then:
In the following, we list some properties of LizassoainMoreno function:
Proposition 2 (Properties of LizassoainMoreno function)
If is a complete lattice, then:

If is such that any pair of coordinates and , with , is comparable, then , where is a permutation on the set such that .

If is a linear order, then for all there is a permutation on the set such that .

If , then .


For any permutation for and for all , .
Proof

Straightforward from Lemma 1.

Since is a linear order, then any satisfies . Thus, for any there is a permutaion such that .

Straightforward from definition of Lizassoain and Moreno funcion.

As , where then, by follows that for all ,

Let be and . By definition, we have to:
Therefore, .
Now that we have defined the LizassoainMoreno function and know some of its properties, it is possible to define a generalized version of OWAs for complete lattices:
Definition 11 (Definition 3.5 of (Lizasoain and Moreno, 2013))
Let be a distributive vector of weights in a complete lattice . For any , consider the totally decreasing ordered vector . The LizasoainMoreno OWA function associated with and the triplet is
(1) 
Examples and properties of LizasoainMoreno OWA can be found in (Lizasoain and Moreno, 2013), it is noteworthy that Yager’s OWA is a particular case of LizasoainMoreno’s OWA:
Theorem 1
Every Yager’s OWA is an LMOWA.
Proof
Let , , and be a vector of weights. Since, and , for all , then is a distributive vector of weights.
To prove that coincides with the Yager’s OWA, first note that the Proposition 2 ensures that for any input vector there is a permutation on such that . Besides,
that is, . Therefore, for all it is verified that:
Remark 3

OWA’s satisfies the properties (IP) and (SP). Futhermore, for any distributive vector of weights and all we have
Now, observe that both: Yager’s and LizasoainMoreno’s OWAs are obtained from a unique fixed vector of weights . In (Farias et al, 2016a, c) we propose a generalization of Yager’s OWA, in such a way that the weights are not fixed. In this sense, we propose here a generalization of LizasoainMoreno’s OWA taking into account nonfixed weights.
5 Dynamic Ordered Weighted Averaging Functions
In the sequel we propose and investigate a generalized form of OWA for complete lattices; they are named Dynamic Ordered Weighted Averaging (DYOWA) functions. The DYOWA functions generalize both Yager’s and LizasoainMoreno’s OWA. In order to to introduce them, we need first to define the notion of weights function.
Definition 12 (Weight function)
Let be the structure , where is a complete lattice, is a tnorm and is a tconorm. A finite family of functions is called of weight function family whenever for all , is a vector of weights on . The natural function , s.t. is called weight function on ^{4}^{4}4Or just weight function whenever is clear in the context.. If furthermore, for all ,
is called distributive weight function on or simply that is a distributive family.
Remark 4
Since then, by (T1), .
The following are some examples of distributive families:
Example 3
Let be a complete lattice, and . Then a weight family of functions must satisfy
In particular, if for all , and for , then is a weight function. Furthermore,
that is, is a distributive family. Analogously, if and for , then is also a distributive family.
Example 4
Consider , and . A distributive family must satisfy:
and
for any and each . In particular, , with
satisfies these properties. Therefore, is a distributive family.
Example 5
Let be , where and is the KulischMiranker partial order (Kulisch and Miranker, 1981), i.e., . Moreover, and are a tnorm and a tcomorm, respectively, on (See Bedregal and Takahashi (Bedregal and Takahashi, 2006)). It is easy to verify that the finite family of functions formed by given by (constant functions) provides a distributive family.
Now we can define our proposed generalized form of OWA.
Definition 13 (DYOWAs)
Given a complete lattice , a tnorm , a tconorm and a weigtht function family , we call of Dynamic Ordered Weighted Averaging Function (DYOWA), the function:
where .
Below we will present some examples of DYOWA functions.
Example 6
Let , , and defined in Example 3, then and are:
and
In addition, and are isotonic functions which satisfy (IP), (SP), (NP), (AP) and (ASP), but do not satisfy (ZD) and (OD).
Example 7
If , and are defined as in Example 4, then
Futhermore, satisfies (IP) and (SP), but do not (NP), (HP), (ZD), (OD) and (AP). It is important to note that, when , this DYOWA function is not monotonic, since and .
Example 8
For , , and provided in Example 5 a formulae to calculate for any dimension is difficult to find. However, for this can be done by observing that for :
and