Compatible elements in partly ordered groups

COMPATIBLE ELEMENTS IN PARTLY ORDERED GROUPS

Received 16 September 2004 and in revised form 28 September 2005

Some conditions equivalent to a strong quasi-divisor property (SQDP) for a partly or-dered groupGare derived. It is proved that ifGis defined by a family oft-valuations of finite character, thenGadmits an SQDP if and only if it admits a quasi-divisor property and any finitely generatedt-ideal is generated by two elements. A topological density con-dition in topological group of finitely generatedt-ideals and/or compatible elements are proved to be equivalent to SQDP.

1. Introduction

LetGbe a partly ordered commutative group (po-group). ThenGis said to have a quasi-divisor propertyif there exist commutative lattice-ordered group (l-group) (Γ,·,∧) and an order isomorphismh(the so-called quasi-divisor morphism) fromGintoΓsuch that for anyα∈Γ, there existg1,. . .,gn∈Gsuch thatα=h(g1)∧ ··· ∧h(gn). Moreover, if this

embeddinghsatisfies the condition

∀α,β∈Γ+ ∃γ∈Γ+ α·γ∈h(G), β∧γ=1, (1.1)

thenG is said to havea strong quasi-divisor property. Many papers have dealt withpo -groups with (strong) quasi-divisor property (e.g., see [1,3,4,5,6,7,8]). It is well known that there are some generic examples of suchl-groupΓ. Namely, ifh:G→Γis a quasi-divisor morphism, thenΓiso-isomorphic to the group (Ᏽtf(G),×t) of finitely generated t-ideals ofG. Recall that at-idealXtofGgenerated by a lower bounded subsetX⊆Gis

a setXt= {g∈G: (∀s∈G)s≤X⇒g≥s}. Then the setᏵtf(G) of all finitely generated t-ideals ofGis a semigroup with operation×tdefined such thatXt×tYt=(X·Y)t(see

[2]). It is clear that a mapd:G→Ᏽf

t(G) defined byd(g)= {g}tis an embedding. Another

example of a groupΓis a group᏷(W) of compatible elements of a defining family oft -valuationsW (see the definitions below). In this note, we want to show that properties of a group ᏷(W) can be used for deriving new conditions under which quasi-divisor property is also a strong quasi-divisor property.

Copyright©2005 Hindawi Publishing Corporation

Let w :G→G1 be an o-homomorphism. Then, w is called t-homomorphism if

w(Xt)⊆(w(X))tfor any lower bounded subsetX⊆G. Moreover, ifG1is a totally ordered

group (i.e.,o-group), thenwis calledt-valuation. Recall that a familyWoft-valuations w:G→Gwis called adefining family forGif

(∀g∈G) g≥1⇐⇒(∀w∈W) w(g)≥1. (1.2)

We say thatWis of finite character if

(∀g∈G) (∀_{w∈W) w(g)=1, (1.3)}

where∀ means “for all but a finite number.” Hence any defining familyW of finite character creates an embedding ofGinto a sumw∈WGwofo-groupsGw,w∈W. Then

a quasi-divisors property ofGis said to be offinite character, if there exists a defining family oft-valuations of finite character forG. Ifw1,w2are twot-valuations of apo-group

G, thenw1is said to be coarser thanw2(w1≥w2) if there exists ano-epimorphismdw1,w2:

Gw1→Gw2such thatw2=dw1,w2w1. It may be then proved that for any twot-valuations

w1,w2, there exists at-valuationw1∧w2 which is the infimum ofw1,w2with respect

to this preorder relation. Then,dw1,w1∧w2(resp.,dw2,w1∧w2) is ano-epimorphism such that

w1∧w2=dw1,w1∧w2,w1=dw2,w1∧w2w2. For simplicity, we setdw1w2=dw1,w1∧w2,dw2w1=

dw2,w1∧w2 (see the difference betweendw1,w2 anddw1w2). IfW is a system oft-valuations

w:G→Gwof apo-groupGandW⊆W, then a system (gw)w∈

w∈WG_wof elements

is calledcompatibleprovided thatdwv(gw)=dvw(gv) for allw,v∈W. Finally, (gw)w∈W

is calledW-completeifw∈WW(g_w)=W, whereW(g_w)= {v∈W:d_wv(g_w)=1}for gw=1wandW(1w)= {w}for anyw∈W.

LetWbe a defining family oft-valuations ofG. Then, we set

᏷(W)=

aw

w∈

w∈W Gw:

aw

wis compatible

. (1.4)

It can be proved that᏷(W) is anl-subgroup inw∈WGw (see [8]). Now we say thatG

with a defining family oft-valuations satisfies the positive weak approximation theorem (PWAT) if for any finite subsetF⊆Wand any compatible system (αw)w∈F∈

w∈FG+w,

there existsg∈G+_{such thatw(g)=α}

w,w∈F. Finally, we say thatGwithWsatisfies

the approximation theorem (AT) if for any finite subsetF⊆Wand any compatible and F-complete system (αw)w∈F∈

w∈FGw, there existsg∈Gsuch that w(g)=αw, w∈F,

w(g)≥1, w∈W\F. (1.5)

2. Results

Lemma2.1. Let(αw)w∈᏷(W)and letW0= {w∈W:αw=1}. Then(αw)w∈WisW

-complete for anyW0⊆W⊆W.

Proof. Letv∈w∈WW(α_w). Then there existsw∈W0such thatv∈W(αw). Because

(αw,αv) is compatible, we have 1=dwv(αw)=dvw(αv) and it follows thatαv=1. Hence,

v∈W0⊆W.

Proposition2.2. LetGbe apo-group with a quasi-divisor property of finite character and let W be a defining family oft-valuations of G. Let (αw)w∈᏷(W). ThenX= {g ∈G:

(∀w∈W)w(g)≥αw}is a finitely generatedt-ideal ofG.

Proof. Because thet-system is defined by a familyW of t-valuations, according to [8, Theorem 2.6], the group᏷(W) iso-isomorphic to a Lorenzenl-groupΛt(G). It follows

that a mapd:G→᏷(W) such thatd(g)=(w(g))wis a quasi-divisors morphism. Then

for any (αw)w∈᏷(W), there existg1,. . .,gn∈Gsuch thatd(g1)∧ ··· ∧d(gn)=(αw)w.

ThenX=(g1,. . .,gn)t. In fact, forg∈X, we havew(g)≥αw and it follows thatw(g)∈

(w(g1),. . .,w(gn))t. Because thet-system is defined byW, we haveg∈(g1,. . .,gn)t,

analo-gously for the other inclusion.

Corollary2.3. LetGbe a po-group with a quasi-divisor property of finite character and letWbe a defining family oft-valuations ofG. Then there exists ano-isomorphism

σ:᏷(W)−→Ᏽf

t(G) (2.1)

such that for(αw)w∈᏷(W)andJ∈Ᏽtf(G),

σαw_w= g∈G: (∀w∈W)w(g)≥αw, σ−1_(J)=∧

x∈Jw(x)

w

. (2.2)

It is well known that the existence of quasi-divisor property is equivalent to the ex-istence of a defining family ofessentialt-valuations (see [3, Theorem 2.1]). Recall that a t-valuation w of G is essential if kerw is a directed subgroup of G and w is an o -epimorphism.

Lemma2.4. Letw,vbe essentialt-valuations ofGand letα∈Gvbe such thatdvw(α)=1.

Then there existsg∈Gsuch thatw(g)=1,v(g)≥α.

Proof. We may assume thatα >1. LetJ= {x∈G:v(x)≥α}. Let us suppose on contrary that the statement of the lemma is not true. Then for anyx∈J, we havew(x)>1. Let H be the largest convex subgroup inGvsuch thatα∈Hand letw:G→v Gv→Gv/Hbe

the composition ofvand canonical morphism. Thenw≤w. In fact, letx∈G,x≥1 be such thatw(x)>1. Becausew(x)=v(x)H, we havev(x)∈H,v(x)>1. Then there exists n∈Nsuch thatv(x)n_{≥α. In fact, ifv(x)}n_{< αfor alln∈N, then the convex subgroup} H generated byH∪ {v(x)}does not contain α andH⊆H. On the other hand, we havev(x)∈H\H, a contradiction. Thenxn_{∈J for somen∈Nand according to the}

assumption, we havew(x)n_{>1. Hencew(x)>1 and we proved the implication}

Letρ:Gw→Gwbe defined byρ(w(g))=w(g). Thenρis well defined. In fact, letw(x)= w(y). Sincewis essential, there existst∈kerwsuch thatt≥1,xy−1_{. Ifw(x)=w(y), we}

have, for example,w(xy−1_{)>1. Thenw(t)≥w(xy}−1_{)>1. According to (2.3), we have}

w(t)>1, a contradiction witht∈kerw. Thusw=ρ·wandw≤w. Then, we have also w≤w∧v. For anyb∈Gsuch thatα=v(b), we obtainw(b)=v(b)H=αH=1 and v∧w(b)=dvw> v(b)=dvw(α)=1, a contradiction, becausev∧w≥w.

Lemma2.5. Letw1,. . .,wnbe essentialt-valuations ofGand let(α1,. . .,αn)∈ni=1G+wi be

compatible elements. Then there existsa1∈G,a1≥1, such that

∀j=1, w1a1=α1, wja1> αj. (2.4)

Proof. The proof will be done by the induction with respect ton. Forn=1, the proof is trivial. Let us assume that the statement is true for any compatible set ofn−1 elements. Let us assume firstly thatw1< wkfor somek=1. According to the induction assumption,

there existsa∈G+such that

∀j=k, 1, wk(a)=αk, wj(a)> αj. (2.5)

Becausew1< wk, there exists ano-epimorphismσ:Gwk→Gw1 such that w1=σ·wk.

Since (α1,αk) is compatible, we haveσ(αk)=α1. Since kerσ= {1}, there existsδ∈kerσ,

δ >1. From the fact thatwkis essential, it follows that there existsg∈G,g >1, such that wk(g)=δ. We seta1=ga. Then, we have

w1

a1

=σ·wk(ga)=σ(δ)·σ

αk

=α1,

wk

a1

=δ·αk> αk, ∀i=k,i≥2, wi

a1

≥wi(a)> αi.

(2.6)

Let us assume now thatw1wj,j≥2. Thenwj=w1∧wjand for any j≥2, there exists δj∈kerdj1,δj>1. According toLemma 2.4, for anyj≥2, there existsgj∈G+such that

w1(gj)=1,wj(gj)≥δj. We setg1=

j≥2gj. Then

∀j≥2, w1

g1

=1, wj

g1

≥wj

gj

≥δj>1. (2.7)

According to the induction assumption, there existsa1∈G+such that

∀2≤j≤n−1, w1

a1

=α1, wj

a1

> αj. (2.8)

Without the loss of generality, we may assume that

∀2≤j, w1

a1

=α1, wj

a1

In fact, ifwn(a1)< αn, thendn1(αn·w−n1(a1))=d1n(α1)·d1n(w1−1(a1))=1 and according

toLemma 2.4, there existsa1∈G+such thatw1(a1)=1,wn(a1)≥α·wn−1(a1). Then for

a1 =a1a1, we have

w1

a1

=w1

a1a1

=α1,

∀n > j≥2, wj

a1

≥wj

a1

> αj, wn

a1

≥αn.

(2.10)

We setc1=a1g1, wherea1satisfies the relation (2.9). Then we have

w1c1=w1a1=α1,

wj

c1

> wj

a1

≥αj, j≥2.

(2.11)

IfGadmits a quasi-divisor property of finite character, the existence of a map

σ:᏷(W)−→Ᏽf

t(G) (2.12)

follows immediately fromProposition 2.2. Between thel-group of compatible elements ᏷(W) and a semigroupᏵtf(G) of finitely generatedt-ideals ofany po-group G, there

exists another naturally defined map, namely,

τ:Ᏽtf(G)−→᏷(W) (2.13)

such thatτ(Xt)=(∧w(X))w∈W=(∧w(Xt))w∈W∈᏷(W).τis well defined and it can be

proved easily thatτis a semigroup monomorphism (becauset-ideals are defined byW). IfGadmits a quasi-divisor property of finite character, thenσandτare mutually inverse o-isomorphisms (seeCorollary 2.3). Moreover, ifh:G→Ᏽf

t(G) andd:G→᏷(W) are

natural embedding maps such thath(g)=(g)tandd(g)=(w(g))w∈W, then the following

diagram commutes:

Ᏽf t(G)

τ

᏷(W) σ _Ᏽtf₍G)

G h G d G h (2.14)

In the group᏷(W), a group topology᐀Wcan be defined such that kerw= {(αv)v∈

᏷(W) :αw=1}is a subbase of neighborhoods of 1 for anyw∈W(clearly,w:᏷(W)→ Gw is the projection map). Then the semigroup monomorphismτ:Ᏽtf(G)→᏷(W)

in-duces a semigroup topologyᏲW onᏵtf(G). If forw∈W, we define a mapw:Ᏽtf(G)→ Gwsuch thatw(Xt)= ∧w(X)(= ∧w(Xt)), then for any finiteF⊆W, we obtain

τ−1

w∈F

kerw

=

w∈F

Hence, the topologyᏲW can be defined by mapsw,w∈W. Moreover, in the ordered

semigroup (Ᏽtf(G),×t,≤t), whereXt≤tYtifYt⊆Xt, at-ideals structure can be defined

analogously as in any po-group. The following lemma shows that the topologyᏲW is

defined also byt-valuations.

Lemma2.6. For anyw∈W,wis a(t,t)-morphism from(Ᏽtf(G),×t,≤t)toGw.

Proof. Letᐄtbe at-ideal inᏵtf(G) generated by a lower bounded subsetᐄand letXt∈

ᐄt. Then there exists a finite setᏲ⊆ᐄsuch thatXt∈Ᏺt. We setS=Ft∈ᏲF. Then,S

is a finite subset inGandSt≤tFt for anyFt∈Ᏺ. Hence,Xt≥tStand we have∧w(X)= ∧w(Xt)≥ ∧w(St)= ∧w(S). Thusw(Xt)∈(w(St))t=(∧Ft∈Ᏺw(Ft))t=(w(Ᏺ))t.

Theorem 2.7. Let G be defined by a family oft-valuations of finite character. Then the following statements are equivalent.

(1)Gadmits a strong quasi-divisor property.

(2)Gadmits a quasi-divisor property and for any(αw)w∈᏷(W)anda∈Gsuch that αw≤w(a) for allw∈W, there existsb∈G such that αw=w(a)∧w(b)for all w∈W.

(3)Gadmits a quasi-divisor property and for anyXt∈Ᏽtf(G)anda∈Xt, there exists b∈Gsuch thatXt=(a,b)t.

IfWis an infinite set, then these statements are equivalent to the following equivalent state-ments.

(4)Gadmits a quasi-divisor property andh(G)is dense in(Ᏽtf(G),ᏲW).

(5)d(G)is dense in(᏷(W),᐀W).

Proof. (1)⇒(2) Let (αw)w∈᏷(W),a∈Gsuch thatw(a)≥αwfor allw∈W. LetW1=

{w∈W:αw=1} ∪ {v∈W:v(a)=1}. According toLemma 2.1, (αw)w∈W1 is

compati-ble andW1-complete and according to AT, there existsb∈Gsuch that

w(b)=αw, w∈W1,

w(b)≥1, w∈W\W1.

(2.16)

Then forw∈W1, we havew(a)∧w(b)=w(a)∧αw=αw, and forw∈W\W1,w(a)∧

w(b)=1∧w(b)=1=αw.

(2)⇒(3) Leta∈Xt∈Ᏽtf(G). Becauset-system is defined byW, we have Xt= {g∈ G:w(g)≥ ∧w(X),w∈W}. According to [3, Lemma 2.9], (∧w(X))w∈᏷(W) and there

existsb∈Gsuch that∧w(X)=w(a)∧w(b), for allw∈W. Then we haveXt= {g∈G: w(g)∈(w(a),w(b))t,w∈W} =(a,b)t.

(3)⇒(1) We show thatGsatisfies the positive weak approximation theorem (PWAT). Let (α1,. . .,αn)∈ni=1G+wibe compatible. According toLemma 2.5, there exista1,. . .,an∈

G+such that

∀i,∀j=i, wi

ai

=αi, wj

ai

We setb=a1···an. Thenb∈(a1,. . .,an)t. Hence, there existsa∈G+such that (a1,. . .,

an)t=(a,b)t. Then for anyi, we have

wi(b)=αi·

j=i

wi(aj)> αni ≥αi. (2.18)

Let us assume that there existsisuch thatwi(b)< wi(a). Sinceai∈(a,b)t, we haveαi= wi(ai)≥wi(a)∧wi(b)=wi(b), a contradiction. Then we haveαi=wi(ai)≥wi(a)∧wi(b) =wi(a). Sincea∈(a1,. . .,an)t, we havewi(a)≥wi(a1)∧ ··· ∧wi(an)=αi∧j=iwi(aj) =αi. Thuswi(a)=αi,i=1,. . .,nandG satisfies the PWAT. According to [7, Theorem

3.5],Gadmits a strong quasi-divisor property. Now letWbe an infinite set.

(1)⇒(4) SinceGadmits a quasi-divisor property, (Ᏽtf(G),×t) is a group and the

sub-base of neighborhoods of unity in topologyᏲW is{kerw:w∈W}. We show that a map σ:᏷(W)→Ᏽf

t(G) is a homeomorphism. Leta,b∈᏷(W). Then there exista1,. . .,an,b1,

. . .,bm∈Gsuch thata=d(a1)∧ ··· ∧d(an),b=d(b1)∧ ··· ∧d(bm) and we haveσ(a)=

(a1,. . .,an)t,σ(b)=(b1,. . .,bm)t. Thena·b=d(a1b1)∧ ··· ∧d(anbm) and σ(a·b)=

(a1b1,. . .,anbm)t=σ(a)×tσ(b). Ifσ(a)=(1)t, then (a1,. . .,an)t=(1)tand it follows

eas-ily thata=1. It is clear thatσ is also homeomorphism. According to [8, Theorem 2.6], there exists ano-isomorphismψsuch that the following diagram commutes:

Λt(G)

w ψ

᏷(W)

w

Gw Gw

(2.19)

wherewis a canonical extension ofw. SinceG→Λt(G) is a strong quasi-divisor

mor-phism, it follows thatd:G→᏷(W) is a strong quasi-divisor morphism as well. Then, according to [5, Theorem 2.9],d(G) is dense in (᏷(W),᐀W) and it follows thath(G) is

also dense in (Ᏽf

t(G),ᏲW).

(4)⇒(5) IfG admits a quasi-divisor property, thenᏵtf(G) iso-isomorphic toΛt(G)

and according to [8, Theorem 6], it is alsoo-isomorphic to᏷(W). It can be proved easily that (Ᏽtf(G),ᏲW) is also homeomorphic to (᏷(W),᐀W).

(5)⇒(1) It follows directly from [5, Theorem 2.9].

References

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[3] J. Moˇckoˇr,t-valuations and the theory of quasi-divisors, J. Pure Appl. Algebra120(1997), no. 1, 51–65.

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Jiˇr´ı Moˇckoˇr: Department of Mathematics, University of Ostrava, CZ-702 00 Ostrava, Czech Republic

E-mail address:[email protected]

Angeliki Kontolatou: Department of Mathematics, School of Natural Sciences, University of Patras, 26500 Patras, Greece