A Generalization of Arrow’s Lemma on Extending a Binary Relation

Abstract

By examining whether the individualistic assumptions used in social choice could be used in the aggregation of individual preferences, Arrow proved a key lemma that generalizes the famous Szpilrajn’s extension theorem and used it to demonstrate the impossibility theorem. In this paper, I provide a characterization of Arrow’s result for the case in which the binary relations I extend are not necessarily transitive and are defined on abelian groups. I also give a characterization of the existence of a realizer of a binary relation defined on an abelian group. These results also generalize the well-known extension theorems of Szpilrajn, Dushnik-Miller, and Fuchs.

1. Introduction

One of the most fundamental results on extensions of binary relations is due to Szpilrajn [1], who shows that any transitive and asymmetric binary relation has a transitive, asymmetric, and complete extension. The Szpilrajn extension theorem has applications in many areas, including mathematical logic, order theory, mathematical social sciences, computer sciences, computability theory, fuzzy mathematics, and other fields in pure and applied mathematics. This result remains true if asymmetry is replaced with reflexivity; that is, any quasi-ordering has an ordering extension. Fuchs [2] obtained a more general theorem on extending a partially ordered group to a linear one. In fact, Szpilrajn’s theorem reduces to Fuchs’s theorem, when . Fuchs’ result has been thoroughly investigated (see [3–6]). In fact, It has been proven that a partially ordered abelian group has a linear extension if and only if is torsion-free. Arrow [7, Page 64] states a generalization of Szpilrajn’s theorem, which was the basis for his famous general impossibility theorem under the individualistic assumptions. This generalization may be stated as follows: Suppose that is a quasi-ordering defined on a set of alternatives , and is the set of alternatives that any two elements in are incomparable according to ( implies ). Then, given any ordering in , there exists an ordering in which is compatible both with the given ordering in and with . In other words, there exists an ordering of all the alternatives of which is compatible with any ordering in . While the property of quasi-ordering satisfying the Arrow’s assumptions is sufficient for the existence of an ordering extension, this is not necessary. As shown by Suzumura [8], -consistency is a necessary and sufficient condition for a binary relation to have an extension satisfying Arrow’s assumptions. The extension theorems of Szpilrajn type have played an important role in social choice and economic and game theories. For example, one way of assessing whether a preference relation is rational is to check whether it can be extended to a transitive and complete relation (see [9, 10]). Another example is the problem of the existence of maximal elements of binary relations. If is a binary relation which has a linear extension , then any maximal element of is also a maximal element of . In a very general sense, if is defined in a topological space, then the existence of a linear extension of satisfying some continuity conditions is equivalent to the existence of a continuous utility function representing (a binary relation defined on is represented by a utility function , if for all ) (see [11–14]). In the field of computer science, a topological sorting of a directed graph is a linear ordering of its vertices (it is well-known that directed graphs are useful in computer science because they serve as mathematical models of network structures) such that every edge between two vertices and , the vertex comes before vertex . That is, topological sorting can be used to convert a directed graph into a linear order such that if any event requires that be completed before is initiated, then will occur before in the ordering. An elementary result in this direction says that a necessary and sufficient condition for a directed graph to have a topological sorting is acyclicity. The first implication is equivalent to the statement that every partial order has a linear order extension, which, as Knuth [15] noted, was proved by Szpilrajn 1930] for infinite as well as finite sets. Linearly ordered groups play an important role in many areas in pure and applied mathematics. For example, in economic and game theories, pairwise comparison matrices over a real linearly ordered group play a basic role in multicriteria decision-making methods such as the Analytic Hierarchy Process. (The Analytic Hierarchy Process provides a comprehensive and rational framework for structuring a decision problem, for representing and quantifying its elements, for relating those elements to overall goals, and for evaluating alternative solutions.) In utility theory, a major unsolved question is what can be said if a given preference relation cannot be represented by a utility function. In this case, one possibility is to seek utility functions with values in a suitable linearly ordered group (see [16]). In computer science, variants of duration calculus have been developed for discrete and abstract time (a system is said to be real-time if the total correctness of an operation depends not only on its logical correctness, but also on the time in which it is performed. In computer science, real-time computing describes hardware and software systems subject to a “real-time constraint”. Duration Calculus is an interval-based logic for the specification of real-time systems), where an arbitrary (commutative) linearly ordered group can be the model of time (see [17]).

Building on the original result of Szpilrajn’s Theorem, Dushnik and Miller [18] proved that any strict partial order is equal to the intersection of its linear ordering extensions and, based on this fact, defined the dimension of a partially ordered set as the smallest number of linear orderings the intersection of which is the partial order. Fuchs [2, Theorem 2] proved a more general Dushnik-Miller extension theorem on partially ordered groups. The extension theorems of Dushnik-Miller and Fuchs have many applications in pure and applied mathematics. Much of economic and social behaviour observed is either group behaviour or that of an individual acting for a group. Group preferences may be regarded as derived from individual preferences, by means of some process of aggregation. For example, if all voters agree that some alternative is preferred to another alternative , then the majority rule will return this ranking. In this case, there is one simple condition that is nearly always assumed called the principle of unanimity or Pareto principle. This declares that the preference relation for a group of individuals should include the intersection of their individual preferences. Another example of the use of intersections is in the description of simple games which can be represented as the intersection of weighted majority games [19].

In this paper, I prove extension theorems for binary relations in a general framework allowing abelian groups, which generalize the well-known extension theorems of Arrow, Suzumura, Szpilrajn, Dushnik-Miller, and Fuchs.

2. Notations and Definitions

Let be a nonempty set, and let be a binary relation on . If is any subset of , the restriction of to is the relation . If and are two binary relations on , then denotes the set that results from removing the elements of from . We denote as . Let be a binary relation on and let . We say that and are -incomparable if and . Let and denote, respectively, the asymmetric part of and the symmetric part of , which are defined, respectively, by and and and . We say that on is (i) reflexive if for each ; (ii) transitive if for all , [ and ] ; (iii) antisymmetric if for each , [ and ] ; (iv) complete if, for each , we have or . The transitive closure of a relation is denoted by , that is for all if there exist and such that for all and . Clearly, is transitive and because the case is included, it follows that . Acyclicity says that and do not exist such that , for all and . The relation is -consistent, if, for all , implies (see [20]). The following combination of properties is considered in the next theorems. A binary relation on is (i) partial order if is reflexive, transitive, and antisymmetric and (ii) linear order if is a complete partial order. We say that is contradictory if any two of , , and can not be satisfied simultaneously.

An abelian group is a set, , together with an operation + that combines any two elements and to form another element, denoted and satisfy the following requirements: (i) for all , ; (ii) for all and in , ; (iii) there exists an element in , such that, for every element , the equation holds; (iv) for each , there exists an element such that ; (v) for all , . We write to mean . The sum ( summands) is abbreviated as (called a multiple of ), and ( summands) as with (where denotes the set of positive integers). A binary relation defined on an abelian group is homogeneous if it satisfies the following requirements: () is contradictory; () if and hold for some , then holds. An ordered group is an abelian group equipped with a homogeneous binary relation . We say that is normal, if for some positive integer implies . An ordered group is cancellative if for all , implies . If an ordered group is cancellative, then means that and , which yields and , and therefore . Every ordered group is automatically cancellative since implies , and therefore . An ordered group is called: (1) partially ordered group if is reflexive and transitive and (2) linearly ordered group if is reflexive, transitive, antisymmetric, and complete. If is an ordered group, then we say that has a linear ordering extension if and only if is a linearly ordered group such that and . In fact, subsumes all the pairwise information provided by and possibly further information.

3. The Main Results

In the context of examining if the individualistic assumptions used in economics can be used in the aggregation of individual preferences ([7, Definition 5, Theorem 2], Arrow proved a key lemma that extends the famous Szpilrajn’s Theorem.

Arrow’s Lemma ([7, pp. 64-68]). Let be a quasi-ordering on , a subset of such that, if and , then , and an ordering on . Then, there exists an ordering extension such that .

In fact, the lemma says that, if is a binary relation defined on a set of alternatives , then given any ordering to any subset of -incomparable elements, there is a way of ordering all the alternatives which will be compatible both with and with the given ordering in . In this case, it is important that the linear extension of inherits the relationship we put between the -incomparable elements of .

Definition 1. A binary relation is called -consistent if .

Clearly, -consistency implies -consistency. The following proposition is evident.

Proposition 2. A binary relation is -consistent if and only if is acyclic.

In the following, denotes the set of all natural numbers.

Theorem 3. Let be an ordered group, be a subset of such that, if and then , and let be a linear ordering on . Then, has a linear ordering extension such that if and only if is a normal -consistent binary relation.

Proof. To prove necessity, let be an ordered group and let be normal and -consistent. Suppose that and be two -incomparable elements of .
We putWe show that is an ordered group with being a normal, reflexive, and -consistent extension of satisfying .
First of all, we note that is never zero, because otherwise we should have for some , whence by normality of we have against hypothesis.
To show that is an ordered group, we show that is contradictory and homogenous.
To prove that is contradictory, we show that any two of the three relations , , and cannot be satisfied simultaneously. We prove the case of and ; the other cases are obvious. We have four cases to consider.
Case 1. and . This is impossible because of the homogeneity of the relation .
Case 2. as well as for some . By adding times the in the second inequality, one obtains ; that is to say, . But then, by normality we are led to , a contradiction.
Case 3. and for some . This is impossible as in the preceding case.
Case 4. In this case we have , , and for some . By adding times the first, times the second inequality, we have ; that is, . If , by normality we have , which is impossible. On the other hand, if , then since we have . It follows that and . Therefore, by normality we have and . Since is contradictory, we conclude that , an absurdity. Therefore, is contradictory.
To prove homogeneity for , let such that and . We have four cases to consider: () and ; () and for some ; () for some and ; () and for some . We only prove the fourth case, as the proof of the others is similar. In this case, we have that and . Thus, .
We proceed now to prove that is -consistent. Indeed, suppose to the contrary that there are such that and . Thus, there exists such that and . Therefore, there exist nonnegative integers , , such that By adding times the first, , times the second, times the n-th and by using the fact that , we have We have that is nonzero. On the other hand, , because, otherwise, implies that , an absurdity. It follows that , . But then, a contradiction to -consistency of . It follows that is -consistent.
To prove normality, let for some integer and some . Then, or there exist nonnegative integers such that . If , then by the normality of we have which implies that . Otherwise, which implies as well.
It remains to prove that is an extension of . Clearly, . On the other hand, if for some , then which implies that . It follows that .
Finally, since and , we conclude that .
Suppose that denotes the set of ordered groups such that, for each , is a normal, reflexive, and -consistent extension of satisfying . Since this set is nonempty. Let be a chain in , and let . It is easy to check that is an ordered group such that is a normal, reflexive, and -consistent extension of satisfying .
By Zorn’s lemma, possesses an element, say , that is maximal with respect to set inclusion. It follows that is an ordered group such that is a normal, reflexive, and -consistent extension of satisfying . Let be the transitive closure of . Then, is an ordered group such that is a normal, reflexive, transitive, and -consistent extension of satisfying . To prove it, we show only the normality for . All the other conditions are easily verified from the fact that they are also satisfied by . Indeed, let for some integer and some . Therefore, there exist such that Since and , by the homogeneity of we conclude that . But then, by the cancellativity of we have that and, by an induction argument based on this logic, we obtain which implies that . To prove that is a linearly ordered group, it remains to show that is complete and antisymmetric. Since antisymmetry of is an immediate consequence of the -consistency of , we prove the completeness of . Suppose to the contrary that there exist such that and . We define Then, as in the case of above, we can prove that is an ordered group such that is a normal, reflexive, and -consistent extension of satisfying , a contradiction of maximality of . Therefore, is complete.
To complete the necessity part we show that . Evidently, . To prove the converse, let for some . Suppose to the contrary that . Since is complete, holds which implies ( since ). Since is a linear order extension of , we have that . But then, and antisymmetry of imply that , a contradiction. The last contradiction shows that .
Conversely, suppose that is an ordered group and has a linear order extension . Suppose to the contrary that is non--consistent. Then, . Thus, there exists , such that and , which contradicts with the fact that is antisymmetric. It remains to prove that is normal. Suppose to the contrary that for some integer and and . Then, as in the case of above, the homogeneity and cancellativity of implies that . On the other hands, implies that . It follows that , because otherwise jointly to implies that which concludes that ( is antisymmetric), a contradiction to . Then, as in the case of above, for ( and are incomparable with respect to ) and there exists a homogeneous extension of such that . It follows that . On the other hand, ( and ) implies , a contradiction to the contradictory of . The last contradiction shows that is normal.

The following result of Arrow [7] and Suzumura (see [1, Main Theorem]) is an immediate consequence of Theorem 3.

Corollary 4. Let be a binary relation on and let be a subset of such that, if and , then , and let be a linear ordering on . Then, has a linear ordering extension such that if and only if is an -consistent binary relation.

Corollary 5 (Szpilrajn’s extension theorem [1]). Every partial order possesses a linear order extension . Moreover, if and are any two -incomparable elements of , then there exists a linear order extension in which and a linear order extension in which .

A consequence of Theorem 3 is also the result of Fuchs for extending partial orders of abelian groups to linear orders. The following corollary shows this fact.

Corollary 6 ([2, theorem 1]). Let be an ordered group and be two -incomparable elements. Then, has a linear ordering extension such that .

Proof. The corollary is an immediate consequence of the necessity part of Theorem 3 for being a partial order, and .

Moreover, if such an extension exists, then is necessarily normal [2, Lemma §3].

Definition 7. Let be an ordered group and let be a collection of linear order extensions of . Then is a realizer of if and only if the following conditions are satisfied: () the intersection of the members of coincides with and () for every pair of -incomparable elements , there exists a with .

The following result generalizes the classical Dushnik-Miller’s type extension theorem for an ordered group (see [2, Theorem 2]).

Theorem 8. Let be an ordered group. Then, has as realizer the set of its linear order extensions if and only if is a normal -consistent binary relation.

Proof. To prove necessity, let be an ordered group and let be a normal -consistent binary relation on . Suppose that be the set of linear order extensions of . By Theorem 3, is nonempty. We show that . Indeed, since , we have to show that . Assume by way of contradiction that there exists an with . It follows that . On the other hand, holds. Indeed, suppose to the contrary that . It follows that for some which jointly to contradicts with the fact that is -consistent. Therefore, . Then, as above (since is homogeneous and cancellative) we conclude that and . Define Clearly, . Then, as in the proof of Theorem 3, for and , there exists a linear order extensions of , , such that . Since is -consistent and we have that , a contradiction to . This contradiction confirms that and thus . To finish the proof of necessity, it remains to show that is a realizer. But, this is an immediate consequence of Theorem 3 for and .
To prove sufficiency, let be an ordered group and let have as realizer the set of all linear order extensions of . Then, . Since for each , is a linear order, we conclude that all the members of are normal and -consistent binary relations. Since the intersection preserve the properties of normality and -consistency we conclude that is a normal and -consistent binary relation.

The following corollary is an immediate consequence of the necessity part of Theorem 8 for being a partial order and .

Corollary 9 (Dushnik-Miller’s extension theorem [18, theorem 2.32]). If is any partial order on af set , then there exists a collection of linear orders on which realize .

The following corollary generalizes a result due to Fuchs [2, Theorem 2]. In fact, it is the Dushnik-Miller’s type extension theorem for partially ordered groups.

Corollary 10 ([2, theorem 2]). A partial order defined on a group has as realizer a certain set of linear orders if and only if is normal.

Proof. The sufficiency part is an immediate consequence of the fact that the intersection of linear orders is normal. The necessity part is obvious by Theorem 3, since a partial order is a -consistent binary relation.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.