# An Abstract Result on Projective Aggregation Functions

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (1)
- full if $\{W({\varphi}^{1}\left(x\right),\dots ,{\varphi}^{n}\left(x\right)):x\in X\}=X$, for every ${\varphi}^{1},\dots ,{\varphi}^{n}\in B\left(X\right)$,
- (2)
- idempotent if $W(x,\dots ,x)=x$, for every $x\in X$,
- (3)
- projective if it is a projection, i.e., if there is $k\in N$ such that $W\left(a\right)={a}_{k}$, for every $a=\left({a}_{j}\right)\in {X}^{n}$.

**Definition**

**2.**

- (1)
- binary independence if for every $({\varphi}^{1},\dots ,{\varphi}^{n}),({\psi}^{1},\dots ,{\psi}^{n})\in B{\left(X\right)}^{n}$, $x,y\in X$ such that ${\varphi}^{j}\left(x\right)={\psi}^{j}\left(y\right)$, for all $j\in N$, it holds that $F({\varphi}^{1},\dots ,{\varphi}^{n})\left(x\right)=F({\psi}^{1},\dots ,{\psi}^{n})\left(y\right)$,
- (2)
- unanimous if for every $({\varphi}^{1},\dots ,{\varphi}^{n})\in B{\left(X\right)}^{n}$, $x,y\in X$ such that ${\varphi}^{j}\left(x\right)=y$, for all $j\in N$, it holds that $F({\varphi}^{1},\dots ,{\varphi}^{n})\left(x\right)=y$,
- (3)
- projective if it is a projection, i.e., if there is $k\in N$ such that $F({\varphi}^{1},\dots ,{\varphi}^{n})={\varphi}^{k}$, for every $({\varphi}^{1},\dots ,{\varphi}^{n})\in B{\left(X\right)}^{n}$.

**Definition**

**3.**

- (i)
- the condition of independence of irrelevant alternatives if for any two profiles of individuals preferences $({\precsim}_{1}^{1},\dots ,{\precsim}_{1}^{n}),({\precsim}_{2}^{1},\dots ,{\precsim}_{2}^{n})\in {\mathcal{R}}^{n}$ and any two alternatives $x,y\in X$ such that $x{\precsim}_{1}^{j}y\iff x{\precsim}_{2}^{j}y$, and $y{\precsim}_{1}^{j}x\iff y{\precsim}_{2}^{j}x$, for all $j\in N$, it holds that $x{\precsim}_{G({\precsim}_{1}^{1},\dots ,{\precsim}_{1}^{n})}y\iff x{\precsim}_{G({\precsim}_{2}^{1},\dots ,{\precsim}_{2}^{n})}y$ and $y{\precsim}_{G({\precsim}_{1}^{1},\dots ,{\precsim}_{1}^{n})}x\iff y{\precsim}_{G({\precsim}_{2}^{1},\dots ,{\precsim}_{2}^{n})}x$,
- (ii)
- the Pareto condition if for any profile $({\precsim}^{1},\dots ,{\precsim}^{n})\in {\mathcal{R}}^{n}$ and any pair of alternatives $x,y\in X$ such that $x{\prec}^{j}y$, for all $j\in N$, it holds that $x{\prec}_{G({\precsim}^{1},\dots ,{\precsim}^{n})}y$.

**Theorem**

**1.**

## 3. The Main Result

**Theorem**

**2.**

- (i)
- W is projective.
- (ii)
- W is idempotent and full.

**Proof.**

**Example**

**1.**

**Remark**

**1.**

- (i)
- The assumption $\left|X\right|\ge 3$ cannot be ruled out from the statement of Theorem 2. Indeed, take $X=\{x,y\}$. Define $W:{X}^{3}\to X$ as follows: $W(x,x,x)=x$, $W(y,y,y)=y$, $W(x,x,y)=x$, $W(x,y,x)=x$, $W(y,x,x)=x$, $W(y,y,x)=y$, $W(y,x,y)=y$, $W(x,y,y)=y$. Clearly, W so-defined is idempotent and full. However, it is not projective.
- (ii)
- Alternative characterizations of projective functions can be shown provided that the concept of fullness is suitably modified. For instance, suppose that Z is a nonempty set such that $\left|Z\right|=\left|X\right|$, and denote by $B(Z,X)$ the set of all bijections from Z onto X. An aggregation function $W:{X}^{n}\to X$ is said to be full with respect to bijections if $\{W({b}^{1}\left(z\right),\dots ,{b}^{n}\left(z\right)):z\in Z\}=X$, for every ${b}^{1},\dots ,{b}^{n}\in B(Z,X)$. Then, following the proof of Theorem 2 with the obvious modifications, the next result is in order: assume $\left|X\right|\ge 3$. Then, W is projective iff it is idempotent and full with respect to bijections.
- (iii)
- In relation to Remark 1(ii) above, suppose now that $\left|Z\right|\ge \left|X\right|$. Let $S(Z,X)$ stand for the set of all surjections from Z onto X. An aggregation function $W:{X}^{n}\to X$ is said to be full with respect to surjections if $\{W({s}^{1}\left(z\right),\dots ,{s}^{n}\left(z\right)):z\in Z\}=X$, for every ${s}^{1},\dots ,{s}^{n}\in S(Z,X)$. Translating into this context the arguments used in the proof of Theorem 2, the following result can be proven: assume $\left|X\right|\ge 3$. Then, W is projective iff it is idempotent and full with respect to surjections (This result requires consideration of a well ordering, say ≤, on X. Then, for every $s\in S(Z,X)$, define the binary relation, denoted by ${s}_{\le}$, on Z as follows: $z\phantom{\rule{4pt}{0ex}}{s}_{\le}\phantom{\rule{4pt}{0ex}}t\iff s\left(z\right)\le s\left(t\right)$, $(z,t\in Z)$. Consider the following subset of binary relations on Z, ${\mathcal{W}}_{\le}:=\{{s}_{\le}:s\in S(Z,X)\}$. Note that, for each $s\in S(Z,X)$, ${s}_{\le}$ is a total preorder on Z that induces a well ordering on the quotient set $Z/{\sim}_{{s}_{\le}}$. Then, the variant of Arrow’s theorem used in this situation states that a partial social welfare function $G:{\left({\mathcal{W}}_{\le}\right)}^{n}\to \mathcal{R}$ that satisfies the condition of independence of irrelevant alternatives and the Pareto condition is projective.). A similar result can be given by adapting the fullness condition to the context of injections from Z into X provided that $\left|Z\right|\le \left|X\right|$.

## 4. Some Consequences of the Main Result

**Corollary**

**1.**

- (i)
- G is projective.
- (ii)
- G admits a selection that is idempotent and full.

**Corollary**

**2.**

- (i)
- F is projective.
- (ii)
- F is unanimous and satisfies binary independence.

**Proof.**

**Remark**

**2.**

- (i)
- It should be noted that, according to Remarks 1(ii) and 1(iii), alternative versions of Corollary 2 can be established by replacing $B\left(X\right)$ with $B(Z,X)$ and $S(Z,X)$, respectively.
- (ii)
- Corollary 2 can be given an interpretation in the fuzzy framework. Assume U is a finite universe, with states $\{{u}_{1},\dots ,{u}_{m}\}$, and denote by X a set of fuzzy numbers defined on U such that $\left|X\right|\ge 3$ (Note that each number in X can be identified with a vector in the m-dimensional unit cube where, for every $l\in \{1,\dots ,m\}$, the l-component might be interpreted as the uncertainty, or vagueness, that state ${u}_{l}$ occurs. Then, X could be thought of as the set of possible uncertain situations to be managed.). Then, $B\left(X\right)$ represents the set of all possible rearrangements of the referenced uncertain situations. Further, suppose that there are n individuals, $n\in \mathbb{N}$, each of them presenting a particular arrangement of the uncertain situations, and a final arrangement needs to be reached as an aggregation procedure of the individual proposals. Then, Corollary 2 provides necessary and sufficient conditions for such an aggregation procedure to coincide with the proposal of some individual.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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Candeal, J.C.
An Abstract Result on Projective Aggregation Functions. *Axioms* **2018**, *7*, 17.
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Candeal JC.
An Abstract Result on Projective Aggregation Functions. *Axioms*. 2018; 7(1):17.
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Candeal, Juan C.
2018. "An Abstract Result on Projective Aggregation Functions" *Axioms* 7, no. 1: 17.
https://doi.org/10.3390/axioms7010017