STRONG AMALGAMATIONS OF LATTICE ORDERED GROUPS AND MODULES

We show that every variety of representable lattice ordered groups fails the strong amalgamation property. The same result holds for the variety of f-modules over an f-ring. However, strong amalgamations do occur for abelian lattice ordered groups or f-modules when the embeddings are convex.

amalgamating two/-groups with a common convex/-subgroup.We show that this is possible in the variety of abelian/-groups even if the amalgamation is required to be strong.A similar result holds for the variety of lattice ordered modules generated by the totally ordered modules.
Let U be a class of /-groups or lattice ordered modules and let F (A, B1,B2,al, a2) be a quintuple with A, B1,B 2 .Uand al A -B and al A --B and a2: A ---, B 2 /-monomorphisms.
Then F is called a V-formation in U.The V-formation F can be amalgamated in U if there exists a triple (C,1,/2) such that C E U, I:B1---C and /2:B2--C are /-monomorphisms, and lCtl 2a2 This is depicted by the following diagram.i///BI 2 C B2.
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 amalgamation of r if it amalgamates F in such a way that/l(b) =/2(b2) implies b E c(A) and b 2 E a2(A).If all V-formations in U can be strongly amalgamated, then U is said to have the strong amalgamation property (StAP).
In our subsequent discussion there are several varieties of/-groups which will play a significant role.These are given distinguished notation as listed below.
L variety of all/-groups; R variety of representable/-groups (defined by (x A /)2 z2 ^/2); and A variety of abelian/-groups (defined by x/= /z).The variety A is the smallest nontrivial variety of/-groups (Weinberg [18]).The representable l- groups are important since they are precisely those/-groups which are subdirect products of totally ordered groups.
Among the lattice ordered modules there is one class which stands out in its significance.This is the class of f-modules, which is the variety generated by all totally ordered modules.This class of modules forms the natural generalization of the important class of vector lattices.In this paper we shall restrict ourselves to lattice ordered modules over rings which are f-rings; i.e., that are subdirect products of totally ordered rings.Given such a ring 5' we let M variety of f-modules over S.
The investigation of the amalgamation property for classes of/-groups was begun by Pierce in [7], [8], and [9].Here he showed among other things that the variety L fails AP while the variety of abehan/-groups satisfies this property.Implicit in his work is a proof that the varieties above and including the non-representable covers of A also fail AP.Subsequently, Powell and Tsinakis showed in [12] and [13] that there exists an uncountable chain of varieties containing R and faihng 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 itself and many varieties contained therein cannot have the amalgamation property.In [15] Powell and Tsinakis extended this result to all representable 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 date no general proof has surfaced to show that AP fails in all nonabehan varieties of/-groups although this result is likely to be true.
For basic information on /-group free products and amalgamations, see Powell and Tsinakis [12], [16], and [17].Background on lattice ordered groups and modules in general can be found in Bigard, Keimel and Wolfenstein [2].The only paper to date investigating free products of fmodules is Cherri and Powell [3], although several papers on free f-modules help introduce the subject (see Bigard  [1], Powell  [10], or Powell and Tsinakis [14]).
2. THE STRONG AMALGAMATION PROPERTY.
We will show in this section that any class of representable/-groups containing/ (the integers) and closed with respect to the formation of/-subgroups and direct products fails StAP.A similar result follows for a class of f-modules containing the ring S and closed with respect to the formation of l-submodules and direct products.Our initial effort will be with/-groups, and we will subsequently point out the analogous proofs for modules.The first step is to relate amalgamations to free products in the given class.General existence theorems guarantee that these structures can be considered in the classes we are examining (see Grtzer [5]).
To avoid repetition of hypotheses we will make some standing definitions here.
i) U is a class of/-groups containing Z and closed with respect to the formation of/-subgroups and direct products; ii) F (A, Bl, al, a2) is a V-formation in U; iii) BI[JB 2 is the free product of B and B 2 in U; iv) AI'B B1LJB2 and A 2" B 2 BI[JB 2 are the natural embeddings; v) N is the/-ideal of BI[JB 2 generated by {Alal(a)A2a2(a)-lla .A }; and vi) " BI[JB 2 -, BI[JB2/N is the natural projection.
The first Lemma puts amalgamations in terms of free products.
LEMMA 1.If r can be strongly amalgamated in U, then it can be strongly amalgamated by the triple (BI[JB2/N,rAI,aA2).
PROOF.Suppose that r can be amalgamated by the triple (C, B1,B2) in U. Then there is a natural map r/-BI[JB 2 C extending the maps/31 and/32.From the relationship BlC O2a2 it is clear that N C_ kerr/.Hence, there exists a map p: BI[JB2/N ---, C such that pa r/.From this it follows that (BI[JB2/N, trAI,trA2) amalgamates r in U.This is depicted in the following commutative diagram.

BIc
"B I. LJB ---g >B LJB/N Now to see that this new amalgamation is indeed strong suppose that rAl(bl)= rA2(b2) for some b E B and b 2 E B 2. Then 31(b) paAl(bl)= pa2(b2) 2(b2).By the properties of C tHs implies that there exists a A such that al(a b d a2(a) b 2.
The preceding mma is MI that is necessy to exine strong Mgations in U.
THEOREM 2. If U R, then U fls StAP.
PROOF.In generM let (x) denote the totMly ordered cyclic group generated by where our notation is [he se h n previously used.By [he nature of N we have 11(,)22(,) -1 e N so l(b)2(b) -1 e N. Now BI[JB 2 is in U R so it is a subdirt pruct of totly ordered oups each of which is so in U. On each of these totly ordered oups, either (b) (b2) or 2(b2) (b) .
The preceding proof also applies to the variety M of f-modules over an f-ring S. The only difference in the proof for the module case is that S is substituted for Z. THEOREM 3. The variety M of ]'-modules over an f-ring S fails the strong amalgamation property.
Vhereas varieties of/-groups have been investigated in some detail with regard to the (general) amalgamation property, the f-modules have not yet received such attention.In Cherri and Powell [3] the free products within such classes are considered in detail.Among other things it is shown there that the special amalgamation property is satisfied by using a representation of the free products.This, together with the congruence extension property, implies that the class M does in fact satisfy the amalgamation property (Gr.tzer and Lakser [6]).
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, we find that strong amalgamations can occur.In fact for the variety A of abelian/-groups it will always be possible.A similar situation holds for the variety M of ]'-modules if the ring S is assumed to be totally ordered and a left Ore domain.These assumptions on S are used in creating the representation of M-free products described below.
The proof of the next theorem will draw on representations of free products in A and M. We describe briefly this process here and refer the reader to Powell and Tsinakis [11] and Cherri and Powell [3] for details.Let G and H be abelian/-groups (or f-modules in M) and consider the sets Pili I} and QjlJ J} of primes of G and H, respectively.For each I and j E J there is at 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 over all I, j J, and appropriate total orders T. Then there is a natural embedding 7: G H --A of the group G H, and G[]H is the sublattice of A generated by 7(G H).That is, GLJH { V k A n 7(xtn) zlm G x H} The terms in the product, that is the terms of the form (G/PixH/Qj, T), are called the components of G[JH.
In the following theorem the notation previously established is continued.THEOREM 4. If F is a V-formation in A or M and if al(A and a2(A are convex in B and B2, respectively, then F can be strongly amalgamated. PROOF.We consider a V-formation in A, leaving the completely analogous proof for M to the reader.We know that an amalgamation of F exists in A since this variety has AP.To see that this amalgamation can be made strong we consider BI[JB2/N and the maps al:B BI[JB2/N and aA2: B 2 BI[JB2/N.Suppose now that b E B and b 2 E B 2 where b al(A).We must show that aAl(bl)# aA2(b2).Since we are dealing with abelian groups, the/-ideal N of BI[JB 2 is just the convex sublattice generated by elements of the form Alal(a)$2o2(a) -1 where a A. Let P be a prime subgroup of B such that al(A C_ P and b P, and let Q be a prime subgroup of B 2 with a2(A C_ Q.Consider the (BLIP x B2/Q,T component in the representation of BI[JB 2 where T is any appropriate total order.Then Alal(a)A2a2(a) -1 is the identity element on this component 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 N and so al(bl) # a2(b2).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.