Free ℓ-groups and free products of ℓ-groups Ton Dao-Rong

Free ℓ -groups and free products of ℓ -groups

Ton Dao-Rong

Abstract. In this paper we have given the construction of free ℓ-groups generated by a po-group and the construction of free products in any sub-product classUofℓ-groups.

We have proved that theU-free products satisfy the weak subalgebra property.

Keywords: lattice-ordered group (ℓ-group), freeℓ-group, free product ofℓ-groups, sub- product class ofℓ-groups

Classification: 06F15

1. Introduction

We use the standard terminologies and notations of [1], [2], [5]. The group operation of anℓ-group is written by additive notation. A po-group is a partially ordered group [G, P] where P = {x ∈ G | x ≥ 0} is the positive semigroup of G. Let Gand H be two po-groups. A mapϕ: G→H is called a po-group homomorphism, if ϕ is a group homomorphism andx≥y impliesϕ(x) ≥ϕ(y) for anyx, y∈G. A po-group homomorphismϕis called a po-group isomorphism, ifϕis an injection andϕ^⊣is also a po-group homomorphism.

A partialℓ-groupGis a set with partial operations corresponding to the usual ℓ-group operations·,^⊣,|,∨and∧such that whenever the operations are defined for elements ofG, the theℓ-group laws are satisfied. Suppose [G, P] is a po-group.

ThenGhas implicit partial operations∨and∧as determined by the partial order.

That is,

x∨y=y∨x=y if and only if x≤y and

x∧y=y∧x=x if and only if x≤y.

Using these partial lattice operations together with the full group operations,G can be considered as a partialℓ-group. Then we have the following definition as a special case of theU-free algebra generated by a partial algebra.

Definition 1.1. LetU be a class of ℓ-groups and [G, P] be a po-group. The ℓ- groupFU[G, P] is called theU-freeℓ-group generated by [G, P] (orU-freeℓ-group over [G, P]) if the following conditions are satisfied:

(1) F_U([G, P])∈ U;

(2) there exists a po-group isomorphismα:G→ F_U([G, P]) such thatα(G) generatesF_U([G, P]) as anℓ-group;

(3) ifK∈ U and β:G→K is a po-group homomorphism, then there exists anℓ-homomorphismγ:FU([G, P])→K such thatγα=β.

A

[G,P] FU([G,P])

K α

β

γ

Definition 1.2. Let U be a class ofℓ-groups and {G_λ |λ∈ Λ} be a family of ℓ-groups inU. The U-free product ofG_λ is anℓ-groupG, denoted by^U`

λ∈ΛG, together with a family of injectiveℓ-homomorphismsα_λ:G_λ →G(called copro- jections) such that

(1) ^U`

λ∈ΛG_λ ∈ U;

(2) S

λ∈Λα_λ(Gλ) generates^U`

λ∈ΛG_λ as anℓ-group;

(3) if K ∈ U and {βλ : G_λ → K | λ∈Λ} is a family ofℓ-homomorphisms, then there exists a (necessarily) unique ℓ-homomorphism γ : G → K satisfyingβ_λ=γα_λ for allλ∈Λ.

A familyU ofℓ-groups is called a sub-product class, if it is closed under taking (1)ℓ-groups and (2) direct products. All our sub-product classes ofℓ-groups are always assumed to contain along with a givenℓ-group all itsℓ-isomorphic copies.

Clearly, all varieties ofℓ-groups are sub-product classes ofℓ-groups. LetL,Rand A be the varieties of all ℓ-groups, representable ℓ-groups and abelianℓ-groups, respectively.

In this paper we will discuss the existence and constructions of free ℓ-groups generated by a po-group and free products in any sub-product classes ofℓ-groups.

In what follows,U is always denoted a sub-product class ofℓ-groups.

2. Construction for aU-free ℓ-group generated by a po-group

In 1963 and 1965, E.C. Weinberg initially considered theA-freeℓ-group gener- ated by a po-group [G, P]. He has given a necessary and sufficient condition for existence and a simple description ofFA([G, P]) in [17], [18].

In 1970, P. Conrad generalized Weinberg’s result as follows.

Lemma 2.1 ([3]).

(1)There exists anL-freeℓ-groupFL([G, P])generated by[G, P], if and only if there exists a po-group isomorphism of[G, P]into anℓ-group, if and only ifP is the intersection of right order onG.

(2)Suppose thatP =T

λ∈ΛP_λwhere{P_λ|λ∈Λ}is the set of all right orders ofGsuch thatP_λ ⊇P. IfG_λisGwith one such right order, then denote byA(G_λ) theℓ-group of order preserving permutations of G_λ. Eachx∈Gcorresponds to an elementρxofA(G_λ)defined byρxg=g+x. ThenF_L([G, P])is the sublattice ofQ

λ∈ΛA(G_λ)generated by the long constantshgi(g∈G).

By Gr¨atzer existence theorem on a free algebra generated by a partial algebra (Theorem 28.2 of [6]) we have

Theorem 2.2. There exists aU-freeℓ-groupF_U([G, P])generated by a po-group [G, P]if and only if[G, P]is po-group isomorphic to anℓ-group inU.

Lemma 2.3 (Lemma 11.3.1 of [5]). LetL andL^′ be ℓ-groups and M be a sub- group of L which generates L as a lattice. Let ϕ : M → L^′ be a group ho- momorphism such that for each finite subset {xjk | j ∈ J, k ∈ K} of M, W

j∈J

V

k∈Kx_jk = 0 implies W

j∈J

V

k∈Kϕ(x_jk) = 0. Then ϕ can be uniquely extended to anℓ-homomorphismϕ^′:L→L^′.

LetU be a sub-product class ofℓ-groups. An ℓ-homomorphic imageH of an ℓ-groupGis said to be a U-homomorphic image, if H ∈ U. Suppose that a po- group [G, P] is po-group isomorphic into an ℓ-groupF₀ ∈ U with the po-group isomorphism δ. By Lemma 2.1 (1) there exists the L-free ℓ-group FL([G, P]) generated by [G, P] with the po-group isomorphismα: [G, P]→ FL([G, P]). By Definition 1.1 there exists an ℓ-homomorphismγ : FL([G, P]) → F₀ such that γα = δ. Let D = {F_λ | λ ∈ Λ} be the set of all U-homomorphic images of F_L([G, P]) with theℓ-homomorphismsγ_λ (λ∈Λ). Thusγ(F_L([G, P]))∈D and D is not empty.

[G,P]A G^′⊆F

Q

λ∈ΛFλ

FL([G,P]) F0

L

π β α γ

δ γλ

β^′ β^∗ γo

For each λ∈Λ, γ_λαis a po-group homomorphism of [G, P] into F_λ. Then the direct productQ

λ∈ΛF_λ is anℓ-group inU. Letπbe the natural map ofGonto the subgroupG^′ of long constants ofQ

λ∈ΛF_λ. That is, π(g) = (. . . , γ_λα(g), . . .)

forg∈G. Sinceγα=δis a po-group isomorphism,πis a po-group isomorphism.

LetF be the sublattice of Q

λ∈ΛF_λ generated byG^′. ThenF is anℓ-subgroup ofQ

λ∈Λ, and soF ∈ U.

Theorem 2.4. Suppose that a po-group[G, P]is po-group isomorphic into an ℓ-group in a sub-product class ofℓ-groups. Then the U-freeℓ-groupF_U([G, P]) generated by[G, P]is the sublatticeF of the direct productQ

λ∈ΛF_λgenerated by the po-group isomorphic imageG^′ofGwhere{F_λ |λ∈Λ}are allℓ-homomorphic images of theL-freeℓ-groupFL([G, P])generated by[G, P].

Proof: We have already known that F ∈ U and [G, P] is po-group isomorphic into F. Suppose that β is a po-group homomorphism of [G, P] into anℓ-group L ∈ U. Then there exists an ℓ-homomorphism γ_o : F_L([G, P]) → L such that γ_oα=β. Soγ^′(FL([G, P]))∈D. Forg^′=π(g)∈G^′ (g∈G), put

β^′(g^′) =β(g).

Then β^′ is a group homomorphism ofG^′ into L and β^′π = β. By Lemma 2.3 we only need to show that for each finite subset {g_jk | j ∈ J, k ∈ K} ⊆ G, W

j∈J

V

k∈Kβ^′π(g_jk)6= 0 impliesW

j∈J

V

k∈Kπ(g_jk)6= 0. In fact, _

j∈J

^

k∈K

γ_oα(gjk) = _

j∈J

^

k∈K

β(gjk) = _

j∈J

^

k∈K

β^′π(gjk)6= 0.

Hence

_

j∈J

^

k∈K

π(g_jk) = _

j∈J

^

k∈K

(. . . , γoα(g_jk), . . .)

= (. . . , _

j∈J

^

k∈K

γoα(g_jk), . . .)6= 0.

Thereforeβ^′ can be uniquely extended to anℓ-homomorphismβ^∗:F →L.

3. Construction of U-free products

Let U be a sub-product class of ℓ-groups and {G_λ | λ ∈ Λ} be a family of ℓ-groups inU. By Corollary 2 of Theorem 2 of [6] theU-free product^U`

λ∈ΛG_λ always exists. Specifically, there exists an L-free product ^L`

λ∈ΛG_λ with the coprojection α_λ. In [7]–[14] J. Martinez, W. Powell and C. Tsinakis have given several descriptions and some properties for the free products in the varietiesR andA. W.C. Holland and E. Scrimger have given a description forL-free product.

LetH be the group free product of{G_λ|λ∈Λ}. LetP ={h∈H |hbe a sum of conjugates in H of elements of S

λ∈ΛG⁺_λ} and P^′ = {Q | Q is the positive cone of a right order onH with P ⊆Q}. Then [H, P^′] is a po-group and its L- free extensionFL([H, P^′]) by theℓ-ideal generated by{g⁻∧g⁺|g∈S

λ∈ΛG_λ} (Theorem 3.7 of [4]). There exists a group homomorphismα:H →^L`

λ∈ΛG_λ which extends everyα_λ (λ∈Λ).

It is clear that the cardinal sum⊞G_λis anℓ-group inU and everyG_λ (λ∈Λ) can naturally be embedded into⊞λ∈ΛG_λas anℓ-group with embeddingδ_λ. Hence there exists a group homomorphismδ:H →⊞λ∈ΛG_λ which extends eachδ_λ

HA H^′⊆F

Q

i∈IFi L^`

λ∈ΛGλ

⊞λ∈ΛGλ

Gλ L

π β α αλ

γo

δ γi

β^′ β^∗ f δλ

(λ∈Λ) and there exists anℓ-homomorphism γ_o :^L`

λ∈ΛG_λ → ⊞λ∈ΛG_λ such that γ_oα_λ = δ_λ for each λ ∈ Λ. Let D = {Fi | i ∈ I} be the set of all U- homomorphic images of^L`

λ∈ΛG_λ with theℓ-homomorphismsγ_i (i∈I). Thus,

⊞λ∈ΛG_λ ∈ D and D is not empty. For each λ∈ Λ and each i∈ I, γ_iα_λ is an ℓ-homomorphism ofG_λ intoF_i. The direct productQ

i∈IF_i is anℓ-group inU. For eachλ∈Λ, letπ_λbe the naturalℓ-homomorphism ofG_λonto theℓ-subgroup G^′_λ ofQ

i∈IF_i. That is,

π_λ(g_λ) = (. . . , γ_iα_λ(g_λ), . . .) forg_λ ∈G_λ. Let H^′ be the subgroup ofQ

i∈IFi generated byS

λ∈ΛG^′_λ. Let π be the group homomorphism ofH ontoH^′ which extends everyπ_λ(λ∈Λ). That is,

π(h) = (. . . , γiα(h), . . .)

forh∈H. Since⊞_λ∈ΛG_λ ∈D and everyδ_λ (λ∈Λ) is anℓ-isomorphism,π_λ is anℓ-isomorphism for each λ∈Λ. LetF be the sublattice ofQ

i∈IFi generated byH^′. For eachh∈H, put h^′ =π(h). SinceQ

i∈IF_i is a distributive lattice, F=





 _

j∈J

^

k∈K

h^′_jk|h_jk∈H, J andK finite





 . Then we have the following construction theorem for^U`

λ∈ΛG_λ.

Theorem 3.1. Suppose that {G_λ | λ ∈ Λ} is a family of ℓ-groups in a sub- product class ofℓ-groups. Then theU-free product ^U`

λ∈ΛG_λ is the sublattice F of the direct productQ

i∈IF_i generated by the group homomorphic imageH^′ of the group free product H of G_λ, where {Fi | i ∈ I} are allU-homomorphic images of theL-free product^L`

λ∈ΛG_λ.

Proof: We have seen that F ∈ U and eachG_λ (λ∈Λ) can be embedded into F as an ℓ-group. Suppose thatL ∈ U and {β_λ : G_λ →L | λ ∈Λ} is a family ofℓ-homomorphisms. We must show that there exists a uniqueℓ-homomorphism β^∗ : F → L such that β^∗π_λ = β_λ. By the universal property of group free product, there exists a group homomorphismβ:H→Lwhich extends everyβ_λ (λ∈Λ). For anyh^′ =π(h)∈H^′, put

β^′(h^′) =β(h).

By the universal property of an L-free product, there exists a unique ℓ-homo- morphismf :^L`

λ∈ΛG_λ→Lsuch thatβ_λ=f α_λ for eachλ∈Λ. Then f α=β^′π=β.

By Lemma 2.3 we only need to show that for each finite subset{h_jk|j∈J, k∈ K} ⊆H,W

j∈J

V

k∈Kβ^′π(hjk)6= 0 impliesW

j∈J

V

k∈Kπ(hjk)6= 0. In fact, _

j∈J

^

k∈K

f α(h_jk) = _

j∈J

^

k∈K

β^′π(h_jk)6= 0.

Becausef(^L`

λ∈ΛG_λ)∈D,W

j∈J

V

k∈Kγ_iα(h_jk)6= 0 for somei∈I. So _

j∈J

^

k∈K

π(h_jk) = _

j∈J

^

k∈K

(. . . , γ_iα(h_jk), . . .)

= (. . . , _

j∈J

^

k∈K

γ_iα(h_jk), . . .)6= 0.

Thereforeβ^′ can be uniquely extended to anℓ-homomorphismβ^∗:F →L.

By using the similar proof as the one for Theorem 3.1 we can get the following result.

Theorem 3.2. Suppose thatU is a sub-product class ofℓ-groups which is con- tained in A and {G_λ | λ ∈ Λ} is a family in U. Then the U-free product

U`

λ∈ΛG_λ is the sublattice of Q

i∈IF_i generated by the group homomorphic image H^′ of the group free product H of G_λ, where {Fi | i ∈ I} are all ℓ- homomorphic images of theA-free product^A`

λ∈ΛG_λ. 4. The weak subalgebra property

LetU be a sub-product class ofℓ-groups. U-free products are said to have the subalgebra property if for any family {G_λ |λ∈Λ} in U withℓ-subgroupsH_λ ∈ G_λ,^U`

λ∈ΛH_λis simply theℓ-subgroup of^U`

λ∈ΛG_λgenerated byS

λ∈ΛH_λ. It is well known thatA-free products satisfy the subalgebra property (Theorem 3.2 of [11]). U-free products are said to have the weak subalgebra property if{G_λ|λ∈ Λ}is a family inUwithℓ-subgroupsH_λ⊆G_λand any family ofℓ-homomorphisms σ_λ :H_λ →L∈ U can be extended to a family of ℓ-homomorphismsσ_λ^′ :G_λ → L^′ ∈ U such thatL is an ℓ-subgroup of L^′ and σ_λ^′|_Hλ =σ_λ, then ^U`

λ∈ΛH_λ is theℓ-subgroup of^U`

λ∈ΛG_λ generated byS

λ∈ΛH_λ.

Theorem 4.1. Suppose thatU is a sub-product class ofℓ-groups which is con- tained inA. ThenU-free products satisfy the weak subalgebra property.

Proof: Suppose that{G_λ |λ∈Λ} is a family inU with ℓ-subgroupsH_λ ⊆G_λ and any family ofℓ-homomorphismsσ_λ:H_λ→L∈ Ucan be extended to a family

ofℓ-homomorphismsσ_λ^′ :G_λ →L^′ ∈ U such thatL is anℓ-subgroup ofL^′ and σ_λ^′|Hλ =σ_λ. We see that^A`

λ∈ΛH_λ is theℓ-subgroup of^A`

λ∈ΛG_λ generated byS

λ∈ΛH_λ.

(1) First we show that any ℓ-homomorphism γ : ^A`

λ∈ΛH_λ → L ∈ U can be extended to an ℓ-homomorphism γ^′ : ^A`

λ∈ΛG_λ → L^′ ∈ U such that L is an ℓ-subgroup of L^′ and γ^′|^A`

λ∈ΛHλ = γ. In fact, any ℓ-homomorphism γ:^A`

λ∈ΛH_λ→L∈ U induces a family ofℓ-homomorphismsσ_λ :H_λ →L∈ U such thatγα_λ=σ_λ for eachλ∈Λ whereα_λ is the inclusion map. Thusσ_λ can

A

Hλ Gλ

A^`

λ∈ΛHλ A^`

λ∈ΛGλ

L^′

αλ L α^′_λ

σλ σ^′_λ

γ γ^′

be extended to a family ofℓ-homomorphismsσ_λ^′ :G_λ →L^′ ∈ U such thatL is anℓ-subgroup of L^′ andσ^′_λ|_Hλ =σ_λ. By the universal property there exists an ℓ-homomorphismγ^′:^A`

λ∈ΛG_λ→L^′such thatγ^′α^′_λ =σ^′_λfor eachλ∈Λ where α^′_λ is the inclusion map. Hence

σ_λ=σ_λ^′|_Hλ = (γ^′α^′_λ)|_Hλ=γ^′|_Hλ for eachλ∈Λ. By the uniquenessγ^′|^A`_λ∈ΛHλ=γ.

(2) Now we show that^U`

λ∈ΛH_λ is the ℓ-subgroup of^U`

λ∈ΛG_λ generated by the S

λ∈ΛH_λ. Let G₀ = ⊕_λ∈ΛG_λ, H₀ = ⊕_λ∈ΛH_λ, G = ^A`

λ∈ΛG_λ and H =^A`

λ∈ΛH_λ. ThenG₀ and H₀ are subgroups ofGandH, respectively, and H₀ is a subgroup ofG₀,H is a subgroup ofG. LetD={F_i|i∈I}be the set of allU-homomorphic images ofGwith theℓ-homomorphismsγ_i^′ (i∈I). For each

A Q

i∈IEi

Q

i∈IFi

H G

H0 G0

H₀^′ G^′0 U^`

λ∈ΛHλ U^`

λ∈ΛGλ

π π^′

i ∈ I, γ_i^′|_H(H) is a U-homomorphic image of H. Conversely, if E is a U- homomorphic image ofH with the ℓ-homomorphismγ. It follows from (1) that γ can be extended to an ℓ-homomorphism γ^′ : G → F ∈ U such that E is an

ℓ-subgroup ofF andγ^′|_H =γ. Hence the set of allU-homomorphic images ofH isC ={E_i |i∈I} where E_i is an ℓ-subgroup of F_i andγ_i^′|_H(H) =E_i for each i∈ I. By Theorem 3.2 we see that U-free product^U`

λ∈ΛG_λ is the sublattice of the direct product Q

i∈IF_i generated by the group homomorphic image G^′₀ of G₀ with the group homomorphism π^′, and the U-free product^U`

λ∈ΛH_λ is the sublattice of the direct product Q

i∈IE_i generated by the group homomor- phic imageH₀^′ of H₀ with the group homomorphismπ. π^′|_Gλ and π|_Hλ are all ℓ-homomorphisms for λ ∈ Λ. Hence ^U`

λ∈ΛG_λ is the ℓ-subgroup of Q

i∈IF_i generated byS

λ∈ΛG^′_λ whereG^′_λ=π^′(G_λ) and ^U`

λ∈ΛH_λ is theℓ-subgroup of Q

i∈IE_i generated byS

λ∈ΛH_λ^′ whereH_λ^′ =π(H_λ). From the above we see that Q

∈IE_i is anℓ-subgroup ofQ

i∈IF_i and π^′|_H0 =π.

Therefore^U`

λ∈ΛH_λ is the ℓ-subgroup ofQ

i∈IF_i generated byS

λ∈ΛH_λ^′, and so^U`

λ∈ΛH_λ is also theℓ-subgroup of^U`

λ∈ΛG_λ generated byS

λ∈ΛH_λ^′. 5. An example

Theorem 2.4 and Theorem 2.1 are applicable to all varieties ofℓ-groups. But here we give an example of a class ofℓ-groups which is not a variety.

Anℓ-groupGis said to be weak Hamiltonian if each closed convexℓ-subgroup ofG is normal. Let WH be the set of all weak Hamiltonianℓ-groups. It is easy to show that WH is a sub-product class ofℓ-groups (see [16]) and

A ⊆WH⊆ R ⊆ L.

So we have the construction theorem for the WH-free product.

Theorem 5.1. Suppose that{G_λ |λ∈Λ}is a family inWH. Then^WH`

λ∈ΛG_λ is the sublattice ofQ

i∈IF_igenerated by the group homomorphic imageH^′of the group free product H of G_λ, where {Fi | i ∈ I} are all weak Hamiltonian ℓ- homomorphic images of^L`

λ∈ΛG_λ.

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29–305, 3 Xikang Road, Nanjing, 210024, China (Received April 28, 1994)