Lattice-ordered Groups

Transcription

1 Chapter 2 Lattice-ordered Groups In this chapter we present the most basic parts of the theory of lattice-ordered groups. Though our main concern will eventually be with abelian groups it is not appreciably harder to develop this material within the class of all groups. Moreover, the additional generality allows us to digress somewhat (if it is possible to digress before one begins) and to present some of the classical theorems in the subject. The fundamental interactions between the lattice structure and the group structure of a lattice-ordered group are dealt with first. Included are the characterization of the lattice structure in terms of the subsemigroup of positive elements as well as the elementary identities which result from the two structures. We then examine the morphisms in the category of lattice-ordered groups and the various kinds of subobjects. What is significant here is that the lattice of kernels in a lattice-ordered group is a distributive lattice and so is the corresponding lattice of subobjects that arises by dropping the normality requirement. These latter subobjects are precisely those for which the corresponding partition of cosets of the group is a lattice homomorphic image of the underlying lattice of the lattice-ordered group. Those lattice-ordered groups that lack bounded subgroups are those that admit a completion, and they will be represented as extended real-valued continuous functions on a compact topological space. This representation bears fruit not only in the study of these kinds of lattice-ordered groups but also in the study of those latticeordered fields of which they are the underlying group. In order to represent other lattice-ordered groups as lattice-ordered groups of functions it is essential to find totally ordered sets that arise naturally from the group. This is accomplished by taking the partition of cosets determined by a not necessarily normal kernel that is maximal with respect to excluding a given element. These totally ordered sets can be joined together and used to represent a lattice-ordered group as a subobject of the lattice-ordered group of all automorphisms of a totally ordered set. They will also be used to represent an abelian lattice-ordered group as a subobject of an antilexicographically ordered lattice-ordered group of real-valued functions. S.A. Steinberg, Lattice-ordered Rings and Modules, DOI / _2, Springer Science + Business Media, LLC

2 34 2 Lattice-ordered Groups 2.1 Basic Identities and Examples Numerous identities and inequalities arise from the coexistence of the group and lattice structures. The two that will become most prominent are the unique decomposition of an element as the difference of two disjoint centralizing elements and a modified triangle inequality that can be replaced by the usual triangle inequality exactly when the group is abelian. Before this is established we will see that the lattice makes the group torsion-free and the group makes the lattice distributive. Let G be a group which is also a partially ordered set. We will denote the group operation in G additively even if G is not assumed to be abelian. G is called a po-group if translations in G are order preserving: x y a + x + b a + y + b x, y, a, b G. (2.1.1) If only the right (respectively, left) translations are isotone, then G is a right (respectively, left) po-group. In a right po-group each right translation is, in fact, an automorphism of the underlying poset since the inverse of a right translation is a right translation. One consequence of this fact is that in a po-group G, if x i (or xi ) exists and each x i commutes with x, then x i (or x i ) commutes with x that is, each centralizer in G contains the sup or inf of any of its subsets that exist in G. For, x + ( x i ) = (x + x i ) = (x i + x) = ( x i ) + x, and dually. Also, in a po-group G, inversion is an anti-automorphism of the underlying poset since x y y = y + x x y + y x = x. The po-group G is called a totally ordered group, a lattice-ordered group (lgroup) or a directed group, accordingly, as its underlying poset is totally ordered, a lattice, or is directed. The positive cone of G is G + = {g G : 0 g} and the elements of G + are called positive. As the following theorem demonstrates the partial orders of the group G that make it into a po-group are in one-to-one correspondence with the positive cones of G. For this reason we will frequently refer to G + as a partial order (or a total order or a lattice-order) of G. Theorem If G is a po-group with positive cone P = G +, then (a) P + P P (P is a subsemigroup of G); (b) g + P + g P g G (P is normal in G); (c) P P = 0; (d) x y y x P x,y G. Conversely, if G is a group and P is a normal subsemigroup of G that satisfies (c), then the relation defined in (d) is a partial order of G which makes G into a po-group with positive cone P.

3 2.1 Basic Identities and Examples 35 Proof. That (a) is true follows from the more general observation that if a b and c d in the po-group G, then a + c b + d; for, a + c b + c b + d. As for (b), if 0 x then 0 g + x + g. If x P P, then 0 x and x = y for some y 0. So x 0 and hence x = 0. This proves (c), and (d) is a consequence of the fact that right translation by either x or x is order preserving. Conversely, given a normal subsemigroup P that satisfies (c), the relation defined in (d) is reflexive since 0 P. Also, if x y and y x, then y x, x y P; so x y = 0 by (c). If x y and y z, then z x = (z y) + (y x) P by (a); so x z and is a partial order of G. Finally, (2.1.1) is satisfied since, by (b), (a + y + b) (a + x + b) = a + (y x) a P, if y x P; and G + = P by (d). Since two elements in the po-group G are comparable precisely when their difference is comparable to 0, G is totally ordered if and only if G = G + G +. Also, since inversion is an order anti-automorphism, G is directed exactly when it is directed down (or up). Moreover, for a,b G (a b) = a b = [0 ( b + a)] a provided any one of the three expressions exists. So G is an l-group if and only if x 0 exists for each x in G. This proves the more transparent parts of Theorem Let G be a po-group. (a) G is totally ordered iff G = G + G +. (b) G is directed iff G is directed up (down). (c) G is directed iff G + generates G. Moreover, if G + generates G, then G = G + G + = {a b : a,b G + }. (d) G is an l-group iff x 0 (respectively, x 0) exists for each x G. (e) G is an l-group iff G + is a lattice and generates G. Proof. By the previous remarks we only have to prove (c) and (e). If G is directed let g G and take x G with x 0, g. Then g = g x ( x) G + G +. Conversely, if G + generates G, then for a,b G there exists g 1,...,g n G + such that a b = g 1 ± g 2 ± ± g n g 1 + g g n = x, and then a b, 0 x. So, a, b x + b. Certainly G + is a sublattice of G and G + generates G (by (c)) provided that G is an l-group. On the other hand, suppose that G + is a lattice and generates G. If a,b G + and g = sup G +{a,b}, then g = sup G {a,b}. For if h a,b, then h G + ; so h g. But then if x G, x = a b with a,b G +, and (a b) b = (a b) 0 = x 0. The proof of (e) in the previous theorem shows that G is an l-group provided that it is directed and each pair of elements in G + has a sup in G +. The dual is also true (Exercise 9). Let us give some examples of po-groups and l-groups. (i) Each group is a po-group with positive cone P = {0}. (ii) The group and poset direct product G = G i of a family of po-groups {G i : i I} is a po-group that is called the direct product of the family. Recall that the group operation in G is (x i )+(y i ) = (x i +y i ) and the partial order is given by (x i )

4 36 2 Lattice-ordered Groups (y i ) if x i y i for each i I. G is an l-group if each G i is an l-group. Also, the group direct sum of the G i, denoted by i I G i, is a po-subgroup of G which is an l-subgroup if each G i is an l-group; it is called the direct sum of the family of po-groups {G i : i I}. It will be denoted by G i i I or by G 1 Gn if I = {1,...,n}. Of course, the I-tuple (x i ) G i iff {i I : x i 0} is finite. If each G i = H then the direct sum will be denoted by H (I). (iii) Let A and B be nonzero po-groups and let ϕ : B Aut(A) be a group homomorphism into the group of automorphisms of the po-group A. Then the ordinal semidirect product A ϕ B is a po-group. Here, (a,b) (c,d) if b < d, or b = d and a c ; and the group operation is (a,b) + (c,d) = (a + b c,b + d) where b c = ϕ(b)(c). A ϕ B is an l-group exactly when B is totally ordered and A is an l-group (Exercise 10). If ϕ is the trivial homomorphism, then we get the ordinary ordinal product A B. (iv) If {G i : i I} is a family of totally ordered groups and I is a totally ordered set which satisfies the maximum condition, then the ordinal product V (I,G i ) of the G i is a totally ordered group which is called a Hahn product. Here, the underlying group of V is the direct product of the G i, and v V + provided that v(i) > 0 if i is maximal with v(i) 0. In particular, each torsion-free abelian group can be made into a totally ordered group. For each such group is embedded in a rational vector space (Exercise 2) and hence in the direct product Q I for some set I (Exercise 1). According to Theorem I can be totally ordered in such a way that it has the maximum condition; hence Q I = V (I, Q) is a totally ordered group. (v) If P is a poset then its automorphism group Aut(P) becomes a po-group when it is given the pointwise partial order : f g if and only if f (p) g(p) for each p in P; so it is a subposet of the direct product P P. If P is totally ordered, then Aut(P) is an l-group (Exercise 5). (vi) Let G be a topological l-group. This means that G is a topological space and an l-group for which the inversion function : G G is continuous, and also the functions +,, : G G G are continuous if G G is given the product topology. If X is a topological space, then C(X,G) = {X f G : f is continuous} is an l-subgroup of the direct product G X. In particular, if G is the additive group of R we get that C(X), the set of all continuous real-valued functions on X, is an l-group. (vii) Let G be the group of real 3 3 matrices of the form A = 1 a c 0 1 b Then G is a totally ordered group if A is defined to be positive if a > 0, or a = 0 and b > 0, or a = b = 0 and c 0. This also works for n n matrices.

5 2.1 Basic Identities and Examples 37 (viii) Various other examples can be found in analysis. For instance, the set of measurable functions on a measure space is an l-group. Two of the most important consequences of the interaction of the lattice and group structures in an l-group are that the underlying group is torsion-free and the underlying lattice is distributive. Moreover, there are several basic identities and inequalities that hold in any l-group. We present these facts next but first we need to give some definitions. The po-group G is said to be semiclosed if nx 0 implies that x 0 for any x in G and any integer n 1. Recall from Exercise that the lattice L is infinitely distributive if whenever {x i } is a subset of L for which x i exists, then, for each y in L, (y x i ) exists and y x i = (y x i ), and the dual also holds. Theorem Let G be an l-group. (a) G is infinitely distributive and hence is distributive. (b) G is semiclosed and hence it has no nonzero elements of finite order. Proof. (a) Suppose that x = x i exists in G. Then, for y G, y x [(x x i ) + y] x = (x x i ) + (y x i ). So But so 0 (y x) (y x i ) x x i. 0 = x x = x + x i = (x x i ); [(y x) (y xi )] = 0, (y x) + (y x i ) = 0, and y x = (y x i ). The dual follows by inversion. (b) To see that G is semiclosed we show by induction that n(x 0) = nx (n 1)x x 0 for any x G and each integer n 1. If this equation holds for some n, then (n + 1)(x 0) = x 0 + [nx x 0] = [x + (nx x 0)] nx x 0 = (n + 1)x 2x x nx x 0 = (n + 1)x nx x 0. So if kx 0 for some k, then k(x 0) = (k 1)(x 0); and hence x 0 = 0. Let x be an element of the l-group G. The positive part, negative part, and absolute value of x are defined by

6 38 2 Lattice-ordered Groups and x + = x 0, x = ( x) 0, x = x ( x), respectively. Since each centralizer in an l-group is a sublattice any pair of elements in {x, x +, x, x } commute. If x,y G, then x and y are said to be disjoint if x y = 0. The subset X of G is called disjoint if any two distinct elements in X are disjoint. Theorem Let G be an l-group. The following hold for all x, y, z G. (a) x (x y) + y = x y. (b) (x x y) (y x y) = 0 = ( (x y) + x) ( (x y) + y). (c) x y y x y. (d) x = x 0. (e) x y = (x y) (x y). (f) x = x + x. (g) x = x + + x. (h) x = 0 x = 0. (i) x + x = 0; consequently, disjoint elements commute. (j) The following are equivalent: ( ) x y = 0; ( ) x + y = x y; ( ) x = (x y) + and y = (x y). (k) (Riesz decomposition property) If x, y, z G + and x y+z, then x = y 1 +z 1 where 0 y 1 y and 0 z 1 z. (l) If x, y, z G +, then x (y + z) (x y) + (x z). (m) x y = x z = 0 x (y + z) = 0. (n) n(x y) = nx ny and n(x y) = nx ny if n N and x and y commute. (o) x + y ( x + y + x ) ( y + x + y ). (p) G is abelian if and only if x + y x + y for all x and y in G. Proof. (a) x (x y) + y = x + ( x y) + y = y x. (b) (x x y) (y x y) = x y x y = 0. (c) x = x ( x) y x, x y y x y. (d) By (c), x x x ; so 2 x 0 and hence x 0 since G is semiclosed. (e) (x y) (x y) = [x + ( x y)] [y + ( x y)] = 0 (x y) (y x) 0 = x y. (f) x + x = x + ( x 0) = 0 x = x +. (g) Put y = 0 in (e). (h) 0 x +, x x by (g). So if x = 0, then x = 0 by (f). (i) x + x = (x 0) ( x 0) = (x x) 0 = x 0 = 0 by (d) and the

7 2.1 Basic Identities and Examples 39 fact that G is a distributive lattice. If x y = 0, then any two distinct elements in {x +,x,y +,y } are disjoint and hence commute by (a). So x and y commute by (f). (j) The equivalence of ( ) and ( ) is an easy consequence of (a), and that ( ) implies ( ) follows from (i). Also, ( ) implies ( ) since x = x + 0 = x + ( x y) = (x y) + and y = (y x) + = (x y). (k) Let y 1 = x y and z 1 = y 1 + x. Then 0 y 1 y, z 1 = ( x y) + x = 0 ( y + x) z, z 1 0, and x = y 1 + z 1. (l) By (k) x (y + z) = y 1 + z 1 with 0 y 1 y and 0 z 1 z; but also, y 1, z 1 y 1 + z 1 x; so x (y + z) (x y) + (x z). (m) This is an immediate consequence of (l). (n) First, let y = 0. Since nx + nx = 0 by (i) and (m), (nx) + = (nx + nx ) + = nx + by (f) and (j). In general, n(x y) = n[(x y) + +y] = (nx ny) + +ny = nx ny. That n(x y) = nx ny follows by the inversion anti-automorphism (or use (a)). (o) and (p). x y x x y x+y x + y x + y + x. So, by (c), x + y x + y + x and x + y x + y if x and y commute. If the triangle inequality holds and x, y G +, then x + y = x + y = y x y + x = y + x. Interchanging x and y we get that x + y y + x; so x + y = y + x, G + is abelian, and hence so is G since it is generated by G +. The statements in the preceding theorem will frequently be used without an explicit reference. Exercises. 1. Use Zorn s Lemma to show that each vector space has a basis. 2. The group D is said to be divisible if nd = D for each n N where nd = {nd : d D}. A subgroup G of the group D is called essential in D if G A 0 for each nonzero subgroup A of D. If G is an essential subgroup of the divisible abelian group D, then D is called a divisible hull of G. (a) Show that a torsion-free abelian group is divisible if and only if it is a vector space over the rationals. (b) Show that any two divisible hulls of G are isomorphic. (c) Show that each torsion-free abelian group G has a divisible hull d(g). (Hint: Imitate the construction of the rationals from the integers.) 3. Let d(g) be the divisible hull of the torsion-free abelian po-group G (Exercise 2), and let P = {x d(g) : n N with nx G + }. (a) Show that P is a positive cone for d(g) and (d(g),p) is semiclosed. (b) Show that G is semiclosed iff G + = P G. (c) If G is an l-group (or is directed or is totally ordered), show that d(g) is an l-group (or is directed or is totally ordered) and G is a sublattice of d(g). 4. Suppose that each element g of the po-group G has a decomposition g = x y where x y = 0. Show that G is an l-group.

8 40 2 Lattice-ordered Groups 5. If T is a totally ordered set show that the group Aut(T ) of all order automorphisms of T is a sublattice of the direct product T T and hence is an l-group. Show that each disjoint set of positive elements in Aut(T ) has a least upper bound in Aut(T ). An l-group with this property is called laterally complete. 6. Let G be a group. Show that there is a bijection between the set of partial orders of G that make G into a right po-group and the set of subsemigroups P of G with the property that P P = 0. In this correspondence (G, ) is a totally ordered right po-group iff P P = G, and then P is called a right order of G and G is called a right O-group. 7. In a torsion-free abelian group show that each partial order is contained in a total order. 8. Prove that an abelian po-group G is semiclosed if and only if G + is the intersection of total orders of G. 9. Let G be a po-group. (a) Suppose that a,b,c,d,e G + and c = inf G +{a,b} and e = inf G +{a+d, b+ d}. Show that e = c + d. (b) Show that G is an l-group if and only if G is directed and any two elements of G + have an inf in G Prove that A ϕ B is an l-group iff B is a totally ordered group and A is an l-group. 11. Let G = (Z Z) ϕ Z where ϕ : Z Aut(Z Z) is given by ϕ(1) = τ where τ(x,y) = (y,x). Then G is an l-group but the subgroup H of G generated by a = (1, 1,0) and b = (0,0,1) cannot be made into an l-group. (Hint: b + a + b = a; show that a + cannot exist in H.) 12. Let R be a commutative integral domain with identity element, and let F be its field of quotients. If a,b F, then a divides b relative to R (notation : a b) if b = ar for some r R. Let F denote the multiplicative group of nonzero elements of F, and let U be the group of invertible elements in R. Define a relation in the quotient group F /U by au bu if a b. Verify each of the following. (a) F /U is a directed po-group. (It is called the group of divisibility of R.) (b) F /U is an l-group iff any two elements in R have a greatest common divisor. (R is called a GCD domain.) (c) F /U is totally ordered iff R is a valuation domain that is, the lattice of ideals of R is a chain. 13. Prove that the following hold in any l-group G. (a) If i x i and j y j exist, then ( i x i ) + ( j y j ) = i, j(x i + y j ), and dually. (b) a 0 iff 2a a 0.

9 2.1 Basic Identities and Examples 41 (c) x y = 0 iff x G + and x+y = x y. (See Exercise 11 for an example where this equation holds with 0 x < y.) (d) If x and y commute or if G can be embedded into a product of totally ordered groups, and x + y = x y, then x y = 0. (e) (x y) (x z) y z. (f) x + y + (x + y) +. (g) (x z) (y z) + (x z) (y z) = x y. (Hint: Let s = x y and t = x y and, using (e) of Theorem 2.1.4, express x y and each of the absolute values in terms of s,t and z.) (h) x + y + x y. (i) x y x y if x and y commute. Give an example of an l-group in which this inequality fails. (j) (x + y) + = x + (x y + ) (x + y )+y + (Hint: (x + y) + = (x y) +y; compute (x y) + and (x y) in terms of x +,x,y + and y.) Also, (u + v) + = u + + v + if and only if (u + v) + u v. (k) If A = {a 1,...,a n } is a finite subset of G +, then infa = 0 if and only if, for each 0 < x G, there exists y G and a i A such that 0 < y x and y a i = This exercise shows that each abelian l-group is a group of divisibility (see Exercise 12). If F is a ring and G is a semigroup the semigroup ring of G over F, denoted by F[G], is defined as the ring with G as an F-basis and with multiplication induced by the semigroup operation in G. More specifically, the elements of F[G] are functions from G into F which are written as α = a g x g g G where a g F and a g = 0 except for finitely many g G. If β = b g x g F[G], then α + β = (a g + b g )x g and ( αβ = g a h b k )x g. h+k=g The support of α is defined to be supp α = {g G : a g 0}. (a) If F is a domain (that is, ab 0 if a 0 and b 0) and G is a right O-group (Exercise 6), then F[G] is a domain. (b) Let F be a domain and let G be an l-group. (G is a right O-group by Exercise 2.4.2(c).) Define the function v : F[G] G { } by v(α) = inf(supp α). Show that v(α +β) v(α) v(β) and v(αβ) = v(α)+v(β). (Hint: For the latter, reduce to the case that v(α) = v(β) = 0 and use Exercise 13(k) three times. Use it twice to show that if 0 < x G, then there exists 0 < z x such that A = z supp α φ and B = z supp β φ, where z = {g G : g z = 0}. Let α F[G] have supp α = A and agree with α on A

10 42 2 Lattice-ordered Groups and similarly for β. Then αβ = α β + γ with supp α β supp γ = φ. Use it again to show that v(αβ) = 0.) (c) Let Q be the field of quotients of F[G] where F is a field and G is an abelian l-group. Show that there is a unique function v : Q G { } that extends v on F[G] and has the two properties in (b). (d) Let R = {γ Q : v(γ) 0}. Show that R is a subring of Q, and Q is the field of quotients of R. (e) Show that the group of divisibility of R is isomorphic to G. (f) Show that each finitely generated ideal of R is principal; that is, R is a Bezout domain. 15. Let x and y be elements of the po-group G. (a) If x + y y + x, then nx + ny n(x + y) n N. (b) If y + x x + y, then ny + nx n(x + y) n N. 16. Show that a group homomorphism ϕ : H D between two po-groups is complete if and only if 0 = ϕ(a i ) in D whenever 0 = a i in H. 17. The po-group G is called n-semiclosed (n N) if nx 0 implies that x 0. (a) If G is abelian and n-semiclosed then it satisfies the condition: ( ) If z G, X G and z nx, then z x 1 + +x n for all x 1,...,x n X. (b) If G satisfies ( ) then multiplication by n, G n G, is a complete map. (c) If X is an upward directed subset of G and a = supx, then na = supnx. (d) If G is abelian and semiclosed and d(g) is the divisible hull of G, then the inclusion map G d(g) is complete. 18. The po-group G is called a Riesz group if it has the following property: For all subsets X = {x 1,x 2 } and Y = {y 1,y 2 } of G (*) X Y z G with X z Y. Show that the following are equivalent for the po-group G. (a) G is a Riesz group. (b) (*) holds if 0 X. (c) (*) holds for all finite nonempty subsets X and Y of G. (d) The set U(X) of upper bounds of a finite nonempty subset X of G is downward directed. (e) If X and Y are finite and nonempty, then U(X) +U(Y ) = U(X +Y ). (f) For all x, y G +, [0,x] + [0,y] = [0,x + y]. (g) If x,y 1,...,y m G + with x y 1 + +y m, then there exist x 1,...,x m G + with x j y j for each j, and x = x x m. If G is abelian show that the following may be added to the previous list.

11 2.2 Subobjects and Homomorphisms 43 (h) If x 1,x 2,y 1,y 2 G + with x 1 +x 2 = y 1 +y 2, then there is a 2 2 matrix (x i j ) with entries in G + such that x i is the sum of the entries in the ith row of (x i j ) and y j is the jth column sum of (x i j ). (i) If x 1,...,x n,y 1,...,y m G + with x x n = y y m, then there is an n m matrix (x i j ) with entries in G + such that x i (respectively, y j ) is the ith row sum (respectively, jth column sum) of (x i j ). (Show the equivalence of (a) with each of (b), (c), (d), and (e), the implications (a) (f) (g) (b), and the equivalence of (f) with (h).) 2.2 Subobjects and Homomorphisms In this section we examine homomorphisms between two l-groups as well as those subobjects of an l-group that arise from various order-theoretic conditions. It is shown that those subgroups of an l-group modulo which the lattice structure is sustained constitute a complete and distributive sublattice of the subgroup lattice which satisfies one of the infinitely distributive equations but not the other. Among these subgroups are the polars, which arise as the closed elements of the Galois connection associated with disjointness. The lattice of polars is a complete Boolean algebra, and in the exercises an investigation is undertaken of when the Boolean algebra of polars of a subobject is naturally isomorphic to that of the full group. In subsequent sections we will see that this occurs under different guises when we are able to form a completion. Let C be a subgroup of the po-group G, and let G/C denote the set of all of the left cosets of C in G. The relation defined on G/C by x +C y +C if x y + c for some c C (2.2.1) is independent of the coset representatives; for, if x y + c, x = x 1 + d and y = y 1 + e where c, d, e C, then x 1 + d y 1 + e + c yields that x 1 y 1 + e + c d. This relation is reflexive and transitive but not necessarily antisymmetric. We are, of course, interested in the situation when it is a partial order of G/C. The subset X of the poset P is called convex if x p y with x,y X implies that p X. It is easily seen that a subgroup C of the po-group G is convex precisely when it contains the closed interval [0,c] whenever c C. The subgroup C of a po-group is called an l-subgroup if it is also a sublattice. Since x y = (x y) + + y and x y = ( x y), C is an l-subgroup exactly when x C implies that x + C (whenever x + exists in the larger group). Theorem Let C be a subgroup of the po-group G. (a) The relation defined by (2.2.1) is a partial order of G/C if and only if C is convex. If C is convex then the natural map G G/C is isotone, and the map G Aut(G/C) induced by the left translations in G is a po-group homomorphism into the po-group of automorphisms of the poset G/C.

12 44 2 Lattice-ordered Groups (b) Suppose that G is an l-group and that C is a convex subgroup of G. Then the following statements are equivalent. (i) G/C is a (distributive) lattice and the natural map G G/C is a lattice homomorphism. (ii) C is an l-subgroup of G. (c) If C is normal and convex, then G/C is a po-group. (d) If G is an l-group and C is a normal convex l-subgroup, then G/C is an l-group. Proof. (a) If C is convex and x +C y +C and y +C x +C, then x y + c and y x + d for some c, d C. But then d y + x c, so y + x C and G/C is a poset. Conversely, suppose that (2.2.1) defines a partial order of G/C. If c x d with c, d C, then C = c +C x +C d +C = C; so x C and C is convex. The last statement is easily verified. (b) Let C be a convex l-subgroup of the l-group G. If z +C x +C, y +C, then z + c x and z + d y for some c,d C. So z + (c d) x y, and hence z +C x y +C. Thus x y +C = sup G/C {x +C, y +C}, and dually. Conversely, let G G/C be a lattice homomorphism. Then if x C, C = (x+c) C = x + +C; so x + C and C is an l-subgroup. (c) If C is a normal convex subgroup of G, then (a) implies that each translation in G/C is isotone, so G/C is a po-group. (d) This is a consequence of (b) and (c). If G and H are l-groups (respectively, po-groups) then a group homomorphism f : G H which is also a lattice homomorphism (respectively, isotone) is called an l-homomorphism (respectively, a po-homomorphism). So a group homomorphism f is a po-homomorphism exactly when f (G + ) H +. If the context is clear an l-homomorphism or po-homomorphism will just be called a homomorphism. If there is an l-homomorphism (respectively, a po-homomorphism) from G to H which is an order isomorphism we say that G and H are isomorphic and we write G = H. It is easily verified that the kernel of an l-homomorphism (respectively, a po-homomorphism) is a convex l-subgroup (respectively, convex subgroup) of its domain. The usual isomorphism theorems hold in the variety of l-groups, but before we state them we give some useful criteria for a group homomorphism to be an l-homomorphism. For the rest of this chapter, unless stated otherwise, all groups will be l-groups. Theorem Let f : G H be a group homomorphism between the l-groups G and H. Then the following statements are equivalent. (a) f is an l-homomorphism. (b) f (x + ) = f (x) + for each x in G. (c) x y = 0 f (x) f (y) = 0 for all x,y in G. (d) f ( x ) = f (x) for each x in G.

13 2.2 Subobjects and Homomorphisms 45 Proof. It is clear that (a) implies (c), and (a) also implies (d) since f ( x ) = f (x x) = f (x) f (x) = f (x). (c) implies (b). Since f (x + ) f (x ) = 0, f (x) + = [ f (x + ) f (x )] + = f (x + ) by (j) of Theorem (b) implies (a). f (x y) = f ((x y) + + y) = f ((x y) + ) + f (y) = [ f (x) f (y)] + + f (y) = f (x) f (y), and f (x y) = f ( x y) = ( f ( x) f ( y)) = f (x) f (y). (d) implies (b). Since 2 f (x + ) = f (2x + ) = f (x + x ) = f (x) + f (x) = 2 f (x) +, and f (x + ) and f (x) + commute, and H is torsion-free, we have f (x + ) = f (x) +. Theorem Let N be a normal convex l-subgroup of the l-group G. (a) If f : G H is an l-homomorphism with kernel N, then f (G) is an l- subgroup of H and G/N = f (G). (b) The mapping A A/N is a lattice isomorphism between the lattice of convex subgroups (respectively, l-subgroups) of G that contain N and the lattice of convex subgroups (respectively, l-subgroups) of G/N. (c) If A is an l-subgroup of G, then A + N is an l-subgroup of G and (A + N)/N = A/A N. (d) If K is a normal convex l-subgroup of G with N K, then (G/N)/(K/N) = G/K. Proof. We will leave it to the reader to check that the lattice isomorphism in (b) between the subgroup lattices restricts to an isomorphism on the indicated sublattices. Each of the isomorphisms in (a), (c), and (d) is the well-known group isomorphism, so it suffices to verify that each is an l-isomorphism. For (d) this follows from (a) (for example). As for (a), if f denotes this isomorphism, then f ((x + N) + ) = f (x + +N) = f (x + ) = f (x) + = f (x+n) + ; so f is an l-isomorphism by the previous theorem. Finally, (c) follows from (a) provided that A+N is an l-subgroup. But, in fact, if A is just a sublattice of G, then A + N is also a sublattice. For if a,b A and n,m N, then (a + n) (b + m) + N = (a + N) (b + N) = a b + N in the lattice G/N; so (a + n) (b + m) A + N, and dually. In the next two results we give some fundamental properties of the subobjects of an l-group. If X is a subset of the l-group G, then C(X) = C G (X) will denote the convex l-subgroup of G generated by X, and [X] will denote the l-subgroup generated by X. The polar of X is defined by X = {a G : a x = 0 x X}. Note that X X, reverses inclusion and X = X. Let C (G) denote the set of all convex l-subgroups of G, and let B(G) denote the set of polars of G. Since C (G) is closed under intersections it is a complete lattice. Theorem Let G be an l-group. (a) A subgroup C of G is a convex l-subgroup if and only if x c with x G and c C implies that x C.

14 46 2 Lattice-ordered Groups (b) If X G, then C(X) = {g G : g x x n for some x 1,...,x n X}. (c) The subgroup of G generated by a family of convex l-subgroups is a convex l-subgroup, and its positive cone is the subsemigroup of G + generated by the corresponding family of positive cones. (d) If X, Y G, and D, E C (G) and a,b G, then (i) C(X) C(Y ) = C(X Y ) = C({ x y : x X, y Y }); (ii) C(X) C(Y ) = C({ x y : x X, y Y }); (iii) C(D,a) C(D,b) = D C( a b ); (iv) D E = 0 iff D E. (e) If X G, then X C (G). (f) The sublattice of G generated by a subgroup S is an l-subgroup and, in fact, { } [S] = s i j : s i j S, 1 i n, 1 j m = i j { i j s i j : s i j S, 1 i n, 1 j m Proof. (a) If C is a convex l-subgroup of G and x c with c C, then 0 x +, x x c gives that x = x + x C. Conversely, if the subgroup C has this property, then it is certainly convex, and it is also an l-subgroup since 0 c + c. (b) Since any convex l-subgroup which contains X must contain g if g x x n with x i X, it suffices to show that the set C of all such g is a convex l-subgroup. But if also h y y m with y j X, then by (o) of Theorem g h g + h + g x x n + y y m + x x n. So C is a subgroup which is a convex l-subgroup, by (a). (c) Let {C i : i I} be a family of convex l-subgroups of G, and let C be the subgroup of G generated by the C i. If x = c 1 + +c n C where each c j is in some C i, then x d d m with each d j in some C + i, by Theorem (o). So if y x, then y + = e e m and y = f f m where 0 e j, f j d j, by Theorem (k). Hence y = y + y C and C is a convex l-subgroup by (a). The last statement follows by specializing to x = x = y. (d) The equalities in (i) are clear and, since x y C(X) C(Y ) if x X and y Y, C({ x y : x X,y Y }) C(X) C(Y ). But if g C(X) C(Y ), then g ( x x n ) ( y y m ) x 1 y x n y m for some x i X, y j Y. This proves (ii) which, together with (i), readily gives (iii) and (iv). }.

15 2.2 Subobjects and Homomorphisms 47 (e) If a, b X and x X, then by (o) and (m) of Theorem 2.1.4, 0 a b x ( a + b + a ) x = 0. So a b X, and hence X is a convex l-subgroup by (a). (f) It clearly suffices to show that the set { } H = s i j : s i j S, I and J finite I J is an l-subgroup of G, and this follows readily from the general distributivity equations (1.2.1) which hold in any distributive lattice. For, ( ) s i j = ( s i j ) = ( s i f (i) ) I J I J J I I and ( ) ( ) s i j + t kl = (s i j + t kl ) = I J K L I J K L I = (s i j + t f ( j)l ). I K J J L Thus, H is a subgroup; it is an l-subgroup since ( ) (s i j + t f ( j)l ) J L K J ( ) + s i j = I J I (s i j 0) H. J Theorem Let G be an l-group. (a) C (G) is a complete distributive sublattice of the lattice of subgroups of G, and, in fact, it satisfies the infinite distributive law ( ) A B i = (A B i ). i i (b) The mapping C C + is a lattice isomorphism between C (G) and the lattice of convex l-subsemigroups of G + which contain 0. (c) B(G) is a complete Boolean algebra. Proof. (a) That C (G) is a sublattice of the subgroup lattice of G follows from part (c) of the previous theorem. If 0 x A ( B i ), then x = b 1 + +b n with each b j in some B + i (again, by (c) of Theorem 2.2.4). But then each b j A since 0 b j x; so x (A B i ). The other inclusion is obvious. (b) Since C + generates C it suffices to verify that the mapping is onto. So, let

16 48 2 Lattice-ordered Groups S be a convex l-subsemigroup of G + with 0 S, and let C be the subgroup that it generates. We claim that C = {a b : a, b S} C (G) and C + = S. Let a, b, c, d S. Then 0 (b c) + b = b 1 b and 0 (b c) + c = c 1 c; so b 1 and c 1 S. Also, b 1 and c 1 are disjoint and so commute. But then c + b 1 c 1 = c c 1 + b 1 = b c + b 1 = b. Thus (a b) + (c d) = a + c 1 b 1 d = (a + c 1 ) (d + b 1 ). So {a b : a, b S} is a subgroup which must be C. If x a b a + b + a, where a, b S, then x + and x S and so x = x + x C. So C is a convex l- subgroup. If a, b S, then 0 a b b = (a b) + a b; so (a b) + S and C + = {x + : x C} = S. (c) B(G) is a complete lattice since i X i = ( ) X i. i To show that B(G) is a Boolean algebra it suffices, by Theorem 1.2.3, to verify that for polars A and B, A B if and only if A B = 0. But this follows from Theorem 2.2.4(d)(iv). An example of an l-group G in which the other infinite distributive equation fails is obtained by letting G be the l-group of all real-valued continuous functions on the closed interval [0,1]. If B a is the set of those functions in G which vanish at a [0,1], then B a is a maximal ideal of the ring G and ( ) B 0 B a = B 0 G = a 0(B 0 B a ). a 0 As we indicate below, the fact that C (G) is half infinitely distributive has an effect on the direct sum decompositions of G. The l-group G is the direct sum of its convex l-subgroups {C i : i I} if the map (c i ) Σc i is an isomorphism of the l-group Ci onto G. In this case we write G = i I C i. The conditions for this to hold are the familiar group theoretic ones: G must be generated by the C i, each C i must be normal in G, and, for each i, ( ) C i C j = 0. j i Under these condition the mapping is certainly a group isomorphism; but it is also an l-isomorphism. For if c c n 0 where c k C ik and i 1,,i n are distinct indices, then c k c c k 1 + c k c n = d k ; so c k d+ k and

17 2.2 Subobjects and Homomorphisms 49 c k C i k C j = 0. j i k Thus, each c k 0 and the mapping is an order isomorphism. These conditions can be weakened slightly. Theorem Let G be an l-group. (a) Suppose that {C i : i I} is a family of convex l-subgroups of G that generates G. Then the following statements are equivalent. (i) G = C i. (ii) C i C j = 0 if i j. (iii) If x x n 0 where x j C i j and i 1,...,i n are distinct indices, then each x j 0. (b) If C i, D j C (G) and G = C i = D j, i I j J then G = C i D j. i, j (c) If A, B C (G) and G = A B, then B = A. Proof. (a) Certainly, (i) implies (iii) and (iii) implies (ii). If (ii) holds, then, as we have seen, the elements in C i are disjoint from those in C j and hence commute with those in C j. So each C i is normal and C i C j = i C j ) = 0; j i j i(c thus (ii) implies (i). As for (b), C i = C i G = (C i D j ) = j j (C i D j ); so G = i (C i D j ). j For the proof of (c) just note that A = A (A B) = A B yields that A B A. Given two decompositions G = C i = D j of G, {D j } is a refinement of {C i } if for each j there is an i with D j C i. An easy induction using (b) of the previous theorem gives that any finite number of decompositions of G have a common refinement. Recall that an l-group G is a subdirect product of the family {G i : i I} of l- groups if there is a monomorphism f : G ΠG i such that each composite π i f is

18 50 2 Lattice-ordered Groups an epimorphism, and G is subdirectly irreducible if, in any such representation of G there is an index i such that π i f is an isomorphism. Now, a function f : G ΠG i on an l-group G is uniquely determined by the family {π i f : i I}, and f is an l- homomorphism if and only if each π i f is an l-homomorphism. In this case ker f = i ker(π i f ) and G/ker(π i f ) = image of π i f. Consequently, each family {N i : i I} of normal convex l-subgroups of the l-group G determines a homomorphism G ΠG/N i with kernel N = i I N i, and G/N is a subdirect product of the family {G/N i : i I}; and all subdirect product representations of G/N essentially arise in this way. Clearly, a nonzero l-group G is subdirectly irreducible if and only if it has a smallest nonzero normal convex l-subgroup. As a specific instance of Birkhoff s theorem for abstract algebras (see Exercise ) we have Theorem Each l-group is a subdirect product of a family of subdirectly irreducible l-groups. Proof. Let G be an l-group. If 0 a G let N a be a normal convex l-group of G which is maximal with respect to excluding a. The existence of N a is given by Zorn s Lemma. Since each normal convex l-subgroup of G that properly contains N a must contain a, G/N a is subdirectly irreducible. But N a = 0, a 0 so G is isomorphic to a subdirect product of the G/N a. Exercises. 1. The following statements are equivalent for the l-group G. (a) G is totally ordered. (b) Each subset of G is a sublattice. (c) Each convex subset is a sublattice. (d) Each convex subgroup is a sublattice. (The first three are equivalent in any poset G.) 2. In an l-group G a minimal element in G + \ {0} is called an atom. Prove: (a) The subgroup of G generated by the atoms is a normal, abelian, convex l-subgroup. (b) The following statements are equivalent: (i) G is generated by its atoms. (ii) G + has the minimum condition. (iii) G is isomorphic to a direct sum of copies of Z. (c) If R is a commutative unital domain with quotient field F, then R is a unique factorization domain if and only if its group of divisibility F /U is an l- group which is generated by its atoms (see Exercise ).

19 2.2 Subobjects and Homomorphisms The following statements are equivalent for the subgroup C of the l-group G. (a) C is a convex l-subgroup. (b) C is a convex directed subgroup. (c) If a, b G and a b = 0, then a (b + c) C for each c C. (d) If a, b G and a b C, then a (b + c) C for each c C. 4. Let A and B be l-groups and let Aut(A)(respectively, l-aut(a)) be the group of automorphisms of the group (respectively, l-group) A. Suppose that ϕ : B Aut(A) is a group homomorphism and let G = A ϕ B be the semidirect product of A by B supplied with the partial order of the direct product: (a,b) (c,d) if a c and b d. Show that the following statements are equivalent. (a) G is an l-group. (b) ϕ(b) = 1. (c) ϕ is isotone and its image is contained in l-aut(a), where the partial order of l-aut(a) is coordinatewise: f g if f (x) g(x) for each x in A. 5. Let C C (G). Prove: (a) If C is finitely generated as a convex l-subgroup, then C has a single generator. (b) If C is finitely generated as an l-subgroup, then it need not have a single generator. 6. (a) Show that each convex l-subgroup of a divisible l-group is divisible (see Exercises and 2.1.3). (b) If d(g) is the divisible hull of the abelian l-group G, show that the map C d(c) is a lattice isomorphism from C (G) onto C (d(g)). (c) Show that the isomorphism in (b) restricts to an isomorphism between the Boolean algebras B(G) and B(d(G)). 7. Let P be the MacNeille completion of the poset P. Show that there is an embedding of po-groups ϕ : Aut(P) Aut(P) that preserves any infs or sups that exist. In particular, if P is totally ordered, then ϕ is an embedding of l-groups (Exercise 2.1.5). 8. Let X be a poset, H a po-group and P the set of isotone maps from X into H. Then the subset P of the po-group H X (the direct product of X copies of H) has the following properties. (a) P + P P. (b) P P = { f H X : x y implies f (x) = f (y)}. (c) If H is abelian and X is a directed po-group and Hom Z (X,H) is the group of homomorphisms from X to H, then, P Hom Z (X,H), the set of po-group homomorphisms from X to H, is a positive cone for the group Hom Z (X,H). 9. The category whose objects are po-groups (respectively, abelian po-groups) and whose morphisms are po-group homomorphisms will be called Pog (respectively, Poag), while the category whose objects are l-groups (respectively,

20 52 2 Lattice-ordered Groups abelian l-groups) and whose morphisms are l-homomorphisms will be called Log (respectively, Loag). (a) The direct product is a product in each of the categories Pog, Poag, Log, and Loag. (b) Let ({G i } i I, {ψ i j } i j ) be an inverse system in any one of these four categories, and let G = {(g i ) ΠG i : ψ i j (g i ) = g j if i j}. Then limg i = (G,{p i }) where p i is the restriction of the projection π i (see Exercise 1.1.6). 10. (a) The direct sum is a free product in Poag (see Exercise ). (b) If G i 0 for two indices, then G i is not a coproduct of {G i } i I in Loag. (If G i 0, then G i has a nonzero totally ordered homomorphic image by Theorem ) (c) Show that the direct limit construction that is given in Exercise , which is valid in the varieties Log and Loag, also gives the direct limit in the categories Pog and Poag. 11. Let Sgp (respectively, L ) denote the category whose objects are semigroups (respectively, lattices) and whose morphisms are semigroup (respectively, lattice) homomorphisms. Suppose that G,H Pog with G directed. (a) Show that the restriction map Pog[G,H] Sgp[(G +,+),(H +,+)] is a bijection. (b) If G,H Log, then the image of Log[G,H] under this bijection is Sgp [(G +,+), (H +,+)] L [G +,H + ] = Sgp[(G +,+), (H +,+)] Sgp[(G +, ), (H +, )] = Sgp[(G +,+), (H +,+)] Sgp[(G +, ), (H +, )]. 12. Let N be a normal subgroup of the group G, and suppose that N and G/N are po-groups. Let P = N + {x G : 0 x + N (G/N) + }. Then P is a positive cone for G iff N + is normal in G. Assume that P is a positive cone. (a) The subgroup (N,N + ) is a convex subgroup of (G,P) and (G/N) + = P/N. (b) If g P \ N, then g > N. (c) The group (G,P) is totally ordered iff (N,N + ) and (G/N,(G/N) + ) are totally ordered. (d) If N 0, then (G,P) is an l-group iff (N,N + ) is an l-group and (G/N, (G/N) + ) is totally ordered. (e) If (G,P) is an l-group, then the inclusion map N G is complete. 13. (a) Let X be a subset of the l-group G. Then X = X = [X] = C(X), where X is the subgroup generated by X. (b) If {B i : i I} B(G), then i B i = ( i B i ). (c) If B B(G), then B = b B b (where b = {b} ). (d) For each n N and a G, a = a = (na). (e) The function a a is a lattice homomorphism from G + to B(G) which preserves all sups that exist in G +. (f) If a,b G +, then (a + b) = (a b).

21 2.2 Subobjects and Homomorphisms Let H be a subset of the l-group D which contains 0. If X D, then X D = X and X H = X H. Let B(H) = {A H : A = A H H }. Let C (H) denote the collection of all those convex subsets of H which contain 0, and define the functions ϕ : B(H) B(D) and ψ : B(D) C (H) by ϕ(a) = A H D and ψ(b) = B H. B(H) and B(D) are called canonically isomorphic if ϕ and ψ are inverse isomorphisms between B(H) and B(D); that is, if ψ(b(d) B(H) and ϕψ = 1 B(D) and ψϕ = 1 B(H). Verify each of the following. (a) B(H) is a complete lattice. If x y H whenever x,y H, then B(H) is a Boolean algebra. (b) ψϕ = 1 B(H). (c) ψ(b(d)) B(H) iff ψ(b(d)) = B(H). (d) If X D, then X D H X H H and X D D X H D. (e) The following statements are equivalent. (i) B(H) and B(D) are canonically isomorphic. (ii) ϕ is surjective. (iii) ψ(b(d)) B(H) and ψ is injective. (iv) If X D, then X H D = X D D (f) If B(H) and B(D) are canonically isomorphic, then X D H = X H H for each subset X of D. An example which shows that the converse fails is given by the direct sum D = H K where H and K are nonzero l-groups. In this example ψ(b(d)) B(H). (g) Suppose that Ω is a set of operators on D + ; so, each w Ω induces a function w : D + D +. Assume: (i) each w Ω is isotone; (ii) a b = 0 = aw b = 0, for all a,b D and every w Ω; (iii) if 0 < a D, then 0 < aw H for some w Ω. Then B(H) and B(D) are canonically isomorphic. (Show that (iv) of (e) holds.) (h) H is called a cl-essential subset of D if H + C 0 for each nonzero convex l-subgroup C of D. If H is a cl-essential subset of D, then B(H) and B(D) are canonically isomorphic. (Let Ω = {n( ) d : n N,d D + }, where a(n( ) d) = na d and use (g).) 15. A monomorphism ϕ : H D of l-groups is called cl-essential if ϕ(h) is a cl-essential l-subgroup of D (Exercise 14). Show that if ϕ is a cl-essential monomorphism, then ϕ is complete if either D is abelian or, for each 0 < d D, there exists h H with 0 < ϕ(h) d (use Exercises and ). 16. Show that the l-subgroup G of the l-group H is cl-essential in H if and only if, for each convex subgroup C of H, G C 0 or C is trivially ordered. 17. Let G be a po-group, let S be a convex subset of G + with 0 S and let H be the subgroup of G that is generated by S. (a) The following conditions are equivalent. (i) S is the positive cone for some po-subgroup of G.

22 54 2 Lattice-ordered Groups (ii) S + S S. (iii) S is directed up, and if a S then 2a S. (iv) H + = S. (v) H = {a b : a,b S}. (b) If the conditions in (a) hold or if G has the Riesz decomposition property (that is, G satisfies the condition in Theorem 2.1.4(k)), then H is convex. (c) If H is convex, then H is closed under any finite sups and infs that exist in the po-group G. (d) If G is an l-group, then H is a convex l-subgroup of G, and the conditions in (a) are equivalent to (vi) H = {a G : a S}. 18. Let G and A be l-subgroups of the l-group H, and suppose that G + A is a subgroup of H. (a) Show that (g+a) + G+A whenever g G, a A and one of them is comparable to 0 iff g a G + A for every g G + and a A +. (Assuming the meet condition show that (g + a) 0 G + A by considering the elements g + (g + a) 0 if a 0 and (g + a) 0 a + g if a < 0.) (b) If A is totally ordered show that G+A is an l-subgroup of H iff g a G+A for every pair (g,a) G + A +. (c) If g a G for every (g,a) G + A + show that G is convex in G + A +. (d) If A is totally ordered show that G is a convex l-subgroup of the l-group G + A iff g a G for every pair of elements (g,a) G + A Let C be a convex directed normal subgroup of the Riesz group G (see Exercise ). Show that if S = {x 1 +C,x 2 +C,...} is a countable subset of G/C, then there exists a subset T = {t 1,t 2,...} of G such that t n +C = x n +C for each n and T S is an order isomorphism. (Given t 1,...,t n 1 let X = {t i : t i +C < x n +C} and Y = {t i : x n +C < t i +C}. Take x,y G with X < x, y < Y and x +C = x n +C = y +C. Then X {x c 2 } t n Y {x + c 1 } where y = x + c 1 c 2 with c 1,c 2 C +.) 20. Let f : L M be a lattice homomorphism from the lattice L onto the lattice M. Suppose S = { f (x 1 ), f (x 2 ),...} is a countable subset of M. Show that there is a subset T = {t 1,t 2,...} of L with f (t n ) = f (x n ) for each n and f : T S is an order isomorphism. (In the previous exercise let x = lub X, y = glb Y and t n = (x x n ) y.) 2.3 Archimedean l-groups In this section we are concerned with those l-groups which, like the additive group of the reals, have the property that they do not have any nonzero bounded subgroups. As we will see, each such l-group is abelian, and its completion is also an l-group. Also, a representation theorem for these l-groups as extended real-valued continu-

23 2.3 Archimedean l-groups 55 ous functions on a topological space will be given. The target of this representation is rather complete and has the property that it is a summand of each l-group in the current category in which it is embedded as a convex l-subgroup. This latter property is also investigated here and other l-groups which have it are identified. In preparation for the representation theorem we will establish various topological results. One of these is the duality between the category of Boolean algebras and the category of those compact Hausdorff spaces in which each open subset is a union of sets which are both open and closed. The topological space on which the representing functions are defined is obtained from the Boolean algebra of polars through this duality. A po-group G is called integrally closed if, for all a,b G, Na b implies that a 0; and it is called archimedean if Za b implies that a = 0. An l-group G is complete if its underlying lattice is conditionally complete. Recall that this means that each nonempty bounded subset of G has an inf and a sup. It is σ-complete if each nonempty bounded countable subset has an inf and a sup. While this terminology is different than that for posets no confusion should arise since the only complete (in the sense of posets) po-group is 0. The basic connections between these concepts are given in Theorem (a) An integrally closed po-group is archimedean. (b) An archimedean po-group need not be integrally closed. (c) An l-group is archimedean iff it is integrally closed. (d) Each σ-complete l-group is integrally closed. Proof. Certainly, (a) is clear, and an example of an archimedean po-group that is not integrally closed is given in Exercise 5. Suppose that the l-group G is archimedean and na b for each n N. Then na + b + for each n Z; so a + = 0 and a 0. If G is σ-complete and na b for each n N, then c = supna G. Since c a na for each n N, c a c and a 0. So G is integrally closed. Let G be a directed po-group, and let D(G) be its Dedekind completion. Recall from Section 1.3 that D(G) = {X G : X = LUX, φ X G} = {LUX : X G, X φ and UX φ}. We wish to extend the group operation from G to D(G). It will be convenient to denote LUX by X, for X G. Since is a closure operator on the power set of G X = {A G : X A and A = A }. (2.3.1) A partially ordered semigroup is a semigroup S which is also a poset in which each left and each right translation is order preserving: x,y,z S x y z + x z + y and x + z y + z. A monoid is a semigroup with identity.