STRONG AMALGAMATIONS OF LATTICE ORDERED GROUPS AND MODULES

Internat.

VOL. 16 NO. (1993) 75-80

STRONG AMALGAMATIONS OF LATTICE ORDERED GROUPS AND MODULES

MONA CHERRI

Department

ofMathematicsand

Computer

Science Texas

Woman’s

University

Denton, Texas

76204 and

WAYNEB. POWELL

Department

ofMathematics Oklahoma

State

University Stillwater, Oklahoma 74078

(Received

September 25,1990 and inrevisedform

January

21,

1992)

ABSTRACT. We

show that every variety ofrepresentablelattice ordered groups fails the strong amalgamation property. The same result holds for the variety of

f-modules

^{over an f-ring.}

However,

strong amalgamations dooccurfor abelian lattice ordered groupsor

f-modules

^{when the}

embeddingsareconvex.

KEY

WORDS

AND PHRASES.

Amalgamation property, ordered group, ordered module, free product.

1991

AMS SUBJECT CLASSIFICATION CODE.

06F15, 06F25; 08B25.

1.

INTRODUCTION.

In

this paper we consider two variations of the amalgamation property for classes of lattice ordered groups

(/-groups)

^{and lattice} ordered modules. The first of these is the strong amalgamationpropertywhichwewillshowfails inevery variety ofrepresentable/-groupsaswellas in a particular class of lattice ordered modules. Secondly, we investigate the possibility of amalgamating two/-groupswithacommon

convex/-subgroup. We

^showthat thisispossibleinthe variety ofabelian/-groupsevenif theamalgamationisrequired tobestrong.

A

similarresultholds for the variety oflatticeordered modulesgenerated bythetotallyordered modules.

Let

U

be a class of /-groups or lattice ordered modules and let

F (A, B1,B2,al, a2)

^{be a}

quintuplewith

A, B1,B

₂

. ^U

^{and al}

^A - ^B

^andal

^A -- ^B

^{and a2:}

^A

^---,

^B

² /-monomorphisms.

Then

F

is calledaV-formationinU. The V-formation

F

canbeamalgamatedin

U

ifthereexistsa triple

(C,1,/2)

^{such that CE}

^U, I:B1---C

^and

/2:B2--C

^are /-monomorphisms, and

lCtl 2a2

^{This isdepictedbythefollowingdiagram.}

i///BI

²

_C

B2.

²

If every V-formation in U can be amalgamated in

U,

then U is said to have the amalgamation property

(AP).

^{The triple}

(C,/1,/2)

^{is called a} strong amalgamationof

r

ifit amalgamates

F

in suchaway

that/l(b) =/2(b2)

^implies

b

^E

c(A)

^{and b}2E

a2(A ).

If all V-formationsin

U

canbe strongly amalgamated, then

U

issaidtohave thestrong amalgamation property

(StAP).

In oursubsequentdiscussionthere areseveralvarietiesof/-groups which willplay ^asignificant role. Thesearegivendistinguished^notationaslistedbelow.

L

varietyofall/-groups;

R

variety ofrepresentable/-groups

(defined

by

(x

^A

/)2 z2 ^ ^/2);

^and

A

variety ofabelian/-groups

(defined

byx/=

/z).

The variety

A

is the smallest nontrivial varietyof/-groups (Weinberg

[18]).

The representablel- groupsareimportant sincethey areprecisely those/-groupswhichare subdirectproductsoftotally ordered groups.

Among

thelatticeordered modules thereisoneclasswhichstands outin itssignificance. This is the class of

f-modules,

^{which is} the varietygenerated by alltotallyordered modules. Thisclass ofmodules forms the naturalgeneralizationof the important class of vectorlattices.

In

thispaper

we shall restrict ourselves to lattice ordered modules over rings which are f-rings; i.e., that are subdirect productsoftotallyordered rings. Given sucharing5’welet

M

variety of

f-modules

^overS.

The investigation of theamalgamationproperty for classes

of/-groups

wasbegun by Pierce in

[7], [8],

^and

[9].

Herehe showed among otherthingsthat the variety

L

fails

AP

whilethevariety of abehan/-groupssatisfies this property. Implicitinhis workis aproof that thevarieties above and including the non-representable covers of

A

also fail AP. Subsequently, Powell and Tsinakis showedin

[12]

^and

[13]

that there existsanuncountable chain ofvarietiescontaining

R

andfaihng

AP

such that their join is the largest proper variety of/-groups. It was later proved by

Glass,

Saracino, and Wood

[4]

^that the variety

R

itselfandmany varieties contained thereincannothave the amalgamation property.

In [15]

Powell and Tsinakis extended thisresult to allrepresentable varieties containing ^one of the two

solvable,

non-nilpotent covers of A.

Further,

ⁱⁿ

[12]

they showed that the varieties of nilpotent /-groups do not satisfy

AP.

To dateno general proof has surfaced to show that

AP

failsinallnonabehanvarietiesof/-groups althoughthisresultislikely to betrue.

For basic information on /-group free products and amalgamations, see Powell and Tsinakis

[12], [16],

and

[17].

Background on latticeordered groups and modules ingeneral canbefoundin Bigard, Keimel and Wolfenstein

[2].

^The only paper to date investigating free products of

f-

modules is Cherri and Powell

[3],

although several _papers on free

f-modules

^{help introduce the}

subject

(see

Bigard

[1],

Powell

[10],

orPowell and Tsinakis

[14]).

2.

THE

STRONG

AMALGAMATION

PROPERTY.

Wewill showin this sectionthatanyclass ofrepresentable/-groupscontaining/

(the

integers) and closed with respect to theformationof/-subgroups and directproducts fails StAP.

A

similar result follows for a class of

f-modules

containing the ring

S

and closed with respect to the formationofl-submodules and direct products.

Our

initialeffort will bewith/-groups, andwewill subsequentlypoint out theanalogous proofsfor modules. The first step is torelate amalgamations tofree productsinthegivenclass. Generalexistencetheorems guaranteethat these structurescan be considered in the classesweareexamining

(see Grtzer [5]).

Toavoidrepetition of hypotheseswewillmakesomestandingdefinitions here.

AND MODULES 77

i) U

is a class of/-groups containing

Z

and closed with respect to the formationof/-subgroups anddirectproducts;

ii) F (A, Bl,

al,

a2)

^isaV-formation in

U;

iii) BI[JB

2isthe free product of

B

and

B

₂in

U;

iv) AI’B B1LJB

2and

A

_2"

B

₂

BI[JB

_{2 are}^thenaturalembeddings;

v)

^{Nisthe/-idealof}

BI[JB

2generated by

{Alal(a)A2a2(a)-lla . ^A};

^and

vi) " ^BI[JB

^2-,

^BI[JB2/N

^isthe natural projection.

The first

Lemma

puts amalgamationsintermsof freeproducts.

LEMMA

1. If

r

canbe

strongly

amalgamatedin

U,

thenit can bestrongly amalgamated by the triple

(BI[JB2/N,rAI,aA2).

PROOF. Suppose

that

r

canbeamalgamated by the triple

(C, B1,B2)

ⁱⁿ

^U.

^{Then thereis a}

natural map r/-

BI[JB

2 C extending the

maps/31 and/32.

^Fromtherelationship

BlC O2a

2it is clearthatNC_

kerr/. Hence,

^{thereexists amap p:}

BI[JB2/N

^---,Csuch that pa r/. Fromthis it follows that

(BI[JB2/N, trAI,trA2)

amalgamates

r

in

U.

This is depicted in the following commutativediagram.

BI -

^"BI.

^LJB

^{---g >B}

_LJB/N ^c

Now to see that this new amalgamation is indeed strong suppose that

rAl(bl)= rA2(b2)

^for

someb E

B

and b₂E

B

_2. Then

31(b) paAl(bl)= pa2(b2) 2(b2). ^By

the properties of

C tHs

implies that thereexistsa

A

such that

al(a

^{b d}

a2(a)

^b2.

Thepreceding

mma

^is

MI

that is necessy to exinestrongMgationsinU.

THEOREM 2. IfU

R,

then

U

flsStAP.

PROOF. In generM

let

(x)

denote the totMly ordered cyclic group generated by x.

Set A (a),B (bl)

^d

2 ^(b2)"

^Define

1"

A B1

^d2"

A B

₂ by

x(a) b

^d

^2(a) b.

r (A, HI, H2,oI,2)

^c strongly Mga[ed in

U,

then

s

c done by

HI[JB2/N

where our notation is [he se h

n

previously used.

By

^[he nature of

N

we have

11(,)22(,)

^-1

^e

^Nso

l(b)2(b)

^-1

^e

^N.

^Now BI[JB

₂^{is in}

^{U R}

^{so it isasubdirt}

^pruct

oftotly orderedoups each of whichis

so

in

U. On

each of thesetotly orderedoups, either

(b) (b2)

^or

2(b2) (b) .

In

the firstce wehave

l(b1)2(b2)

^{-1 d}

2(b2)-ll(bl)

^{1. Thisimpliesthat}

Al(bl) A2(b2)

^-2

1(bl) [l(bl)2(b2) -1] [2(b2)

^-1

1(bl)]

2(b2)

^-1

1(bl)

Al(bl) -1A2(b2)

On the other hand, if

A2(b2) ^>_ Al(bl) ^>_

^{1, then}

A2(b2)Al(bl)

^-1

^>_

^{1 and}

Al(bl) -1A2(b2) ^>_

^1.

^In

^this

casewehave

Al(bl) -1A2(b2)

²

Al(bl)

^-1

[Al(bl) -1A2(b2)][A2(b2)Al(bl) -1]

> Al(bl) -1A2(b 2)

> 2(2)- (b).

Togetherthese last two inequalitiesmeanthatin

BI[JB

2wehave

.1(bl) A2(b2)

^-2

Al(bl) [Al(bl) A2(b2)

^-2

Al(bl)]

^V

A1(b1)-lA2(b2)

²

Al(bl) -1]

_> A1(bl)-1A2(b2)]

^V

[A2(b2)-lAl(bl)]

> (b) - ^(b).

But

l(bl)22(b2)

^{-2EN and so} by normality

l(bl)2(b2)-2Al(bl)

^N. The preceding inequality together with the convexity of N implies that

2(b2)-ll(bl)N.

^{Finally, we have}

al(bl)=a2(b2)

^while

blC,l(A

^and

b2a2(A),

contradicting the strongness of the amalgamation.

The preceding proof also applies to the variety

M

of

f-modules

^{over an} f-ring S. The only difference intheprooffor the modulecaseis thatSissubstitutedfor Z.

THEOREM 3. The variety

M

of

]’-modules

^{over an}f-ring S fails thestrongamalgamation property.

Vhereasvarietiesof/-groupshave beeninvestigatedinsomedetail withregardto the

(general)

amalgamationproperty, the

f-modules

^{havenotyetreceivedsuch attention.}

^In

^{Cherri andPowell}

[3]

^{the free products within}such classes are consideredin detail.

Among

other things it isshown there that the special amalgamation property ^is satisfied by using a representation of the free products. This, together withthe congruenceextension property, implies that the class

M

doesin factsatisfy the amalgamation property

(Gr.tzer

^{and Lakser}

[6]).

3. CONVEX

AMALGAMATIONS.

In

the proof of Theorem 2, the image of

A

was not convex in

B

or

B

_2. ^{If we} consider V- formations where this is the case, wefind that strong amalgamations can occur.

In

fact for the variety

A

ofabelian/-groups itwillalwaysbepossible.

A

similar situationholdsfor thevariety

M

of

]’-modules

^{if the} ring S is assumed to be totally ordered and a left Ore domain. These assumptionsonSareusedincreating the representation of M-free productsdescribedbelow.

Theproofof the next theorem will drawonrepresentations of freeproducts in

A

andM. We describebriefly this processhere and refer the reader to Powell and Tsinakis

[11]

and Cherri and Powell

[3]

fordetails. Let G and

H

beabelian/-groups

(or f-modules

ⁱⁿ

M)

and consider the sets

Pili ^I}

^and

QjlJ ^J}

of primes ofGand

H,

respectively. For each

I

and j E

J

thereisat least one total order

T

on

G/PiH/Q

_j ^which extends the orders on

G/P

^and

H/Qj.

^Let

A II(G/PiH/Qj, ^T

be the direct product of all such totally ordered groups

(respectively, modules),

where the product is taken overall

I,

j

J,

and appropriate total orders T. Then there is a natural embedding 7:G H

-- ^A

^{ofthe groupG}

^H,

^and

^G[]H

^{is the sublattice of}

^A

generated by

7(G H).

That is,

GLJH ^{

^V_k ^A_n

7(xtn)

zlm Gx

H}

The terms in the product, that is the terms of the form

(G/PixH/Qj, ^T),

^{are called the}

componentsof

G[JH.

Inthefollowing theorem thenotationpreviously establishediscontinued.

THEOREM 4. IfF is aV-formation in

A

or

M

and if

al(A

^and

a2(A

^{are convex inB and}

B2,

respectively, thenFcanbestrongly amalgamated.

PROOF. We consider a V-formation in

A,

leaving the completely analogous prooffor

M

to thereader. Weknow thatanamalgamationof

F

exists in

A

since thisvariety hasAP. Toseethat thisamalgamation can be made strongweconsider

BI[JB2/N

^{and themaps}

al:B ^BI[JB2/N

and

aA2:

^B2

BI[JB2/N. ^Suppose

^{now that b EB and b}2E

B

₂ where b

al(A ).

^{We must}

showthat

aAl(bl)# aA2(b2).

^{Since we aredealing} with abelian groups, ^the/-ideal Nof

BI[JB

2is just theconvexsublatticegenerated byelements of the form

Alal(a)$2o2(a)

^{-1 wherea}

^A.

^Let

^P

bea prime subgroup of

B

suchthat

al(A

^C_

^P

^{and b P, and let}

^Q

^{be aprime subgroup of}

^B

2 with

a2(A

^C_

^Q.

^Considerthe

(BLIP

^x

B2/Q,T

^{component in}the representation of

BI[JB

2where Tisanyappropriatetotalorder. Then

Alal(a)A2a2(a)

^{-1is theidentityelementonthiscomponent}

for any ^a A.

However, Al(bl)A2(b2)

^{-1 is nontrivial on this component since b Po But this}

means that

l(bl)2(b2)

^{-1 Nandso}

al(bl) # a2(b2).

ACKNOWLEDGEMENT. The first author’s research was partially supported by ^a grant from Texas

Woman’s

University. The second author’s research was partially supported by ^a National

S,

cience Foundation

(EPSCOR)

^grant.

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