FULLY PREORDERABLE GROUPS Efe A. Ok and Gil Riella Abstract. We say that a group is fully preorderable if every (left- and right-) translation invariant preorder on it can be extended to a translation invariant total preorder. Such groups arise naturally in applications, and relate closely to orderable and fully orderable groups (which were studied extensively since the seminal works of Philip Hall and A. I. Mal’cev in the 1950s). Our …rst main result provides a purely group-theoretic characterization of fully preorderable groups by means of a condition that goes back to Ohnishi (1950). In particular, this result implies that every fully orderable group is fully preorderable, but not conversely. Our second main result shows that every locally nilpotent group is fully preorderable, but a solvable group need not be fully preorderable. Several applications of these results concerning the inheritance of full preorderability, connections between full preorderability and full orderability, vector preordered groups, and total extensions of translation invariant binary relations on a group, are provided.

1. Introduction A classic result of set theory, Szpilrajn’s theorem, says that every partial order on a set X can be extended to a linear order on that set. In many applications, however, this theorem alone is not enough, because X has some algebraic structure, and the problem is to extend a given partial order on X that satis…es a certain algebraic property to a linear order which also possesses that property. In particular, a classic problem of algebraic order theory is to determine those groups on which every translation invariant partial order can be extended to a translation invariant linear order. Such groups are referred to as fully orderable (or as O -groups), and have been investigated extensively in the literature. (See Minassian (1973) and Kokorin and Kopytov (1974) for surveys of this literature.) In many applications, it is also the case that one needs to carry out the extension analysis in terms of a preorder (that is, a re‡exive and transitive binary relation) on X instead of a partial order.1 This is readily handled in the case of Szpilrajn’s 2000 Mathematics Subject Classi…cation. Primary 06F15; Secondary 20F60, 20F18. Key words and phrases. Orderable groups, bi-invariant orders, nilpotent groups. The comments of Andrew Glass, Andrés Navas Flores and Cristóbal Rivas have improved the content of this work; we gratefully acknowledge our intellectual debt to them. The second author thanks the CNPq of Brazil for …nancial support. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1 In economics, for instance, X usually stands for some alternative space (such as the space of commodity bundles (real vectors of a given length), or of through-time consumption streams (in…nite real sequences), or of lotteries (probability measures on a given set). The (potentially 1

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EFE A. OK AND GIL RIELLA

theorem. Indeed, by passing on to the quotient set induced by the symmetric part of the given preorder, one easily proves the following generalization of this theorem: Every preorder on X can be extended to a total preorder on X. As simple as it is, this set-theoretical fact has proved of importance in various applied mathematical …elds, ranging from optimization theory to the theory of individual and social choice theory. Given these observations, it is natural to inquire into the nature of groups on which every translation invariant preorder can be extended to a translation invariant total preorder. We refer to such groups as fully preorderable. Not only is that such groups are relevant for applications, they are also of mathematical interest. For, the method of “passing on to the quotient”is of little use in deducing the structure of such groups from that of fully orderable groups. Indeed, a factor group of a fully orderable group need not be torsion-free, and hence, it may well be not orderable (while, as we shall see, such groups are always fully preorderable). Consequently, this method does not even help checking whether or not every fully preorderable group is fully orderable. It appears that one needs to develop the theory of fully preorderable groups separately, and this is the primary objective of the present paper. After introducing some nomenclature in Section 2, we recall two main results from the theory of partially ordered groups. The …rst of these, which goes back to Ohnishi (1950), says that a group is fully orderable i¤ it is generalized torsion-free and satis…es an algebraic condition, which we call the Ohnishi property. Our …rst main …nding shows that the e¤ect of switching focus to full preorderability in this result brings in only the elimination of the torsion-freeness condition. That is: A group is fully preorderable i¤ it satis…es the Ohnishi property (Theorem 3.2). In particular, this result implies that every fully orderable group is fully preorderable, but not conversely. Moreover, it shows that every abelian group is fully preorderable and that every factor group of a fully preorderable group is itself fully preorderable. (These properties are not valid for fully orderable groups.) We also use Theorem 3.2 to prove that the class of fully preorderable groups is locally closed, and that full preorderability is not a hereditary property but it is preserved under taking …nite (but not arbitrary) direct products. (These properties are valid for fully orderable groups.) Another major result of the theory of partially ordered groups we consider here, which goes back to Mal’cev (1951), says that every torsion-free locally nilpotent group is fully orderable. In turn, the second main result of the present paper shows that this fact modi…es for fully preorderable groups again simply by omitting the torsion-freeness condition. That is: Every locally nilpotent group is fully preorderable. (Theorem 5.2). This result furnishes a rich class of concrete examples of fully preorderable groups. To prove it, we …rst factor a given locally nilpotent group X by its torsion subgroup, and use Mal’cev’s theorem to show that the resulting factor group is fully orderable. Next, we use this fact and Ohnishi’s theorem to show that X satis…es the Ohnishi condition, and hence it is, by Theorem 3.2, fully preorderable. incomplete) preferences of an individual, or a group of individuals, on X are then represented as a preorder (but not as a partial order, because it is natural to leave room for indi¤erence across some alternatives). As the eventual decisions must be able to rank all alternatives, one often looks for a “completion” of such a preorder.

FULLY PREORDERABLE GROUPS

3

In Sections 7 and 8, we consider some applications of our main results to algebraic order theory by studying when a given translation invariant preorder on a group X can be extended to such a total preorder, and when it can be represented as the intersection of a collection of total group preorders on X: Finally, in Section 9, we sketch an application to (economic) decision theory by studying when a given (possibly non-transitive) translation invariant binary relation (viewed as an individual, or social, preference relation) on an abelian group can be extended to a total group preorder (viewed as a “rational”extension of that preference relation). 2. Preliminaries Preordered Sets. Let X be a nonempty set and R a binary relation on X. In what follows, we adopt the usual convention of writing x R y instead of (x; y) 2 R. Similarly, if R0 is another binary relation on X; we write x R y R0 z to mean x R y and y R0 z; and so on. The asymmetric part of R is the binary relation R> on X de…ned by x R> y i¤ x R y but not y R x: The symmetric part of R is then de…ned as RnR> : A binary relation S is said to extend R; in which case we refer to S as an extension of R; if the symmetric and asymmetric parts of R are contained in the symmetric and asymmetric parts of S, respectively. That is, S extends R i¤ R S and R> S > . By a preorder % on X, we mean a re‡exive and transitive binary relation on X. Two elements x and y of X are said to be %-comparable if either x % y or y % x; and non-%-comparable otherwise. On the other hand, if (x; y) belongs to the symmetric part of %; that is, x % y % x; we say that x and y are %-equivalent. If every x and y in X are %-comparable, % is said to be a total preorder, and if no two distinct elements x and y of X are %-equivalent, we say that % is a partial order. In turn, a total partial order on X is said to be a linear order on X: The ordered pair (X; %) is called a preordered set if % is a preorder on X, a poset if % is a partial order on X; and a loset if % is a linear order on X: (Throughout the paper, a generic preorder is denoted as %; and a generic partial order as <. We denote the asymmetric part of such a relation as .) A subset S of X is said to be %-convex if, for any x; y 2 S and z 2 X with x % z % y, we have z 2 S: A preorder % on X is said to be a totalization of a preorder % on X if % is total and it extends %. Note that a partial order on X extends another partial order on X i¤ the former contains the latter. In this case, if the former partial order is a linear order, we refer to it as a linearization of the latter. Groups. Let X be a group, which we notate multiplicatively. (1 corresponds to the identity of X: We adopt the usual notation of denoting conjugates, that is, xa := axa 1 for any x and a in X. Also, for any x 2 X and A; B X, we set AB := fab : (a; b) 2 A Bg, Ax := Afxg, xB := fxgB, and A 1 := fa 1 : a 2 Ag.) Recall that a subset C of X is said to be a subsemigroup of X if CC C; a submonoid of X if 1 2 C and CC C; and a normal subset of X if xCx 1 C for every x 2 X: The following result is elementary, and its proof is omitted. Lemma 2.1.2 Let A and B be two normal submonoids of a group X: Then, AB 1 is a normal submonoid of X: If A \ A 1 = B \ B 1 = A \ B = f1g; then AB 1 \ BA 1 = f1g as well. 2 This lemma appears as Lemma 1 on p. 8 of Kokorin and Kopytov (1974).

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EFE A. OK AND GIL RIELLA

In what follows, for any x in X; we denote by [[x]] the normal subsemigroup of X that is generated by x: It is easily veri…ed that y 2 [[x]] i¤ there is a positive integer k and elements a1 ; :::; ak of X such that y = xa1 xak . (In particular, 1 1 2 [[1]]= f1g; [[x]] = [[x ]]; and when X is abelian, [[x]] = fx; x ; :::g:) In turn, X is said to be generalized torsion-free if [[x]] does not contain 1 for any x 2 Xnf1g: (An abelian group is generalized torsion-free i¤ it is torsion free, that is, xk 6= 1 for any x 2 Xnf1g and k 2 N. Finally, we de…ne [[1;x]] := f1g [ [[x]]; which is nothing but the normal submonoid generated by x: We say that an element x of X satis…es the Ohnishi property if [[y]] \ [[z]] 6= ;

for every y; z 2 [[x]]:

(See Ohnishi (1950).) We note that every element of the center of X is sure to satisfy the Ohnishi property. Indeed, if x 2 X commutes with every element of X; then, for any z; y 2 [[x]]; we have y = xm and z = xn for some m; n 2 N; and hence xmn 2 [[y]] \ [[z]]: On the other hand, a non-abelian group may or may not contain elements that fail to satisfy the Ohnishi property. In what follows, as is standard, we say that a property holds for a group X locally if it holds for every …nitely generated subgroup of X: In particular, by a locally nilpotent group, we mean a group every …nitely generated subgroup of which is nilpotent. Preordered Groups. Let X be a group and % a preorder on X: If % is both leftand right-translation invariant, that is, x % y implies !x % !y and x! % y! for every x; y; ! 2 X; we say that % is a group preorder, and refer to the ordered pair (X; %) as a preordered group. (Note that if % is a group preorder on X; then the asymmetric part of % is also translation invariant.) If % is a group preorder which is antisymmetric, then it is called a partial group order (or a bi-invariant partial order ), and (X; %) is said to be a po-group. If it is a linear order, then % and (X; %) are referred to as a group order and an ordered group, respectively. Let (X; %) be a preordered group. The set fx 2 X : x % 1g is called the positive cone of (X; %): In what follows, we denote this set as X+ when % is clear from the context, and as X+ (%) when there is reason to be explicit about the underlying preorder. It is plain that X+ is a normal submonoid of X that characterizes the preorder % as follows: x % y i¤ xy 1 2 X+ , for every x and y in X: (It is readily veri…ed that (i) [[1;x]] X+ for every x 2 X+ ; and (ii) x 1 implies that y % 1 does not hold for any y 2 [[x 1 ]].) The following elementary observation will be useful in the sequel. Lemma 2.2. Let (X; %) be a preordered group and x 2 X. If % is total and 1 2 [[x]], then any two elements of [[x]] are %-equivalent. Proof. Suppose 1 2 [[x]] so that 1 = xa1 xak for some …nitely many elements a1 ; :::; ak of X. Then, by translation invariance, x 1 implies 1 1; a contradiction. Similarly, 1 x is impossible. If % is total, therefore, x and 1 must be %-equivalent, and this entails that any two elements of [[x]] are %-equivalent. For any submonoid C of a group X, we de…ne the binary relation %C on X as x %C y

i¤

xy

1

2 C;

FULLY PREORDERABLE GROUPS

5

which is readily checked to be a preorder on X: We refer to %C as the preorder induced by C: When C is normal, this preorder is translation invariant, that is, (X; %C ) is a preordered group. We note that (X; %C ) is a po-group i¤ C \ C 1 = f1g: Let (X; %) be a preordered group and Z a normal subgroup of X: We de…ne the binary relation %Z on the factor group X=Z by xZ %Z yZ

i¤

x % yz for some z 2 Z:

It is readily checked that %Z is well-de…ned. Furthermore, (X=Z; %Z ) is a preordered group, which is a po-group i¤ Z is %-convex.3 We note that ordering X=Z in this manner is quite reasonable, for the natural homomorphism ' from X onto X=Z carries the positive cone of (X; %) onto that of (X=Z; %Z ); that is, '(X+ (%)) = (X=Z)+ (%Z ). A group X is said to be fully preorderable if every group preorder on X can be extended to a total group preorder on X: It is said to be fully orderable if every partial group order on X can be extended to a group order on X; and it is orderable if there is at least one group order on X: In the literature on partially ordered groups, fully orderable groups are often referred to as O -groups and orderable groups as O-groups. Following this tradition, one may choose to refer to fully preorderable groups as O -groups. In this jargon, every O -group is an O-group. We will show below that every O -group is also an O -group, but an O -group need not even be an O-group. The following classic result of the theory of partially ordered groups provides a group-theoretic characterization of fully orderable groups. Ohnishi’s Theorem. A group is fully orderable if and only if it is generalized torsion-free and every element of it satis…es the Ohnishi property. Another famous result of this theory identi…es a fairly large collection of fully orderable groups: Mal’cev’s Theorem. Every torsion-free locally nilpotent group is fully orderable. The two main results of this paper show how these theorems modify for fully preorderable groups. 3. Ohnishi Property and Fully Preorderable Groups The importance of the Ohnishi property for full preorderability of a group stems largely from the following observation: Lemma 3.1. Let (X; %) be a preordered group, and D a maximal group preorder on X that extends %. If x is an element of X that satis…es the Ohnishi property, then it is D-comparable to 1: Proof. To derive a contradiction, suppose there is an x in X that satis…es the Ohnishi property, but it is not D-comparable to 1: (In what follows, we denote the asymmetric part of D by B.) Claim 1. There is a y 2 [[x]] such that y B 1: 3 The second part of this assertion is well-known. The …rst part appears as Lemma 3.3.1 in Ok and Riella (2013).

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Proof of Claim 1. Suppose our claim is not true, that is, y B 1 is false for every y 2 [[x]]: Let D stand for the group preorder on X induced by the normal submonoid X+ (D)[[1;x 1 ]] of X: Clearly, X+ (D) X+ (D ), which means that D D , while x 1 2 X+ (D )nX+ (D). To see that this inclusion is also valid in terms of the asymmetric parts of these preorders, take any a and b in X with a B b (so that ab 1 B 1): Then, a D b; so if a B b is false (where B is the asymmetric part of D ), we must have b D a: It then follows from the de…nition of D that ba 1 D c for some c 2 [[x 1 ]]: But then c 1 2 [[x]] while c 1 D ab 1 B 1; that is, c 1 B 1; which contradicts our hypothesis. We conclude: % D D and B B : This contradicts the maximality of D. Claim 2. There is a z 2 [[x]] such that 1 B z: Proof of Claim 2. Precisely the same argument applies, but we now use [[1; x]] in de…ning D0 instead of [[1; x 1 ]]. Putting Claims 1 and 2 together, we …nd y B 1 B z for some y; z 2 [[x]]: As x satis…es the Ohnishi property, therefore, there is a w in [[y]] \ [[z]]: But, as D is translation invariant, y B 1 implies [[y]] B 1, while 1 B z implies 1 B [[z]]: It follows that w B 1 B w; a contradiction. We now prove that a group is fully preorderable i¤ every element of it satis…es the Ohnishi property. In view of Ohnishi’s theorem, therefore, full preorderability is a less stringent requirement than full orderability. Theorem 3.2. A group X is fully preorderable if, and only if, every element of X satis…es the Ohnishi property. Proof. Let % be a group preorder on X; and let P stand for the set of all group preorders on X that extend %. Then, (P; ) is an inductive poset, so by Zorn’s Lemma, there is a maximal element D of P (with respect to inclusion). If every element of X satis…es the Ohnishi property, Lemma 3.1 ensures that x and 1 are D-comparable for every x 2 X: Then, D is a total group preorder that extends %; so we may conclude that X is fully preorderable. Conversely, suppose that X is fully preorderable, and take any x in X: Suppose …rst that 1 2 [[x]]: We claim that 1 2 [[y]] for all y 2 [[x]]; and hence x satis…es the Ohnishi property. To derive a contradiction, suppose 1 does not belong to [[y]] for some y 2 [[x]]. Let % be the group preorder on X induced by the normal submonoid [[1; y]]. Since 1 2 = [[y]]; it cannot be the case that y 1 2 [[1; y]]; and it follows that y 1: But then there cannot be a total group preorder that extends %, because, by Lemma 2.2, 1 and y must be equivalent with respect to any such total preorder. This contradicts the full preorderability of X: Conclusion: Every x 2 X with 1 2 [[x]] satis…es the Ohnishi property. To complete the proof, suppose that there is an x in X that fails to satisfy the Ohnishi property so that there exist y and z in [[x]] with [[y]] \ [[z]] = ;: Put C := [[1; y]][[1; z 1 ]], which is easily checked to be a normal submonoid of X: Let < denote the group preorder induced by C; and note that y < 1 < z: Notice that we must have [[1; y]] \ [[1; y 1 ]] = f1g; for, otherwise, [[y]] \ [[y 1 ]] 6= ;; which entails 1 2 [[y]] [[x]]; contradicting what we have found in the previous paragraph. Similarly, we have [[1; z]] \ [[1; z 1 ]] = f1g: Furthermore, [[y]] \ [[z]] = ; implies that [[1; y]] \ [[1; z]] = f1g; so it follows from Lemma 2.1 that C \ C 1 = f1g: Thus: < is a partial order on X. In particular, as neither y nor z equals 1; we have y 1 z: As X is fully preorderable, there is a total group preorder D on X such that y B 1 B z:

FULLY PREORDERABLE GROUPS

7

(Here B stands for the asymmetric part of D.) But, as both z and y belong to [[x]]; we must have fy; zg D 1 if x D 1; and 1 D fy; zg if 1 D x: It follows that x and 1 are non-D-comparable, contradicting the totalness of D. Conclusion: Every x 2 X satis…es the Ohnishi property. Remark. Just like Ohnishi’s theorem, Theorem 3.2 is valid in the context of an arbitrary operator group X. Indeed, the proof goes through verbatim, provided that we now view [[x]] as the smallest admissible subsemigroup of X that contains x: Various su¢ ciency results obtain easily from Theorem 3.2. In particular, as every element of the center of a group has the Ohnishi property, an immediate implication of this result is: Corollary 3.3. The center of any group is fully preorderable. In particular, every abelian group is fully preorderable. Corollary 3.4. Every group preorder on a subgroup of an abelian group X can be extended to a total group preorder on X. Proof. Let Z be a subgroup of X and take any group preorder % on Z: De…ne the binary relation D on X by x D y i¤ xy 1 2 Z and xy 1 % 1. Using the fact that X is abelian, one checks readily that D is a group preorder on X that extends %. Applying Corollary 3.3 yields the claim. Another easy consequence of Theorem 3.2 is that, just like that of fully orderable groups, the class of fully preorderable groups is locally closed. Put precisely: Corollary 3.5. If every …nitely generated subgroup of a group X is fully preorderable, then X is fully preorderable. 4. Inheritance of Full Preorderability Inheritance by Factor Groups. Full preorderability behaves well with respect to passing to factor groups. More generally: Proposition 4.1. Every homomorphic image of a fully preorderable group is fully preorderable. Proof. Let ' be a surjective homomorphism from a group X onto a group Y: It is routine to verify that '([[x]]) = [['(x)]] for any x 2 X: It follows from this that if every element of X has the Ohnishi property so does every element of Y: Applying Theorem 3.2 completes the proof. Corollary 4.2. If X is a fully preorderable group, then so is any factor group of X: Proof. As X=Z is a homomorphic image of X (under the natural homomorphism) for any normal subgroup Z of X; this claim follows readily from Proposition 4.1. This observation draws a stark contrast between the notions of full orderability and full preorderability. Indeed, it is well-known that a factor group of a fully orderable group need not be fully orderable.

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Non-inheritance by Subgroups. A famous counterexample due to Kopytov (1966) shows that a subgroup of a fully orderable group need not be fully orderable. Unfortunately, the situation is not di¤erent for full preorderability. Proposition 4.3. A subgroup of a fully preorderable group need not be fully preorderable. Proof. To derive a contradiction, suppose that every subgroup of every fully preorderable group is itself fully preorderable. Now let X be a fully orderable group, and Z a subgroup of X which is not fully orderable. By Ohnishi’s Theorem, X is generalized torsion-free and every element of X satis…es the Ohnishi property. Then, by Theorem 3.2, X is fully preorderable, so by our hypothesis, Z is fully preorderable. Again by Theorem 3.2, then, every element of Z satis…es the Ohnishi property. But it is readily checked that the property of generalized torsion-freeness is inherited by subgroups, so Z is generalized torsion-free. By Ohnishi’s Theorem, therefore, Z is fully orderable, a contradiction. Inheritance by Finite Products. It was independently proved by Kargapolov (1963) and Kokorin (1963) that the direct product of …nitely many fully orderable groups is fully orderable. We now show that the same is true for fully preorderable groups by using the following two preliminary observations. (The method of proof is due to Kokorin (1963).) Lemma 4.4. Let X be a group, and Y and Z two normal subgroups of X with Y \ Z = f1g: Then, any y 2 Y commutes with any z 2 Z: Proof. For any (y; z) 2 Y Z; we have y(zy ZZ Z; so yzy 1 z 1 = 1:

1

z

1

) 2 YY

Y and (yzy

1

)z

1

2

Lemma 4.5. Let X be a group, and Y and Z two fully preorderable normal subgroups of X with Y \ Z = f1g and X = Y Z. If % is a group preorder on X and D is a maximal group preorder on X that extends %, then D \ (Y Y ) is a total group preorder on Y and D \ (Z Z) is a total group preorder on Z. Proof. Let x be any element of Y . As Y is fully preorderable, Theorem 3.2 says that x satis…es the Ohnishi property in Y . Now notice that, because of Lemma 4.4, for any (a; b) 2 (Y; Z), abxb 1 a 1 = axa 1 . In turn, this implies that the normal subsemigroups generated by x are the same in Y and in X, so x satis…es the Ohnishi property in X as well. By Lemma 3.1, then, x and 1 are D-comparable. The argument that shows that D \ (Z Z) is a total group preorder on Z is analogous. Proposition 4.6. The direct product of …nitely many fully preorderable groups is fully preorderable. Proof. It is enough to show that the direct product of two fully preorderable groups is fully preorderable. We will use the internal representation of direct groups for this purpose. Let X be a group, and Y and Z two normal subgroups of X such that X = Y Z and Y \ Z = f1g: Assume that Y and Z are fully preorderable. Let % be a group preorder on X; and use Zorn’s Lemma to …nd a maximal group preorder D on X that extends %. To derive a contradiction, suppose there is an x in X that is

FULLY PREORDERABLE GROUPS

9

not D-comparable to 1: Then, as in the proof of Lemma 3.1, we can …nd elements u and v of [[x]] such that u B 1 B v: Obviously, this implies u B1Bv

(4.1)

for every ;

Now, x = yz for some (unique) (y; z) 2 Y b1 ; :::; bl of X such that u = (yz)a1

(yz)ak

and

2 X:

Z; so there are elements a1 ; :::; ak ; v = (yz)b1

(yz)bl :

Without loss of generality, we may take k = l here (otherwise we would work instead with ul and v k .) By using the normality of Y and Z; and Lemma 4.4, we may write u = y a1

y ak z a1

z ak

and

v = y b1

y bk z b1

z bk :

We put y1 := y a1 y ak ; z1 := z a1 z ak ; y2 := y b1 y bk and z2 := z b1 z bk so that u = y1 z1 and v = y2 z2 : Claim. We have y2b D y1a for some a; b 2 Y and z2d D z1c for some c; d 2 Z: Proof of Claim. By Lemma 4.5, D \ (Y Y ) is a total group preorder on Y: Therefore, there is an a in fa1 ; :::; ak g such that y a D y ai for every i = 1; :::; k: Then, (y k )a D y1 ; and it follows that y k D a 1 y1 a : Similarly, there is a b in fb1 ; :::; bk g such that y bi D y b for every i = 1; :::; k: Then, y2 D (y k )b ; and it follows that b 1 y2 b D y k : Finally, there exist ay 2 Y and az 2 Z such that ay az = a 1 . Similarly, there exist by 2 Y and bz 2 Z such that by bz = b 1 . Now de…ne a := ay and b := by and note that by2 b 1 = b 1 y2 b D y k D a 1 y1 a = ay1 a 1 . The second assertion is analogously proved. Now let a; b; c and d be as found in this claim. Then, y2b z2d D y1a z1c : But ay1 a 1 cz1 c 1 = cay1 z1 a 1 c 1 by Lemma 4.4, so y1a z1c = (y1 z1 )ca ; and similarly, y2b z2d = (y2 z2 )db : Consequently, v db D uca ; but this contradicts (4.1). As we shall show in Section 6, this fact does not extend to the case of an arbitrary family of fully preorderable groups. 5. Full Preorderability of Nilpotent Groups Let X be a group, and de…ne T (X) := fx 2 X : xk = 1 for some k 2 Ng;

that is, T (X) is the set of all periodic elements on X: It is plain that this set is fully invariant, that is, f (T (X)) T (X) for every endomorphism f on X: In particular, T (X) is normal. Furthermore, if T (X) is a subgroup of X and (xT (X))k = T (X) for some x 2 X and k 2 N; we have xk y = 1 for some y 2 T (X): Then, y l = 1 for some l 2 N; and hence, xkl = y l = 1; that is, x 2 T (X), and hence, xT (X) = T (X): In other words, if T (X) is a subgroup of X; we have T (X=T (X)) = fT (X)g. Conclusion: X=T (X) is a torsion-free group, provided that T (X) is a subgroup of X: Now, it is well-known that nilpotence of X implies that T (X) is a subgroup of X (cf. Theorem 5.2.7 of Robinson (1996)). It readily follows from this fact that T (X) is a subgroup of X so long as X is locally nilpotent (cf. Theorem 12.1.1 of Robinson (1996)). On the other hand, it is well-known, and is easily veri…ed, that every factor group of a nilpotent group is nilpotent. But then, if X is locally nilpotent and Z is a normal subgroup of X; for any …nitely many x1 ; :::; xn 2 X, the subgroup generated by fx1 Z; :::; xn Zg in X=Z must be nilpotent, because this subgroup equals fxZ : x 2 hx1 ; :::; xn ig; so we …nd that X=Z is locally nilpotent.

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EFE A. OK AND GIL RIELLA

(Here, as usual, hx1 ; :::; xn i stands for the subgroup of X generated by x1 ; :::; xn .) Combining these observations with that of the previous paragraph, we conclude: If X is a locally nilpotent group, then X=T (X) is a locally nilpotent and torsion-free group. Applying Mal’cev’s theorem, therefore, we …nd: Lemma 5.1. If X is a locally nilpotent group, then X=T (X) is fully orderable. We now use this observation to determine exactly how Mal’cev’s theorem modi…es for fully preorderable groups. Theorem 5.2. Every locally nilpotent group is fully preorderable. Proof. Let X be a locally nilpotent group and x an arbitrary element of X: In view of Theorem 3.2, we will be done if we can show that x satis…es the Ohnishi property. Pick any y and z in [[x]]. Then, yT (X) 2 [[x]]T (X) = [[xT (X)]] (in X=T (X)), and similarly zT (X) 2 [[xT (X)]] (in X=T (X)). By Lemma 5.1 and Ohnishi’s Theorem, every element of X=T (X) – in particular, xT (X) – satis…es the Ohnishi property, so we have [[y]]T (X) \ [[z]]T (X) = [[yT (X)]] \ [[zT (X)]] 6= ;: It follows that there is a w in T (X) such that y 0 w = z 0 for some (y 0 ; z 0 ) 2 [[y]] [[z]]. But, by de…nition of T (X); we have wk = 1 for some k 2 N: Clearly, (y 0 w)k = (z 0 )k 2 [[z]]. On the other hand, (y 0 w)k

= y 0 wk w1 0

= y (w 2 [[y]];

k 0

y wk

1 k 0

yw

k 1

1

w2

)(w

k 0

y

2 k 0

yw

y 0 w2 w

k 2

)

1 0

(w

yw 1 0

y w)

where the second equality follows from wk = 1. It follows that [[y]] \ [[z]] 6= ;, and we are done. 6. Full Orderability vs. Full Preorderability Combining Theorem 3.2 with Ohnishi’s theorem yields: Corollary 6.1. Every fully orderable group is fully preorderable, but not conversely. In the case of abelian groups, Everett (1950) has shown that full orderability (as well as orderability) is equivalent to being torsion-free. (This fact is actually an immediate consequence of Ohnishi’s theorem.) Consequently, in view of Corollary 3.3, every abelian group which is not torsion-free is an example of a fully preorderable, but not fully orderable group. For instance, any nontrivial cyclic group, as well as the Klein 4-group, is fully preorderable, but it is not fully orderable. In these examples, full preorderability is trivial, because no group preorder on any one such group can have a nonempty asymmetric part. The following is a slightly less trivial example. Example 6.1. Let F denote the multiplicative group of nonzero elements of an F where F is viewed as the ordered …eld F and consider the direct product F additive group of the elements of F. As it is abelian, this group is fully preorderable. For instance, consider the partial order on F F de…ned as (x1 ; x2 ) < (y1 ; y2 ) i¤ x1 = y1 and x2 y2 ; which is a partial group order on F F with a nonempty asymmetric part. There are many total preorders on F F that extend <. (For

FULLY PREORDERABLE GROUPS

11

instance, the relation D with (x1 ; x2 ) D (y1 ; y2 ) i¤ x2 y2 is one such preorder.) It is impossible to …nd a linear order that extends <, however. Indeed, F F is not torsion-free (because ( 1; 0) is of order 2), and hence, it is not orderable. In the context of generalized torsion-free groups, however, full preorderability and full orderability is one and the same. Theorem 6.2. Let X be a generalized torsion-free group. Then, the following are equivalent: (a) X is fully preorderable; (b) Every partial order on X can be extended to a total preorder on X; (c) Every x in X satis…es the Ohnishi property; (d) X is fully orderable. Proof. While (a) implies (b) trivially, Ohnishi’s theorem ensures that (c) implies (d). In view of Corollary 6.1, therefore, it remains to show that (b) implies (c). Suppose (b) holds, but there is an x in X that fails to satisfy the Ohnishi property. Then, there exist y and z in [[x]] such that [[y]] \ [[z]] = ;: Put C := [[1; y]] \ [[1; z 1 ]], and let < denote the group preorder induced by C: As X is generalized torsion-free, both [[1; y]] \ [[1; y 1 ]] and [[1; z]] \ [[1; z 1 ]] equal f1g; and the proof is concluded exactly as in the last paragraph of the proof of Theorem 3.2. One may utilize Theorem 6.2 to construct examples of groups that are not fully preorderable. Remark. (On the non-triviality of full preorderability property) It is plain that every orderable group is generalized torsion-free. Therefore, by Theorem 6.2, any orderable group which is not fully orderable cannot be fully preorderable. It is well known that such groups exist. Some examples of them were provided …rst by Holland (1961); see Section 6.5.3 of Kopytov and Medvedev (1999). For a concrete example, we note that any free group generated by at least two generators is generalized torsion-free, but each generator of such a group fails the Ohnishi property. Combining Theorem 6.2 with Ohnishi’s theorem, therefore, we conclude that no free group generated by two or more generators can be fully preorderable. In fact, there are examples of solvable orderable groups which are not fully orderable; see Kargapolov (1963) and Kargapolov, Kokorin and Kopytov (1965). Thus: A solvable group need not be fully preorderable. Remark. (On direct products of fully preorderable groups) As the direct product of any collection of generalized torsion free groups is generalized torsion-free, it follows from Ohnishi’s theorem that the direct product of an arbitrary collection of fully orderable groups is generalized torsion-free. By Theorem 6.2, therefore, the direct product of any collection of fully orderable groups is fully orderable i¤ it is fully preorderable. But a (rather involved) counterexample due to Kargapolov (1963) shows that the direct product of an in…nite collection of fully orderable groups need not be fully orderable. It follows that the …niteness requirement cannot be dropped from the statement of Proposition 4.6. There are other situations in which full preorderability reduces to full orderability. In particular, this happens in the context of certain simple groups.

12

EFE A. OK AND GIL RIELLA

Proposition 6.3. Let X be a simple group such that [[1; x]] 6= X for some x 2 Xnf1g: Then, X is fully preorderable if and only if it is fully orderable. Proof. Assume that X is fully preorderable. Let < be a partial group order on X, and pick any x 6= 1 in X with [[1; x]] 6= X: Besides, since [[1; x]] \ [[1; x 1 ]] is a normal subgroup of X; it equals either f1g or X: In the latter case, we see that x 1 2 [[1; x]] and x 2 [[1; x 1 ]]; so [[1; x]] = [[1; x 1 ]]; and hence, [[1; x]] = X, a contradiction. Thus, [[1; x]] \ [[1; x 1 ]] = f1g: Now, we de…ne < as < if = 6 ;; and as the preorder < on X induced by [[1; x]] otherwise. Clearly, < is a partial group order on X with 6= ;: But, by hypothesis, there is a total group preorder D on X that extends < ; and hence, <. Let Z stand for the set of all x 2 X which are D-equivalent to 1: It is plain that Z is a normal subgroup of X; so either Z = f1g or Z = X: But the latter case is impossible because 6= ; and D extends < : It follows that Z = f1g; that is, no element of Xnf1g is D-equivalent to 1: As this implies that D is a linear order on X, we are done. Put another way, for a simple group on which there exists at least one nontrivial partial group order, the notions of full orderability and full preorderability coincide. 7. On Totalizations of Group Preorders Corollary 3.3 ensures that every group preorder on an abelian group can be extended to a total group preorder on X: The following result makes use of this fact to obtain a generalization. Let (X; %) be a preordered group. We say that (X; %) is pseudo-abelian if all commutators of X are %-equivalent to the identity of X: In the context of such groups, x 2 [[w]] entails that x is %-equivalent to wk for some positive integer k: Theorem 7.1. Let (X; %) be a pseudo-abelian preordered group. Then, % can be extended to a total group preorder on X. Proof. Denote the symmetric part of % by , and put Z := fx 2 X : x 1g: It is readily checked that Z is a %-convex normal subgroup of X. Therefore, (X=Z; %Z ) is a po-group. Furthermore, X=Z is abelian, because, for any x and y in X; we have yx 2 xyZ (because (X; %) is pseudo-abelian), and this implies xZyZ = xyZ = yxZ = yZxZ: Consequently, by Corollary 3.3, there is a total group preorder D on X=Z that extends %Z . We next consider the binary relation % on X; de…ned by x % y i¤ xZ D yZ; which is readily veri…ed to be a total group preorder on X: Now, for any x and y in X; if x y; then xZ 6= yZ; and hence, x Z y, where Z is the asymmetric part of %Z . As D extends %Z ; therefore, x y implies that (xZ; yZ) is contained in the asymmetric part of D; and hence, x y: If, on the other hand, x y; then xZ = yZ; so (xZ; yZ) is trivially contained in the symmetric part of D, and hence, x y: It follows that % extends %, and we are done. 8. On Vector Preordered Groups We say that a binary relation R on a group X is unperforated if xk R 1 implies x R 1 for any (x; k) 2 X N: A well-known result, due to Fuchs (1950) shows that, for abelian groups, this property characterizes those partial group orders that are realized as the intersection of a collection of group orders. It was later shown by Mal’cev (1962) that the abelian requirement can be replaced with full orderability

FULLY PREORDERABLE GROUPS

13

in this statement. While Corollary 6.1 shows that totalizing a group preorder is in general a di¤erent process than linearizing a partial group order, we will show below that Fuchs’characterization extends to the case of group preorders without alteration, even in the case of pseudo-abelian preordered groups. However, we do not have a counterpart of Mal’cev’s said result, that is, we do not know if this characterization is valid for fully preorderable groups. As for terminology, we say that a preordered group (X; %) is a vector preordered group if this group can be embedded (both group- and order-theoretically) in the direct product of a collection of totally preordered groups (where the preorder of the latter group is de…ned coordinatewise). More precisely, (X; %) is a vector preordered group i¤ there is a nonempty family Q f(Xi ; %i ) : i 2 Ig of totally preordered groups and a group embedding ' : X ! fXi : i 2 Ig such that x % y i¤ '(x)(i) %i '(y)(i) for each i 2 I: Proposition 8.1. Let (X; %) be a pseudo-abelian preordered group. Then, the following are equivalent: a. % is unperforated ; b. % is the intersection of a nonempty collection of total group preorders on X; c. (X; %) is a vector preordered group.4 Proof. (a) ) (b). Suppose that % is unperforated, and let P stand for the collection T of all total group preorders on X that extend %. We wish to show that % = P. The part of this claim is obvious. To prove the converse containment, take any x and y in X such that x % y is false. We are to …nd an element of P that fails to contain (x; y): If y x; this is obviously true, so we may assume that x and y are not %-comparable. Put w := yx 1 ; and note that X+ (%)[[1; w]] is a normal submonoid of X: Let % be the group preorder induced by this submonoid. Clearly, % % , because a % b implies ab 1 2 X+ (%) while, obviously, ab 1 2 (ab 1 )[[1; w]]: Now suppose a b. Then, as we have just seen, a % b; that is, ab 1 % 1: To derive a contradiction, suppose we also have 1 % ab 1 . Then, given that (X; %) is pseudo-abelian, we have (ab 1 ) 1 % zwk for some z 2 X+ (%) and nonnegative integer k: But, since ab 1 1; 1 (ab 1 ) 1 % zwk % 1wk = wk ; which shows that k 1: But 1 wk implies (w 1 )k 1; so, as % is unperforated, we …nd w 1 % 1; that is, x % y; contradicting the non-%-comparability of x and y: Therefore, a b must hold, so we may conclude that . Finally, we x; that is, w 1: Obviously, w % 1; so if this were not true, we claim that y would have 1 % w; that is, w 1 % 1; which implies that w 1 % zwk for some z 2 X+ (%) and nonnegative integer k: But then, we would have (w 1 )k+1 % z % 1; so as (X; %) is unperforated, w 1 % 1; that is, x % y; a contradiction. Conclusion: % is a group preorder on X that extends % and that satis…es y x: We now apply Theorem 7.1 to …nd a total group preorder D on X that extends % . Clearly, D 2 P and y B x; and our proof is complete. (b) )T(c). Suppose there is a collection P of total group preorders on X such that % = P. De…ne ' : X ! X P by '(x)(D) := x for every D 2 P: Then, ' is a

4 The statements (b) and (c) remain equivalent even if we dropped the pseudo-abelian requirement from the statement of the proposition.

14

EFE A. OK AND GIL RIELLA

group embedding from X into X P such that x % y i¤ '(x)(D) D '(y)(D) for each D 2 P: (c) ) (a). Let I be a nonempty index set, Q suppose (Xi ; %i ) is a totally preordered group for each i 2 I; and let ' : X ! fXi : i 2 Ig be a group embedding such that x % y i¤ '(x)(i) %i '(y)(i) for each i 2 I: For each i in I; de…ne the binary relation Di on X by x Di y i¤ '(x)(i) T %i '(y)(i). Clearly, each Di is a total group preorder on X; and we have % = fDi : i 2 Ig: But every total group preorder on X; and in particular, each Di , is unperforated, and it follows from this that % is unperforated as well. 9. Application: Consistent Extension of Binary Relations on Groups The classical Szpilrajn’s theorem says that every partial order on a set can be extended to a linear order on that set. By passing to the quotient sets, one easily deduces from this fact that every preorder on a set can be extended to a total preorder on that set.5 This fact is used quite frequently in social choice theory and the theory of individual decision-making, for it provides one with a criteria of “rationality”for incomplete preference relations (which, in mathematics, are referred to as non-total preorders). In the context of social choice, incomplete preferences arise as social preferences that aggregate individual (possibly complete) preference relations, and in individual choice theory, they correspond to preferences that exhibit some indecisiveness on the part of the decision-maker (cf. Aumann (1962), Seidenfeld, Schervish and Kadane (1995), Evren and Ok (2011), and Ok, Ortoleva and Riella (2012)). A natural problem in this setting is to determine to what extent this “rationality” criterion can be extended to the case of binary relations that need not be preorders. Put di¤erently, we would like to identify those binary relations on a given set can be extended to a total preorder on that set. A complete answer to this question was provided by Suzumura (1976), whose work has since then been extended in various directions. To state Suzumura’s theorem properly, we recall that the composition of two binary relations R and S on a nonempty set X is de…ned as R S := f(x; y) 2 X X : x R z S y for some z 2 Xg: In turn, we let R1 := R and Rk := R Rk 1 for any integer k > 1; here Rk is said to be the kth iterate of R: Then, the transitive closure of a binary relation R on nonempty set X, denoted by tran(R); is the smallest transitive relation on X that contains R; and is given by tran(R) := R [ R2 [ . Following Suzumura (1976), we say that R is consistent if x tran(R) y

implies

not y R> x

for every x; y 2 X: (Recall that R> stands for the asymmetric part of R.) Equivalently, R is consistent i¤ tran(R) is an extension of R: Clearly, every preorder is consistent. The converse is, however, false: Suzumura’s Theorem. A binary relation R on a nonempty set X can be extended to a total preorder on that set if and only if R is consistent. In many economic contexts, one needs to apply this result for X that has a particular algebraic (or otherwise) structure. (For instance, in models of commodity spaces, X is often set as a Euclidean space, in those of intertemporal choice, 5 To the best of our knowledge, this fact was stated for the …rst time in Arrow (1951, p. 64).

FULLY PREORDERABLE GROUPS

15

a classical sequence space, and in …nancial contexts, a function space.) In such contexts, Suzumura’s theorem remains rather abstract, for it does not relate the binary relation at hand to the structure of X: In particular, we may ask in what way this theorem would modify if we are interested in extending a given translation invariant binary relation on a group. The following result, which is a straightforward consequence of Theorem 3.2, answers this question in the case of abelian groups. Proposition 9.1. Let X be an abelian group (or more generally, a fully preorderable group) and R a translation invariant binary relation on X: Then, there is a total group preorder on X that extends R if and only if R is consistent. Proof. Necessity is immediate from Suzumura’s theorem.6 To see the su¢ ciency, suppose R is consistent. Then, tran(R) is an extension of R: Besides, translation invariance of R implies that of tran(R); so tran(R) is a group preorder on X: Applying Corollary 3.3 completes the proof. In the context of divisible abelian groups (such as rational, or real, vector spaces), we can sharpen Proposition 9.1 as follows: Proposition 9.2. Let X be a divisible abelian group and R an unperforated and translation invariant binary relation on X: Then, tran(R) is the intersection of a collection of total group preorders on X: Moreover, this collection can be chosen to contain an extension of R if and only if R is consistent. Proof. We claim that tran(R) is unperforated. To prove this, take any x 2 X and k 2 Z+ such that xk tran(R) 1: Then, there exist …nitely many elements a0 ; :::; an of X such that xk = a0 R a1 R R an R 1: If n = 0; we readily …nd x R 1 because R is unperforated, so we may suppose n > 0: As X is divisible, for each i in f1; :::; ng there is some yi 2 X such that yik = ai ; and hence xk R y1k R R ynk R 1: As X is abelian and R is translation invariant, therefore, (xy1 1 )k R 1 and 1 k (yi yi+1 ) R 1; i = 1; :::; n 1; and hence, as R is unperforated, we …nd x R y1 R R yn R 1; that is, x tran(R) 1; and our claim is proved. We may now apply Proposition T 8.1 to …nd a collection P of total group preorders on X such that tran(R) = P. As tran(R) is translation invariant, Corollary 3.3 implies that there T is a total group preorder % on X that extends tran(R): Obviously, tran(R) = (P[f%g): When R is consistent, % is also an extension of R; and we are done. As a less trivial application of Theorem 3.2, we now prove a theorem about extending two group preorders simultaneously. Theorem 9.3. Let (X; %) be an abelian preordered group and Y a nonempty subgroup of X such that no two distinct elements of Y are %-comparable. Then, for every group preorder D on Y; there is a total group preorder % on X that extends both % and D. Proof. Put R := % [ D, and note that R> = [ B under the hypotheses of the theorem. (Here B and are asymmetric parts of D and %, respectively.) We 6 To provide a direct proof, suppose there is a total preorder % that extends R: It is reasily veri…ed that % extends tran(R) as well. But if tran(R) did not extend R, there would exist tran(R)-equivalent elements x and y of X such that x R> y, which means % could not have been an extension of both R and tran(R).

16

EFE A. OK AND GIL RIELLA

wish to show that R is consistent. For that, we …rst note that there cannot exist distinct elements a; b; c; d 2 X such that a D b % c D d, since a D b and c D d imply that b; c 2 Y and, consequently, are not %-comparable. Because of that, for every x; y 2 X, we have x tran(R) y only if x % y or there exist a; b 2 Y with x % a D b % y (where we allow for a = x and b = y). If x % y, it is clear that we cannot have y R> x. In the later case, we cannot have y x, because this would imply b a, which contradicts a; b 2 Y . We also cannot have y B x, as this would imply that y = b and x = a. Conclusion: R is consistent. We may now invoke Proposition 9.1 to …nd a total group preorder % on X that extends R: It is routine to verify that % must in fact extend % and D simultaneously. References Arrow. K., Social Choice and Individual Values, Wiley, New York, 1951. Aumann, R., Utility theory without the completeness axiom, Econometrica 30 (1962), 445-462. Everett, C., Note on a result of L. Fuchs on ordered groups, Amer. J. Math. 72 (1950), 216. Evren, O. and E. A. Ok, On the multi-utility representation of preference relations, J. Math. Econ. 47 (2011), 554-563. Fuchs, L., On the extension of the partial order of groups, Amer. J. Math. 72 (1950), 191-194. Holland, Ch. W., Extensions of ordered algebraic structures, Doctoral Thesis, Tulane University, 1961 Kargapolov, M., Fully orderable groups (Russian), Algebra i Logika 2 (1963), 5-14. Kargapolov, M., A. Kokorin and V. Kopytov, On the theory of orderable groups (Russian), Algebra i Logika 6 (1965), 21-27. Kokorin, A., On fully orderable groups (Russian), Dokl. Akad. Nauk SSSR 151 (1963), 31-33. Kokorin, A. and V. M. Kopytov, Fully Ordered Groups, Halsted Press (John Wiley & Sons), New York - Toronto, 1974. Kopytov, V. M., On the theory of fully orderable groups (Russian), Algebra i Logica 5 (1966), 27-31. Kopytov, V. M. and N. Ya. Medvedev, The Theory of Lattice-Ordered Groups, Kluwer Academic Publishers, Dordrecht, 2010. Levi, F., Ordered groups, Proc. Indian Acad. Sci. A. 16 (1942), 256-263. Mal’cev, A., On the full ordering of groups (Russian), Trudy Mat. Inst. Steklov 38 (1951), 173-175. Mal’cev, A., On partially ordered nilpotent groups (Russian), Algebra i Logika 2 (1962), 5-9. Minassian, D., Types of fully ordered groups, Amer. Math. Monthly 80 (1973), 159-169.

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Neumann, H., An embedding theorem for algebraic systems, Proc. London Math. Soc. 41 (1954), 138-153. Ohnishi, M., On linearization of ordered groups, Osaka Math. J. 2 (1950), 161-164. Ohnishi, M., Linear order on a group, Osaka Math. J. 4 (1952), 17-18. Ok, E. A. and G. Riella, Topological closure of translation invariant preorders, Math. Oper. Res., forthcoming. Ok, E. A., P. Ortoleva, and G. Riella, Incomplete preferences under uncertainty: indecisiveness in beliefs vs. tastes, Econometrica 80 (2012), 1791-1808. Robinson, D., A Course in the Theory of Groups, Springer-Verlag, New York, 1996. Seidenfeld, T., M. Schervish and J. Kadane, A representation of partially ordered preferences, Ann. Statist. 23 (1995), 2168-2217. Suzumura, K., Remarks on the theory of collective choice, Economica 43 (1976), 381-390. Department of Economics and Courant Institute of Mathematical Sciences, New York University E-mail address : [email protected] Department of Economics, Universidade de Brasilia E-mail address : [email protected]