Amalgamation and extensions of summand absorbing modules over a semiring
aa r X i v : . [ m a t h . R A ] J u l AMALGAMATION AND EXTENSIONSOF SUMMAND ABSORBING MODULESOVER A SEMIRING
ZUR IZHAKIAN AND MANFRED KNEBUSCH
Abstract.
A submodule W of V is summand absorbing, if x + y ∈ W implies x ∈ W, y ∈ W for any x, y ∈ V . Such submodules often appear in modules over (additively) idempotentsemirings, particularly in tropical algebra. This paper studies amalgamation and extensionsof these submodules, and more generally of upper bound modules. Contents
Introduction 11. Exchange equivalence and amalgamation 32. Additivity of exchange equivalences 63. Pairs of SA-submodules with amalgamation 74. Multiple amalgamation 85. Formal properties of multiple exchange equivalence and amalgamation 116. D -complements 137. SA-extensions and their saturations by complementary submodules 148. The complementary modules of a fixed SA-extension 179. SA-extensions with a fixed complementary module 1810. Minimal D -cosets 2011. D -isolated vectors 2312. SA-submodules induced by actions on upper bound modules 2713. Amalgamation in the category of upper bound monoids 29References 32 Introduction
This paper continues the development of module theory over semirings [1, 2], along thelines of classical module theory. Our approach to this theory was introduced in [11] andhas been proceeded in [12, 13], starting with decompositions and generations of particularmodules, termed summand absorbing modules. The present paper focuses on amalgamationsand extensions of these modules.Semirings are extensively involved in recent studies due to increasing interest in tropicalalgebra and its applications to discrete mathematics and automata theory. Although ouroriginal aim was to understand modules in tropical algebra, there are many other important
Mathematics Subject Classification.
Primary 14T05, 16D70, 16Y60 ; Secondary 06F05, 06F25,13C10, 14N05.
Key words and phrases.
Semiring, lacking zero sums, direct sum decomposition, free (semi)module, pro-jective (semi)module, indecomposable, semidirect complement, amalgamation, extension. examples where these modules appear, e.g., additive semigroups, which can be viewed asmodules over the semiring N of natural numbers, or sets of positive elements in an orderedring or a semiring.The underlying property of these modules is lack of zero sums: An R -module V over asemiring R lacks zero sums (abbreviated LZS ) or V is zero-sum-free [2, p.150], if ∀ x, y ∈ V : x + y = 0 ⇒ x = y = 0 . (LZS)LZS is closed for taking submodules, direct sums, direct products, and holds for modules offunctions Fun( S, V ) from a set S to a module V [11, Examples 1.6]. For example, any moduleover an idempotent semiring is LZS [11, Proposition 1.8], establishing a large assortment ofexamples.The notion of LZS leads to the next related type of submodules: A submodule W is of V summand absorbing (abbreviated SA ) in V (termed “strong” in [2, p. 154]), if ∀ x, y ∈ V : x + y ∈ W ⇒ x ∈ W, y ∈ W ; (SA) W is then called an SA-submodule of V . An SA-left ideal of a semiring R is an SA-submodule of R , viewed as R -module by left multiplication. An R -module V is LZS ifand only if { V } is an SA-submodule of V , thereby enhancing interest in SA-submodules.Nevertheless, the notion of SA-submodules itself retains sense for any semiring R and (left) R -modules V .SA-submodules arise in tropical geometry [13, § § V by Mod( V ), and thesubposet consisting of the SA-submodules (= summand absorbing) of V by SA( V ). Moregenerally, given submodules A ⊃ C of V , Mod( A, C ) denotes the set of all submodules B of A containing C and SA( A, C ) (resp. SA V ( A, C )) denotes the set of SA-submodules of A (resp. SA-submodules of V ) containing C , i.e.,SA( A, C ) = SA( A ) ∩ Mod(
A, C ) , SA V ( A, C ) = SA( V ) ∩ Mod(
A, C ) . A collection of submodules A , . . . , A n of V has amalgamation AM V , if the product A × · · · × A n modulo an additive exchange equivalence EC VA ×···× A n injects in A + · · · + A n (Definition 1.2). This amalgamation induces amalgamation of SA-submodules W i ⊂ A i ,with W + · · · + W n ∈ SA( A + · · · + A n ) (Theorems 4.5 and 4.6).We proceed in § x + D of a given submonoid D of V .A D - complement of a submodule W in V is a submodule T , such that W + T = V , W ∩ T = D , and ( W + T ) ∩ T = ∅ (Definition 6.1). When D ∈ SA( W ), the D -complementof a submodule W is unique (Theorem 6.5).A submodule A is an SA-extension of D ⊂ A , if D is summand absorbing in A (Defini-tion 7.1). A submodule T is complementary to A over D , if D is a D -complement of A inthe sum A + T (Definition 7.4). The SA-extension B := [( A \ D ) + T ] ∪ D is the saturation of A by the complementary module T . A is T - saturated , if A = B . Theorem 7.7 links MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 3 these notions: If A an SA-extension of D , and T is complementary to A over D for which A is T -saturated, then the pair ( A, T ) has amalgamation in V .A submodule T ⊂ V is subtractive , if for any t , t ∈ T and x ∈ V with x + t = t , also x ∈ T . Any submodule D ⊂ V has a unique minimal subtractive module T , the subtractivehull of D . Theorem 9.5 lays the connections to SA-extensions: The subtractive hull T ofany submodule D ⊂ V is complementary over D to every SA-extension of D .Given a submodule D of an R -module V , we have a hands a D - quasiordering on V definedas ( x, y ∈ V ) x ≤ D y ⇔ x + d = y for some d ∈ D. The module V is called upper bound , if the relation ≤ V is antisymmetric and so is a(partial) ordering on V . (“upper bound” refers to the fact that then x + y is a built in upperbound of the set { x, y } .)Our results pertain to upper bound R -modules V and include the study of minimality andmaximality with respect to the additive relation (cid:22) D for a given submodule D of V , definedby x (cid:22) D y ⇔ y + D ⊂ x + D, as well as stable sets X , i.e., sets X with X + D ⊂ X .A special attention is dedicated to a class of bipotent additive monoids (Definition 11.2)which can be characterized in terms of contraction maps and convexity (Theorem 11.7).The paper ends with a construction of a hierarchy of families of summand absorbingsubmonoids (so called archimedean classes , cf. Definition 12.6) in a suitably preparedadditive monoid V (Theorem 13.8)), which can be associated to any given additive monoid V – a kind of “resolution” of V (cf. § Exchange equivalence and amalgamation
Given a pair ( A , A ) of submodules of V , we look for an equivalence relation ∼ on theset A × A with the following two properties.(1) ( d, ∼ (0 , d ) for all d ∈ A ∩ A .(2) The relation ∼ is additive , i.e., for pairs ( a , a ), ( a ′ , a ′ ), ( b , b ), ( b ′ , b ′ ) in A × A with ( a , a ) ∼ ( a ′ , a ′ ) , ( b , b ) ∼ ( b ′ , b ′ ) , also ( a + b , a + b ) ∼ ( a ′ + b ′ , a ′ + b ′ ) . We state an immediate consequence of these two properties.(3) Given a , a ′ ∈ A , a , a ′ ∈ A , d ∈ A ∩ A , the following holds.3.a) a = a ′ + d ⇒ ( a , a ) ∼ ( a ′ , a + d ),3.b) a = a ′ + d ⇒ ( a , a ) ∼ ( a + d, a ′ ).Property 3) leads to an explicit construction of such an equivalence relation on A × A ,named “exchange equivalence”.We consider finite sequences of pairs ( a , a ) , ( a ′ , a ′ ) , . . . , ( a ( k )1 , a ( k )2 ) in A × A with con-stant sum a + a = a ′ + a ′ = · · · = a ( k )1 + a ( k )2 . First we deal with such sequences of length k = 2. Z. IZHAKIAN AND M. KNEBUSCH
Definition 1.1.
Given d ∈ A ∩ A a (1,2)-exchange of d (in V ) is a sequence ( a , a ) , ( a ′ , a ′ ) in A × A with a = a ′ + d , a ′ = a + d, while a (2,1)-exchange of d is such a sequence with a ′ = a + d , a = a ′ + d. We denote these such exchanges symbolically by ( a , a ) d −−−−−−→ (1 , ( a ′ , a ′ ) , ( a ′ , a ′ ) d −−−−−−→ (2 , ( a , a ) , respectively. We name these processes basic exchanges . They are very simple. In the case of a(1,2)-exchange we split off from a a summand d ∈ A ∩ A (in A ) and add it to a (in A ). Definition 1.2. a) We call two pairs ( a , a ) , ( b , b ) in A × A exchange equivalent (in V ) if thereexists a finite sequence ( a , a ) , ( a , a ) , . . . , ( a k , a k ) in A × A starting with ( a , a ) = ( a , a ) and ending with ( b , b ) = ( a k , a k ) , in which any two consecutivemembers ( a ,i − , a ,i − ) , ( a ,i , a ,i ) , are either a (1,2)-exchange or a (2,1)-exchange ofsome d i ∈ A ∩ A . We then write ( a , a ) ∼ A × A ( b , b ) . and we call such sequences chains of basic exchanges . b) It is obvious that in this way we obtain an equivalence relation on the set A × A ,which we name “ exchange equivalence ” (in V ), and denote by EC A × A , or moreelaborately by EC VA × A . Clearly EC A × A has the properties (1) and (3) from above. Given two (1,2)-exchanges( a , a ) d −−−→ (1 , ( a ′ , a ′ ) and ( b , b ) e −−−→ (1 , ( b ′ .b ′ )it is plain that ( a + b , a + b ) d + e −−−→ (1 , ( a ′ + b ′ , a ′ + b ′ ) . The same holds for (2,1)-exchanges. It follows by an easy argument, which we defer to § A × A is additive, i.e. has the property (2) from above. Moreover EC A × A s respectsscalar multiplication, i.e., for any λ ∈ R ( a , a ) ∼ A × A ( a ′ , a ′ ) ⇒ ( λa , λa ) ∼ A × A ( λa ′ , λa ′ ) . This holds since when, say, ( a , a ) d −−→ , ( a ′ , a ′ ), then ( λa , λa ) λd −−→ , ( λa ′ , λa ′ ). Summariz-ing these observations we obtain Proposition 1.3.
The exchange equivalence relation EC A × A on the R -module A × A is R -linear. It is the finest additive equivalence relation on A × A with ( d, ∼ (0 , d ) forevery d ∈ A ∩ A . MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 5
We denote the EC A × A -class of a pair ( a , a ) ∈ A × A by [ a , a ] A × A , or [ a , a ] forshort, and the set of all the classes by A ∞ V A . In consequence of Proposition 1.3 we havean obvious structure of an R -module on A ∞ V A by defining[ a , a ] A × A + [ b , b ] A × A = [ a + b , a + b ] A × A ,λ · [ a , a ] A × A = [ λa , λa ] A × A , (1.1)where a , b ∈ A , a , b ∈ A , λ ∈ R. Definition 1.4.
We call the R -module A ∞ V A = A × A / EC A × A the amalgamation of A and A (in V ). We furthermore have a well defined surjective R -module homomorphism π = π A ,A : A ∞ V A − ։ A + A ⊂ V mapping [ a , a ] to a + a , and obtain a natural commuting square A j / / A ∞ V A A ∩ A i / / ?(cid:31) i O O A ?(cid:31) j O O (1.2)of R -module homomorphisms. Here i and i are the inclusion homomorphisms of A ∩ A in A and A , and j , j are given by j ( a ) = [ a , , j ( a ) = [0 , a ] . (1.3)Clearly π j = id A , and π j = id A . Thus A and A embed via j and j into A ∞ V A .In all the following we identify A k with j k ( A k ) ( k = 1 ,
2) and so regard A and A also assubmodules of A ∞ V A . We have A + A = A ∞ V A and the four modules A , A , A ∩ A , A ∞ V A always constitute a pushout diagram in the category of R -modules via thereinclusion homomorphisms in A ∞ V A . Definition 1.5.
We say that a pair ( A , A ) of submodules of V has amalgamation (in V ), abbreviated AM V or AM for short, if the map π A ,A : A ∞ V A −→ A + A ⊂ V is injective, hence bijective. This means that the four R -submodules A , A , A ∩ A , A + A of V constitute a pushoutdiagram. In explicit terms ( A , A ) has AM V if for any pairs ( a , a ), ( b , b ) in A × A ( a , a ) ∼ A × A ( b , b ) ⇔ a + a = b + b . (1.4)If ( A ′ , A ′ ) is a second pair of R -submodules of V with A ′ ⊂ A , A ′ ⊂ A , then it isimmediate from the pushout properties of A ′ ∞ V A ′ and A ∞ V A that there is a unique R -module homomorphism κ A ′ × A ′ ,A × A : A ′ ∞ V A ′ −→ A ∞ V A (1.5)with π A ,A ◦ κ A ′ × A ′ ,A × A = π A ′ ,A ′ . (1.6) Z. IZHAKIAN AND M. KNEBUSCH
It sends an element [ a , a ] A ′ × A ′ to [ a , a ] A × A . In other terms, for any ( a , a ), ( b , b ) in A × A ( a , a ) ∼ A ′ × A ′ ( b , b ) ⇒ ( a , a ) ∼ A × A ( b , b ) . (1.7)This is also immediate from our explicit description of the exchange equivalence relation.The map (1.6) is injective iff the restriction EC A × A | A ′ × A ′ of EC A × A to the subset A ′ × A ′ of A × A coincides with EC A ′ × A ′ . We then write A ′ ∞ V A ′ ⊂ A ∞ V A , regarding A ′ ∞ V A ′ as a submodule of A ∞ V A . Proposition 1.6.
When ( A ′ , A ′ ) and ( A , A ) are pairs of R -submodules of V with A ′ ⊂ A , A ′ ⊂ A , and ( A , A ) has AM V , then ( A ′ , A ′ ) also has AM V iff E A × A | A ′ × A ′ coincideswith E A ′ × A ′ .Proof. We have a commuting square A ′ ∞ V A ′ π ′ (cid:15) (cid:15) κ / / A ∞ V A π (cid:15) (cid:15) A ′ + A ′ (cid:31) (cid:127) / / A + A with π ′ = π A ′ ,A ′ , π = π A ,A , κ = κ A ′ × A ′ ,A × A , and the inclusion map of the submodule A ′ + A ′ of V into A × A . By assumption π is injective. Then π ′ is injective iff κ isinjective. (cid:3) We state some obvious facts about exchange equivalence.
Remark 1.7.
Assume that ( A , A ) is a pair of submodules of V and a , b ∈ A , a , b ∈ A . i) ( a , a ) ∼ A × A ( b , b ) ⇔ ( a , a ) ∼ A × A ( b , b ) .Thus A ∞ V A = A ∞ V A . The pair ( A , A ) has AM V iff ( A , A ) has AM V . ii) If A ⊃ A , then ( a , a ) ∼ A × A ( a + a , , and so ( A , A ) has AM V . iii) Let ϕ : V → V ′ be an R -module homomorphism. Then ( a , a ) ∼ A × A ( b , b ) ⇒ ( ϕ ( a ) , ϕ ( a )) ∼ ϕ ( A ) × ϕ ( A ) ( ϕ ( b ) , ϕ ( b )) . Additivity of exchange equivalences
We verify the additivity of EC A × A asserted in § A × A is given, connecting ( a , a ) to ( b , b ), we cansimplify this chain in various ways. First note that if, say( a , a ) d −−−−−→ , ( a ′ , a ′ ) , ( a ′ , a ′ ) e −−−−−→ , ( a ′′ , a ′′ ) , then ( a , a ) d + e −−→ , ( a ′′ , a ′′ ). Thus we can achieve that in the chain from ( a , a ) to ( b , b ) thebasic exchanges alternate between type (1 ,
2) and (2 , zig-zag .Furthermore, we have the trivial basic exchange ( a , a ) −−→ , ( a , a ) which coincides with( a , a ) −−→ , ( a , a ). By employing trivial basic exchanges, we can achieve that the zig-zag MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 7 from ( a , a ) to ( b , b ) also starts with a basic exchange of type (1 ,
2) and ends with one oftype (2 , normalized zig-zag (an admittedly ad hocnotion). Finally we can increase the length of the normalized zig-zag by 1, adding a trivialbasic exchange, or by 2, adding a chain ( a , a ) d −−→ , ( a ′ , a ′ ) d −−→ , ( a , a ) of length 2 with d ∈ A ∩ A . Proposition 2.1.
Assume that ( a , a ) , ( b , b ) , ( u , u ) , ( w , w ) are pairs in A × A with ( a , a ) ∼ A × A ( b , b ) , ( u , u ) ∼ A × A ( w , w ) . Then ( a + u , a + u ) ∼ A × A ( b + w , b + w ) .Proof. We can choose two normalized zig-zags in A × A of same length connecting ( a , a )to ( b , b ) and ( u , u ) to ( w , w ). By adding these zig-zags in the obvious way we obtain anormalized zig-zag in A × A connecting ( a + u , a + u ) to ( b + w , b + w ). (cid:3) Pairs of SA-submodules with amalgamation
Recall that, assuming A is a submodule of V , a submodule W of A is SA in A if for anytwo elements u, v ∈ A with u + v ∈ W , also u ∈ W and v ∈ W . More generally, then a finitesum a + · · · + a n of elements of A is in W iff every a i ∈ W . We are ready for a central resultof this paper. Theorem 3.1.
Assume that ( A , A ) , ( W , W ) are pairs of submodules of V with W ⊂ A , W ⊂ A and W ∩ A ⊂ W , A ∩ W ⊂ W , and hence W ∩ A = A ∩ W = W ∩ W . Assume furthermore that ( A , A ) has amalgamation in V and that W ∈ SA( A ) , W ∈ SA( A ) . Then ( W , W ) has amalgamation in V .Proof. We will verify that EC A × A | W × W = EC W × W . Then, we know by Proposition 1.6that ( W , W ) has AM V . Due to our explicit description of exchange equivalence in § a , b ∈ W , a , b ∈ W , d ∈ A ∩ A with either( a , a ) d −−−−−→ , ( b , b ) or ( a , a ) d −−−−−→ , ( b , b )in A × A , then these moves are exchanges in W × W . In the first case we have a = b + d , b = a + d. From a ∈ W we conclude that b , d ∈ W . It follows that d ∈ W ∩ A = W ∩ W . Thus( a , a ) d −−→ , ( b , b ) is a (1,2)-exchange in W × W . The second case is settled in the sameway. (cid:3) The question arises whether W + W is again SA in A + A . Theorem 3.2.
Under the assumptions of Theorem 3.1 the following also holds.
Z. IZHAKIAN AND M. KNEBUSCH a) If ( a , a ) ∈ A × A , ( w , w ) ∈ W × W and ( a , a ) ∼ A × A ( w , w ) , then ( a , a ) ∈ W × W . In other words, the subset W × W of A × A is a union of EC A × A -equivalence classes. b) W + W ∈ SA( A + A ) .Proof. a): It suffices to verify that ( a , a ) ∈ W × W in the special cases that( a , a ) d −−−−−→ , ( w , w ) or ( a , a ) d −−−−−→ , ( w , w )due to our description of EC A × A in §
1. In the first case we have w = a + d , a = w + d ,and d ∈ A ∩ A . Since W is SA in A , we conclude that a ∈ W and d ∈ W ∩ ( A ∩ A ) = A ∩ W = W ∩ W , whence a = w + d ∈ W + W ∩ W = W , as desired. In the secondcase w = a + d , a = w + d , and again d ∈ A ∩ A . Since W ∈ SA( A ), we have a ∈ W , d ∈ W ∩ ( A ∩ A ) = W ∩ A = W ∩ W , and so a = w + d ∈ W + W ∩ W = W , asdesired.b): Given u, v ∈ A + A with u + v ∈ W + W , we need to verify that u, v ∈ W + W . Wewrite u = a + a , v = b + b , w = w + w with a , b ∈ A , a , b ∈ A , w ∈ W , w ∈ W .Now ( a + b ) + ( a + b ) = w + w . Since ( A , A ) has AM V , this implies( a + b , a + b ) ∼ A × A ( w , w ) , and thus, as proved above, a + b ∈ W , a + b ∈ W . Since W ∈ SA( A ), W ∈ SA( A ), we conclude that a , b ∈ W and a , b ∈ W . Thus u = a + a ∈ W + W and v = b + b ∈ W + W . (cid:3) Corollary 3.3.
Assume that ( A , A ) has AM V and that A ∩ A ∈ SA( A ) . Then A ∈ SA( A + A ) .Proof. Apply Theorem 3.2.b with W = A , W = A ∩ A . (cid:3) Multiple amalgamation
We expand the exchange equivalence relation on A × A for pairs ( A , A ) in Mod( V ) toan “exchange equivalence“ on A × · · · × A n for n -tuples ( A , . . . , A n ) in Mod( V ), n ≥ Definition 4.1.
Let ( a , . . . , a n ) and ( b , . . . , b n ) be tuples in A × · · · × A n . a) We say that n -tuples ( a , . . . , a n ) and ( b , . . . , b n ) are binary exchange equivalent (in A × · · · × A n ) , if there exist i, j ∈ { , . . . , n } , i = j , such that a k = b k for k = i, j and ( a i , a j ) ∼ A i × A j ( b i , b j ) , (as defined in § ( i, j ) -(exchange)-equivalent, and write ( a , . . . , a n ) i,j ∼ ( b , . . . , b n ) , or more elaborately ( a , . . . , a n ) i,j ^ A ×···× A n ( b , . . . , b n ) . MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 9 b) We say that ( a , . . . , a n ) and ( b . . . . , b n ) are exchange equivalent (in A ×· · ·× A n ) ,and write ( a , . . . , a n ) ∼ A ×···× A n ( b , . . . , b n ) , if there is a finite chain of tuples in A × · · · × A n starting with ( a , . . . , a n ) andending with ( b , . . . , b n ) , such that any two consecutive members are binary exchangeequivalent.This is clearly an equivalence relation on the set A × · · · × A n . We denote it by EC VA ×···× A n or EC A ×···× A n for short. Proposition 4.2. EC A ×···× A n is additive. In other words, when tuples ( a , . . . , a n ) , ( b , . . . , b n ) , ( u , . . . , u n ) , ( w , . . . , w n ) in A × · · · × A n are given with ( a , . . . , a n ) ∼ A ×···× A n ( b , . . . , b n ) and ( u , . . . , u n ) ∼ A ×···× A n ( w , . . . , w n ) , then ( a + u , . . . , a n + u n ) ∼ A ×···× A n ( b + w , . . . , b n + w n ) . Proof.
We pick chains of binary equivalences in A × · · · × A n from ( a , . . . , a n ) to ( b , . . . , b n )and from ( u , . . . , u n ) to ( w , . . . , w n ). For any ( x , . . . , x n ) ∈ A × · · · × A n we have thetrivial equivalence ( x , . . . , x n ) ∼ ( x , . . . , x n ), which is a binary ( i, j )-equivalence for anytwo different i, j ∈ { , . . . , n } . Inserting trivial equivalences in both chains, we refine themto chains of some length with ( i, j )-exchanges at the same places. Adding the refined chainsin A × · · · × A n , we obtain a chain of binary equivalences from ( a + u , . . . , a n + u n ) to( b + w , . . . , b n + w n ), as desired. (cid:3) It is evident from the case n = 2 that ( a , . . . , a n ) i,j ∼ ( b , . . . , b n ) implies ( λa , . . . , λa n ) i,j ∼ ( λb , . . . , λb n ) for any λ ∈ R . We conclude that EC A ×···× A n is an R -linear equivalencerelation on the R -module A × · · · × A n . The following is now obvious, as in the case n = 2(Proposition 1.3), and tells us that we have been on the right track. Theorem 4.3. EC A ×···× A n is the finest additive equivalence relation on A × · · · × A n withthe property that for any (different) i, j ∈ { , . . . , n } and d ∈ A i ∩ A j (0 , . . . , , d i , , . . . , ∼ (0 , . . . , , d j , , . . . , . This relation is R -linear. We denote the EC A ×···× A n -equivalence class of a tuple ( a , . . . , a n ) ∈ A × · · · × A n by[ a , . . . , a n ] A ×···× A n , or [ a , . . . , a n ] for short. The rules for addition and multiplication byscalars λ ∈ R for n = 2 (cf. §
1) generalize in the obvious way, and establish the structure ofan R -module on the set A × · · · × A n / EC ∨ A ×···× A n of equivalence classes, which we call againthe amalgamation of ( A , . . . , A n ) (in V ). We denote this R -module by A ∞ V · · · ∞ V A n .As in the case n = 2 we have a well defined R -linear surjection κ A ×···× A n : A ∞ · · · ∞ A n − ։ A + · · · + A n ⊂ V (4.1)mapping [ a , . . . , a n ] to a + · · · + a n and R -linear injections j k : A k −→ A ∞ · · · ∞ A n When the ambient module V of the A i is fixed, we often omit the suffix “ V ”. mapping a ∈ A k to [0 , . . . , , a, , . . . , a at the place k . It allows us to identify A k with the submodule j k ( A k ) of A ∞ · · · ∞ A n whenever this is appropriate. Note that A ∞ · · · ∞ A n = n X k =1 j k ( A k ) . (4.2) Definition 4.4.
As in the case n = 2 , we say that the sequence ( A , . . . , A n ) has amalga-mation in V , abbreviated AM V , if κ A ,...,A n is injective, and so is an R -module isomorphismfrom A ∞ V · · · ∞ V A n to A + · · · + A n ⊂ V . Intuitively we regard the map κ A ,...,A n : A ∞ . . . ∞ A n − ։ A + · · · + A n as a kind of “resolution” of the submodule V := A + · · · + A n of V with respect to thefamily ( A , . . . , A n ) of modules generating V . Theorem 4.5.
Assume that ( A , . . . , A n ) and ( W , . . . , W n ) are n -tuples of submodules of V with W k ∈ SA( A k ) for k = 1 , . . . , n and W i ∩ A j ⊂ W j , whence W i ∩ A j = W i ∩ W j forany two (different) i, j ∈ { , . . . , n } . Then, if ( A , . . . , A n ) has AM V , also ( W , . . . , W n ) has AM V .Proof. Follows from the Definition 4.1 of exchange equivalence and Theorem 3.1. (cid:3)
Theorem 4.6.
Under the assumption of Theorem 4.5 on ( A , . . . , A n ) and ( W , . . . , W n ) thefollowing holds. a) If the tuples of vectors ( a , . . . , a n ) ∈ A × · · · × A n , ( w , . . . , w n ) ∈ W × · · · × W n ,are exchange equivalent in A × · · · × A n , then ( a , . . . , a n ) ∈ W × · · · × W n , andboth tuples are exchange equivalence in W × · · · × W n . b) W + · · · + W n ∈ SA( A + · · · + A n ) .Proof. a): We know by Theorem 3.2.a that, if ( a , . . . , a n ) i,j ∼ ( w , . . . , w n ) in A × · · · × A n ,then ( a , . . . , a n ) ∈ A × · · · × A n and both tuples are ( i, j )-equivalent in W × · · · × W n .Thus any chain of binary exchanges in A × · · · × A n , which meets W × · · · × W n , is a chainof binary exchanges in W × · · · × W n .b): The argument used in the proof of Theorem 3.2.b works also here with the obviouschanges. (cid:3) We clarify the contents of Theorems 4.5 and 4.6 in a special case.
Corollary 4.7.
Assume that A , . . . , A n are submodules of V and that W is an SA-submoduleof A + · · · + A n . a) If ( A , . . . , A n ) has amalgamation in V , then ( W ∩ A , . . . , W ∩ A n ) has amalgamationin V . b) Two n -tuples ( a , . . . , a n ) , ( b , . . . , b n ) of ( W ∩ A ) × · · · × ( W ∩ A n ) are exchangeequivalent in A × · · · × A n iff these tuples are exchange equivalent in ( W ∩ A ) ×· · · × ( W ∩ A n ) . c) ( W ∩ A ) ×· · ·× ( W ∩ A n ) is a union of E A ×···× A n equivalence classes of A ×· · ·× A n .Proof. Apply Theorems 4.5 and 4.6 to ( A , . . . , A n ) and W i := W ∩ A i (1 ≤ i ≤ n ). (cid:3) MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 11
For later use we restate Theorem 4.3 more categorically. This is possible, since dividingout an R -linear equivalence relation on an R -module X means to establish a surjectivehomomorphism ϕ : X ։ Z , which is unique up to composition with an isomorphism δ : Z ∼ −→ Z ′ replacing ϕ by δ ◦ ϕ . Definition 4.8.
Let A , . . . , A n ∈ Mod( V ) . The amalgamation diagram of ( A , . . . , A n ) (in V ) is the diagram of R -module homomorphisms consisting of all inclusion maps A i ∩ A j ֒ → A i for different i, j ∈ [1 , n ] (in V ). Scholium 4.9.
A tuple ( A , . . . , A n ) has amalgamation in V iff the amalgamation diagramof ( A , . . . , A n ) in V admits the submodule A + · · · + A n of V as colimit with the inclusionmaps A i ֒ → A + · · · + A n (1 ≤ i ≤ n ) as morphisms. More explicitly, given R -linear maps α i : A i → Z (1 ≤ i ≤ n ) into some R -module Z with α i | A i ∩ A j = α j | A i ∩ A j for i = j , thereexists a (unique) R -linear map γ : A + · · · + A n → Z such that γ | A i = α i for ≤ i ≤ n . Formal properties of multiple exchange equivalence and amalgamation
The following fact is trivial but useful.
Proposition 5.1.
Let A , . . . , A n ∈ Mod( V ) , n ≥ , be given and let σ be a permutation of [1 , n ] := { , . . . , n } . a) If ( a , . . . , a n ) , ( b , . . . , b n ) are n -tuples in A × · · · × A n , then ( a , . . . , a n ) ∼ A ×···× A n ( b , . . . , b n ) iff ( a σ (1) , . . . , a σ ( n ) ) ∼ A σ (1) ×···× A σ ( n ) ( b σ (1) , . . . , b σ ( n ) ) . b) The tuple ( A , . . . , A n ) has amalgamation in V iff A σ (1) ×· · ·× A σ ( n ) has amalgamationin V .Proof. Evident, since the amalgamation diagram of ( A σ (1) , . . . , A σ ( n ) ) arises from the amal-gamation diagram of ( A , . . . , A n ) simply by relabeling the A i using σ . (cid:3) We briefly discuss the functorial behaviour of exchange equivalence and amalgamationunder a module homomorphism.
Proposition 5.2.
Assume that ϕ is a linear map from an R -module V ′ to V , and that A , . . . , A n are submodules of V . a) If ( a ′ , . . . , a ′ n ) and ( b ′ , . . . , b ′ n ) are tuples in ϕ − ( A ) × · · · × ϕ − ( A n ) with ( a ′ , . . . , a ′ n ) ∼ ϕ − ( A ) ×···× ϕ − ( A n ) ( b ′ , . . . , b ′ n ) , (5.1) then ( ϕ ( a ′ ) , . . . , ϕ ( a ′ n )) ∼ A ×···× A n ( ϕ ( b ′ ) , . . . , ϕ ( b ′ n )) . (5.2)b) Assume that ϕ is injective on ϕ − ( A + · · · + A n ) , then (5.2) implies (5.1) . Then ( ϕ − ( A ) , . . . , ϕ − ( A n )) has amalgamation in V ′ iff ( A , . . . , A n ) has amalgamationin V .Proof. a): Evident from the description of the exchange equivalence relation in Definitions 1.2and 4.1.b): Without loss of generality we may replace the R -modules V and V ′ by A + · · · + A n and ϕ − ( A + · · · + A n ). Now ϕ is an isomorphism from V ′ onto V and ϕ − ( A + · · · + A n ) = ϕ − ( A ) + · · · + ϕ − ( A n ), whence ϕ provides an isomorphism between the amalgamationdiagrams of ( ϕ − ( A ) , . . . , ϕ − ( A n )) in V ′ and ( A , . . . , A n ) in V . Due to step a), applied to ϕ and ϕ − , it is now obvious that for fixed tuples ( a ′ , . . . , a ′ n ) and ( b ′ , . . . , b ′ n ) in ϕ − ( A ) ×· · ·× ϕ − ( A n ) the assertions (5.1) and (5.2) are equivalent. We furthermore have a commutingsquare ϕ − ( A ) ∞ · · · ∞ ϕ − ( A n ) ϕ ∞···∞ ϕ n (cid:15) (cid:15) κ ′ / / V ′ ϕ (cid:15) (cid:15) A ∞ · · · ∞ A n / / V with κ ′ := κ ϕ − ( A ) ×···× ϕ − ( A n ) , κ := κ A × · · · × κ A n (cf. (4.2)) and ϕ ∞ · · · ∞ ϕ n theisomorphism from ϕ − ( A ) ∞ · · · ∞ ϕ − ( A n ) to A ∞ · · · ∞ A n induced by the isomorphisms ϕ i : ϕ − ( A i ) ∼ −→ A i obtained from ϕ by restriction. Since the vertical arrows in this diagramare isomorphisms, the map κ ′ is an isomorphism iff κ is an isomorphism. This means that( ϕ − ( A ) , . . . , ϕ − ( A n )) has amalgamation in V ′ iff ( A , . . . , A n ) has amalgamation in V . (cid:3) We now delve deeper into the theory of amalgamation.
Theorem 5.3.
Let ( A , . . . , A n ) be an n -tuple of submodules of V and A a submodule of A . a) If ( A , . . . , A n ) has AM V , then ( A , A , . . . , A n ) has AM V . b) Given tuples ( a , a , . . . , a n ) , ( b , b , . . . , b n ) in A × · · · × A n we have ( a , a , . . . , a n ) ∼ A ×···× A n ( b , b , . . . , b n ) ⇔ ( a + a , a , . . . , a n ) ∼ A ×···× A n ( b + b , b , . . . , b n ) . Proof. a): Suppose R -module homomorphisms α i : A i → Z (0 ≤ i ≤ n ) are given with α i | A i ∩ A j = α j | A i ∩ A j (0 ≤ i, j ≤ n ). Then (exploiting this for 1 ≤ i, j ≤ n ) wehave a (unique) R -module homomorphism γ : A + · · · + A n → Z with γ | A i = α i for1 ≤ i ≤ n . Since A ∩ A = A we have α = α | A . Since A ∩ A i ⊂ A i for i ≥
2, we have α | A ∩ A i = α | A ∩ A i = α i | A ∩ A i . Thus A + A + · · · + A n = A + · · · + A n is thecolimit of the amalgamation diagram of ( A , A , . . . , A n ) in V (with the obvious inclusionmorphisms).b): ( A , . . . , A n ) has AM in A ∞ · · · ∞ A n , so ( A , A , . . . , A n ) also has AM in A ∞ · · · ∞ A n ,and A ∞ A ∞ · · · ∞ A n = A ∞ · · · ∞ A n . Now b) is obvious, since the inclusion mapsof the A i into A ∞ · · · ∞ A n = A ∞ · · · ∞ A n give us E A ×···× A n and E A × A ×···× A n , and( a , a , . . . , a n ), ( b , b , . . . , b n ) have the same images as ( a + a , a , . . . , a n ), ( b + b , b , . . . , b n )respectively in A ∞ · · · ∞ A n .In short, b) follows from a) by working in A ∞ · · · ∞ A n instead of A + · · · + A n . (cid:3) Given a sequence A , . . . , A n of submodules of V , as before, we define for any set J ⊂ [1 , n ] A J := X i ∈ J A i . (5.3) Theorem 5.4 (Contraction Theorem) . Assume that ( A , . . . , A n ) has amalgamation in V ,and that [1 , n ] = J ˙ ∪ · · · ˙ ∪ J m (5.4) is a partition of [1 , n ] into m (disjoint) subsets. Then ( A J , . . . , A J m ) has amalgamation in V .Proof. In view of Proposition 5.1, we assume without loss of generality that the J i areconsecutive subintervals [1 , r ] , [ r + 1 , r ] , . . . , [ r m − + 1 , r m ] with 1 < r < r < · · ·
Corollary 5.5.
Given submodules A , A , . . . , A n of V , assume that ( A , A , . . . , A n ) has AM V . Then ( A + A , A + A , . . . , A + A n ) has AM V .Proof. By iterated application of Theorem 5.3 (and using the trivial Proposition 5.1) weinfer that the 2n-tuple ( A , A , A , A , . . . , A , A n ) has AM V . It follows by the ContractionTheorem 5.4, that ( A + A , A + A , . . . , A + A n ) has AM V . (cid:3) D -complements In the papers [11] and [12] our main focus has been on the lattice SA( V ) of all SA-submodules of a module V over a semiring R . As a counterpart to this we study for any SA-submodule D of V in § A ⊃ D of V . Here the amalgamationtheory will come into play.In preparation for this endeavor we present general results about submodules of V . Definition 6.1.
Let D ⊂ W be submodules of an R -module V . We call a submodule T of V a D - complement of W in V , if W + T = V , W ∩ T = D , and ( w + T ) ∩ T = ∅ for every w ∈ W \ D . In the case of D = { } these are the weak complements studied in [11]. We generalizepart of the theory of weak complements in [11] to D -complements. Proposition 6.2.
Assume that T is an SA -submodule of V . Then every submodule W of V with W + T = V has T as a D -complement in V for D := W ∩ T . (N.B.: D ∈ SA( W ) ).Proof. Let w ∈ W \ D and t ∈ T . Then w + t / ∈ T since w + t ∈ T would imply that w ∈ W ∩ T = D . This proves that ( w + T ) ∩ T = ∅ . (cid:3) In the following, through Theorem 6.5, D and W are submodules of V with D ⊂ W . Proposition 6.3.
Assume that T is a D -complement of W in V , and that D ∈ SA( W ) .Then T ∈ SA( V ) . Proof.
Let v , v ∈ V with v + v ∈ T . Write v = w + t , v = w + t ( w i ∈ W , t i ∈ T ).Then ( w + w ) + ( t + t ) ∈ T. This forces w + w ∈ D , since otherwise ( w + w + T ) ∩ T = ∅ . We conclude that w ∈ D ⊂ T ,and so w i + t i ∈ T for i = 1 , (cid:3) Proposition 6.4.
Assume that D ∈ SA( W ) , that T is a D -complement of W in V , andthat U is a submodule of V with W + U = V . Then T ⊂ W .Proof. Let t ∈ T . Write t = w + u with w ∈ W , u ∈ U . We infer from Proposition 6.3 that T ∈ SA( V ), and conclude that w ∈ T ∩ W = D ⊂ U . Thus t ∈ U . (cid:3) Theorem 6.5.
Assume that T and U are D -complements of W in V and that D ∈ SA( W ) .Then T = U .Proof. By Proposition 6.4, T ⊂ U and U ⊂ T . (cid:3) Note that up to this point no amalgamation hypothesis has been used.
Remark 6.6. If ( A , A ) is a pair of submodules of V with amalgamation in V , where D := A ∩ A is SA in A , then Corollary 3.3 states that A is a D -complement of A in A + A , the unique one by Theorem 6.5, and moreover A is SA in A + A (which is alsoclear by Proposition 6.4). If also D ∈ SA( A ) , it follows that D ∈ SA( A + A ) , and that A is a D -complement of A in A + A , which again is unique. A + A A ❥❥❥❥❥❥❥❥❥ SA A D ✐✐✐✐✐✐✐✐✐✐✐✐ SA-extensions and their saturations by complementary submodules
As before, V is a module over a semiring R . Definition 7.1. An SA - extension in V is a pair of submodules ( A, D ) of V where D ⊂ A and D is SA in A . We then also say that A is an SA - extension of D . We state some easy facts about these extensions.
Remark 7.2. a) Given submodules D ⊂ A ⊂ B of V , the following holds: If ( B, D ) is an SA-extension, then ( A, D ) is an SA-extension. If both ( A, D ) and ( B, A ) are SA-extensions,then ( B, D ) is an SA-extension. b) Assume that D is a submodule of V and ( A α | α ∈ I ) is a family of SA-extensionsof D such that for any α, β ∈ I there exists γ ∈ I with A α ⊂ A γ , A β ⊂ A γ . Then S α ∈ I A α is an SA-extension of D . c) It follows by Zorn’s Lemma that for any given SA-extension A of D there exists amaximal SA-extension C of D with A ⊂ C . MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 15
We characterize SA-extensions in the following intrinsic way.
Proposition 7.3.
Let D ⊂ A be submodules of A . Then A is an SA-extension of D iff A \ D is closed under addition and ( A \ D ) + D ⊂ A \ D . In this case ( A \ D ) + D = A \ D .Proof. D is SA in A iff ∀ a , a ∈ A : a + a ∈ D ⇒ a ∈ D, a ∈ D. Since both subsets A and D of V are closed under addition, this condition can be rewrittenas ( A \ D ) + ( A \ D ) ⊂ A \ D, ( A \ D ) + D ⊂ A \ D. Furthermore ( A \ D ) + D = A \ D , since 0 ∈ D . (cid:3) We slightly extend the definition of a D -complement in V (Definition 6.1) as follows. Definition 7.4.
Given submodules D ⊂ A in V , we call a submodule T of V complemen-tary to A over D , if D is a D -complement of A in A + T , i.e., A ∩ T = D, [( A \ D ) + T ] ∩ T = ∅ . (7.1)This situation will be studied in this and the following sections in the case that A is anSA-extension of D . By Remark 6.6, A + T is an SA-extension of T , and T is uniquelydetermined by the submodules A and A + T of V .We can enlarge an SA-extension without changing a given complementary module asfollows. Theorem 7.5.
Assume that D ⊂ T are submodules of V and that A is an SA-extension of D with complementary module T . Then B := [( A \ D ) + T ] ∪ D is also such an SA-extension,and B + T = A + T .Proof. We have A ∩ T = D and [( A \ D ) + T ] ∩ T = ∅ . Since D ⊂ T , this implies[( A \ D ) + T ] ∩ D = ∅ , whence B \ D = ( A \ D ) + T . This set is closed under addition, since A \ D is closed under addition. Furthermore B + T = [( A \ D ) + T ] ∪ ( D + T )= [( A \ D ) + T ] ∪ T = A + T, and ( B \ D ) + D = ( A \ D ) + T + D = ( A \ D ) + T = B \ D. We verify that for given z ∈ B , λ ∈ R , also λz ∈ B .If z ∈ D , then λz ∈ D ⊂ B . Otherwise z = x + t with x ∈ A \ D , t ∈ T . Now λz = λx + λt . We have λx ∈ A ⊂ B and λt ∈ T ⊂ B , whence again λz ∈ B . Thus B isan R -submodule of V . Since B \ D is closed under addition and ( B \ D ) + D = B \ D , weconclude by Proposition 7.3 that B is an SA-extension of D . We have [( B \ D ) + T ] ∩ T =[( A \ D ) + T ] ∩ T = ∅ , and so T is complementary to B over D . (cid:3) Note that B is the smallest submonoid of ( V, +) containing A , such that B \ D is a unionof cosets x + T of T in V . Definition 7.6. a) We call the SA-extension B := [( A \ D ) + T ] ∪ D the saturation of A by thecomplementary module T . b) If B = A , i.e., ( A \ D ) + T = A \ D , we say that A is T - saturated (or saturatedw.r.t. T ). Our interest in saturated SA-extensions is due to the following fact.
Theorem 7.7.
Assume that ( A, D ) is an SA-extension in V , and that T is a complementarymodule to A over D , for which A is T -saturated. Then the pair ( A, T ) has amalgamationin V .Proof. Given additive maps α : A → Z , β : T → Z into an R -module Z with α | D = β | D weneed to establish an additive map γ from A + T = ( A \ D ) ˙ ∪ T to Z with γ | A = α , γ | T = β .We define a map γ from the set ( A \ D ) ˙ ∪ T to Z by the rule γ ( a ) = α ( a ) for a ∈ A \ D,γ ( t ) = β ( t ) for t ∈ T. (7.2)Then γ | T = β , γ | A \ D = α | A \ D , γ | D = β | D = α | D . Given x = a + t , x = a + t with a , a ∈ A \ D , t , t ∈ T ,we have x + x = ( a + a ) + ( t + t ). Since A \ D is closedunder addition, the summand a + a is in A \ D , while t + t is in T . Thus γ ( x + x ) = α ( a + a ) + β ( t + t )= α ( a ) + α ( a ) + β ( t ) + β ( t )= [ α ( a ) + β ( t )] + [ α ( a ) + β ( t )]= γ ( x ) + γ ( x ) . If x = a + t and x = t with a ∈ A \ D , t ∈ T , t ∈ T , then x + x = a + ( t + t ),and so γ ( x + x ) = α ( a ) + β ( t + t )= α ( a ) + β ( t ) + β ( t )= γ ( x ) + γ ( x )again. If x , x ∈ T , then γ ( x + x ) = γ ( x ) + γ ( x ), since γ | T = β is additive. Thus γ isindeed additive. (cid:3) Remark 7.8. If ( A, D ) is an SA-extension in V and T is complementary to A over D ,then any submodule T ′ of V with D ⊂ T ′ ⊂ T is again complementary to A over D , since [( A \ D ) + T ] ∩ T = ∅ implies [( A \ D ) + T ′ ] ∩ T ′ = ∅ . If moreover ( A, D ) is T -saturated,then ( A, D ) is T ′ -saturated, since ( A \ D ) + T = A \ D implies ( A \ D ) + T ′ = A \ D . Thus ( A, T ′ ) has AM V . The latter fact is noteworthy, since in general, when ( A , A ) is pair of submodules of V with AM V , and A ′ is a submodule of A containing A ∩ A , the pair ( A , A ′ ) can fail tohave AM V .In good cases it is possible to desend the results above for SA-extensions of D in V toSA-extensions of { V } in V . Theorem 7.9.
Assume that every element of the semiring R is a sum of units of R (e.g. R = N ). Assume furthermore that ( A, D ) is an SA -extension in V and T is a complementarymodule of A over D . Then the subset A := ( A \ D ) ∪ { } MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 17 of V is again a submodule of V and ( A , { } ) is an SA -extension of { } in V with com-plementary module T . The SA -extension ( A , { } ) is T -saturated iff ( A, D ) is T -saturated.Then the pair ( A , T ) has AM V .Proof. We have ( A \ { } ) = A \ D , and so the set A \ { } is closed under addition in V .Let λ ∈ R be given. Then λ = P i ∈ I λ i with finitely many λ i ∈ R ∗ . (We admit I = ∅ . Thenread λ = 0.) For every i ∈ I the map x λ i x is an automorphism of the monoid ( A, +)which restricts to an automorphism of ( D, +). Thus λ i ( A \ D ) = A \ D . We conclude that λ i A = A for every i ∈ I , whence λA ⊂ A . Thus A is an R -submodule of V .Since T is complementary to A over D , i.e., A ∩ T = D, [( A \ D ) + T ] ∩ T = ∅ , we have( A \ { } ) + T = ( A \ D ) + T, ( ∗ )and so ( A \ { } + T ) ∩ T = ∅ . Furthermore A + T = [( A \ { } ) ∪ { } ] + T = [( A \ { } ) + T ] ∪ ( { } + T )= [( A \ D ) + T ] ∪ T = A + T, and A ∩ T = [( A \ { } ) ∩ T ] ∪ ( { } ∩ T ) = [( A \ D ) ∩ T ] ∪ { } = { } . We conclude from ( ∗ ) that A is T -saturated iff A is T -saturated. Then ( A , T ) has AM V by Theorem 7.7. (cid:3) The complementary modules of a fixed SA-extension
We want to get a hold on the set of submodules of V which are complementary to a givenSA-extension in V . In the beginning we work without an SA-assumption. Proposition 8.1.
Let D ⊂ A be submodules of V , and assume that T is a complementarymodule to A over D in V . Further assume that U ′ is a submodule of U := A + T containing D .Then T ′ := { x ∈ T | A + Rx ⊂ U ′ } (8.1) is a submodule of V which is again complementary to A over D , and A + T ′ = U ′ .Proof. If x , x ∈ V and A + Rx ⊂ U ′ , A + Rx ⊂ U ′ , then for any λ , λ ∈ RA + R ( λ x + λ x ) ⊂ A + Rx + Rx ⊂ U ′ , and so λ x + λ x ∈ T ′ . Thus T ′ is an R -submodule of V with T ′ ⊂ T . Clearly A + T ′ ⊂ U ′ .Given x ∈ U ′ , we have A + Rx ⊂ U ′ , and so x ∈ T ′ . Thus U ′ ⊂ A + T ′ . This proves A + T ′ = U ′ . Since D ⊂ U ′ , also D ⊂ T ′ , and so D ⊂ A ∩ T ′ ⊂ A ∩ T = D , which provesthat A ∩ T ′ = D . We have ( A \ D ) + T ′ ⊂ ( A \ D ) + T , and this is disjoint from T , all themore from T ′ . Thus T ′ is complementary to A over D . (cid:3) Remark 8.2. a) It is now obvious from (7.2) that T ′ is the largest submodule F of T with A + F ⊂ U ′ . b) In the case R = N we have the following alternative description of T ′ . T ′ = { x ∈ T | A + x ⊂ U ′ } . (8.2) To verify this, let Φ denote the set on the right hand side. Clearly ∈ Φ . If x , x areelements of T with A + x ⊂ U ′ , A + x ⊂ U ′ , then A + x + x = A + A + x + x ⊂ U ′ + U ′ = U ′ . Thus Φ is an N -submodule of T with A +Φ ⊂ U ′ . It is clear from (8.1) that T ′ ⊂ Φ . We conclude by a) that T ′ = Φ . Proposition 8.1 leads us to the following picture. Given submodules D ⊂ A of V we definetwo sets of submodules of V .Compl ′ D ( A ) := { T ∈ Mod(
V, D ) | A ∩ T = D, [( A \ T ) + T ] ∩ T = ∅} Compl ′′ D ( A ) := { U ∈ Mod(
V, A ) | A has a D -complement in U } We regard these sets as subposets of Mod(
V, D ) and Mod(
V, A ) respectively. It is evident thatCompl ′ D ( A ) is a lower set in Mod( V, D ), and it follows from Proposition 8.1 that Compl ′′ D ( A )is a lower set in Mod( V, A ). We have a surjective map T A + T from Compl ′ D ( A ) toCompl ′′ D ( A ), which respects the partial orderings of these sets.Assume now that ( A, D ) is an SA-extension, i.e. D ∈ SA( A ). Then this map is bijective,since the D -complement T of A in U = A + T is uniquely determined by A and U . Thus wehave an isomorphism of posetsCompl ′ D ( A ) ∼ −−−−→ + A Compl ′′ D ( A ) . Compl ′ D ( A ) and Compl ′′ D ( A ) have the bottom elements D and D + A = A respectively.Note also that the union of the modules in a chain in Compl ′ D ( A ) is again an element ofCompl ′ D ( A ), and so by Zorn’s Lemma every element of Compl ′ D ( A ) is contained in a maximalelement of Compl ′ D ( A ). These are the maximal modules complementary to A over D .9. SA-extensions with a fixed complementary module
We now fix submodules D ⊂ T in V and search for the SA-extensions A of D , for which T is a complementary module over D , i.e. (cf. Definition 7.4) A ∩ T = D, [( A \ D ) + T ] ∩ T = ∅ . Remark 9.1. a) If ( A, D ) is such an SA-extension then every subextension ( A , D ) , D ⊂ A ⊂ A isagain an SA-extension with complementary module T . b) If ( A i | i ∈ I ) is a chain of SA-extensions of D with complementary module T , thenthe union S i ∈ I A i is again such an SA-extension. Thus every SA-extension of D withcomplementary module T is contained in a maximal such extension. Of course suchan extension is T -saturated. Our next goal is, to exhibit a module T ⊃ D which serves as a complementary module for every SA-extension of D . Definition 9.2 ([2, p.154]) . A submodule T of V is subtractive (in V ), if for any twoelements t , t ∈ T and x ∈ V with x + t = t , also x ∈ T , in other terms ∀ x ∈ V : ( x + T ) ∩ T = ∅ ⇒ x ∈ T. It is evident that the intersection of any family of subtractive submodules of V is againsubtractive. Thus for a submodule D of V there is a unique minimal subtractive module T ⊃ D in V , namely the intersection of all subtractive modules containing D . We call thismodule T the subtractive hull of D (Golan uses the term “subtractive closure” [loc.cit.,p. 155]).We look for a more explicit description of the subtractive hull of a given submodule D of V . MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 19
Definition 9.3.
We say that a vector x ∈ V has no D -access to D if ( x + D ) ∩ D = ∅ ,and denote the set of all these vectors by Nac D ( D ) . Thus Nac D ( D ) is the union of all cosets x + D in V which are disjoint from D . In the following we denote the set Nac D ( D ) briefly by N and its complement V \ N in V by N c . Clearly N c is the set of all x ∈ V with ( x + D ) ∩ D = ∅ , i.e., the set of all x ∈ V such that there exist d, d ′ ∈ D with x + d = d ′ . Proposition 9.4 (Michihiro Takahashi, cf. [2, p.155]) . N c = { x ∈ V | ( x + D ) ∩ D = ∅} isthe subtractive hull of D in V .Proof. a) We verify that N c is a submodule of V . Given x , x ∈ N c , we have elements d , d ′ , d , d ′ of D with x + d = d ′ , x + d = d ′ , and so ( x + x ) + ( d + d ) = d ′ + d ′ ,whence x + x ∈ N c . Given x ∈ N c and λ ∈ R we have some d, d ′ ∈ D within x + d = d ′ .It follows that λx + λd = λd ′ . Thus λx ∈ N c .b) We verify that the module N c is subtractive in V . Let x , x ∈ N c , v ∈ V , and x + v = x .There are vectors d , d ′ , d , d ′ ∈ D with x + d = d ′ , x + d = d ′ . Adding d to the equation x + v = x we obtain d ′ + v = x + d . Then adding d , we obtain d ′ + v + d = d + d ′ . This proves that v ∈ N c .c) Since for any x ∈ N c there are elements d, d ′ of D with x + d = d ′ , it is obvious that x ∈ T for any subtractive module T ⊃ D , whence T ⊃ N c . (cid:3) The papers of M. Takahashi cited in [2, p.155] have been inaccessible for us. Thus we feltobliged to give a detailed proof of Proposition 9.4. Golan [loc.cit.] uses the notation E VD ( D )for the subtractive hull of D . Theorem 9.5.
Let D be any submodule of V . Then the subtractive hull T of D in V iscomplementary over D to every SA-extension of D .Proof. We write again N := Nac D ( D ). Then N = V \ T by Proposition 9.4. Let A be anSA-extension of D . Since D is SA in A , we have [( A \ D ) + D ] ∩ D = ∅ , i.e., A \ D ⊂ N .This means that ( A \ D ) ∩ T = ∅ . It follows that A ∩ T = [( A \ D ) ∩ T ] ∪ ( D ∩ T ) = D. This proves that T is complementary to A over D (cf. (7.1)). (cid:3) It is natural to ask for a given semiring R and modules D ⊂ V , whether the subtractivehull of D is the maximal submodule T ⊃ D which is complementary over D for everySA-extension of D in V . We exhibit a situation where this is true. Proposition 9.6.
Assume that R is a zerosumfree semifield. Assume furthermore that T is the subtractive hull of D in V and T ′ is a submodule of V , which properly contains T .Let x ∈ T ′ \ T . Then the module A := D + Rx is an SA-extension of D , for which T ′ is not complementary to A over D (while of course T is complementary to A over D byTheorem 9.5). Proof.
It suffices to verify that D is SA in A , which by Proposition 7.3 means, that ( A \ D )+ D is disjoint from D , and A \ D is closed under addition. Then T ′ is certainly not complementaryto A over D since A ⊂ T ′ .Let z = λx + d ∈ A \ D with λ ∈ R , d ∈ D . Then λ ∈ R \{ } , and so λ ∈ R ∗ . Suppose that z + d = d ′ for some d , d ′ ∈ D . Then λx + d + d = d ′ . Since D ⊂ T and T is subtractive,this implies λx ∈ T and then x ∈ T , a contradiction. Thus [( A \ D ) + D ] ∩ D = ∅ .If z = λ x + d , z = λ x + d are elements of A \ D , then z + z = ( λ + λ ) x + d + d . We have λ = 0, λ = 0, and so λ + λ = 0, whence z + z ∈ A \ D . Thus A \ D is closedunder addition. (cid:3) Similar results about SA-extensions of D can be obtained by starting with the set N := Nac V ( D ) := { x ∈ V | ( x + V ) ∩ D = ∅} of vectors in V “without V -access to D ”, instead of N = Nac D ( D ). Then N c = { x ∈ V | ∃ v ∈ V : x + v ∈ D } is again an R -submodule of V containing D . It is the downset D ↓ of the set D with respectto the minimal preordering (cid:22) V of V discussed in [11, §
6] (in equivalent terms, the convexhull of D in this preordering.) Since the convex submodules w.r. to (cid:22) V are precisely theSA-submodules of V [11, Prop. 6.7] it is clear that N c is the SA- closure of D , i.e., thesmallest SA-submodule T of D which contains D . This SA-closure appears in [2, p.155](citing M. Takahashi) under the name strong closure of D , notated there by E VV ( D ). Since N c = D ↓ , N = { x ∈ V | x D ↓ } , (9.1)and we conclude easily (cf. [2, p.155]) that N + D ↓ = N + V = N . (9.2)Of course, N ⊂ N , more precisely N = { x ∈ N | ( x + V ) ∩ D = ∅} = { x ∈ N | x + V ⊂ N } . In close analogy to the arguments in the proofs of Proposition 9.4 and Theorem 9.5, andusing Theorem 7.5, we obtain
Theorem 9.7. A := N ˙ ∪ D is an SA-extension of D with A ∩ D ↓ = D , and D ↓ is the(unique) D -complement of A in N + D ↓ = V . It is now plain that A is the unique maximal SA-extension C of D with C \ D ⊂ N .Thus the SA-hull D ↓ is the D -complement in V of just one maximal SA-extension, namely N ∪ D , while usually the subtractive hull N c of D shows up as the D -complement of severalmaximal SA-extensions A of D in A + N c .10. Minimal D -cosets Given a submodule D of an R -module V , R any semiring, we introduce a binary rela-tion ≤ D on V as follows: x ≤ D y ⇔ ∃ d ∈ D : x + d = y. (10.1) MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 21
This relation is a quasiordering, i.e., it is transitive and reflexive, but not necessarily antisym-metric. Obviously ≤ D is compatible with scalar multiplication: If x ≤ D y , then λx ≤ D λy for any λ ∈ R . We call ≤ D the D -quasiordering on V .Below we restrict to the case that ≤ D is antisymmetric, and so is a partial ordering on V ,named the D -ordering on V . We then also say that V is D -ordered. The case of D = V has attracted interest for long. Such an R -module V is called upper bound (abbreviatedu.b.), since then the sum x + y of two vectors x, y is an upper bound of the set { x, y } . It iswell known that V is u.b. iff the R -module V lacks zero sums (LZS), i.e., is zero-sum-freein the terminology of [2, p. 156]. (A long list of such modules is given in [12, § R -module is obviously D -ordered for any submodule D of V . Thus our focus on D -orderedmodules is a rather mild restriction.We turn to D -cosets x + D in a D -ordered R -module V . Remark 10.1.
Let x, y ∈ D . Then, obviously, y + D ⊂ x + D ⇔ y ∈ x + D ⇔ x ≤ D y. Thus x + D = y + D ⇔ x ≤ D y and y ≤ D x ⇔ x = y , since ≤ D is antisymmetric. Definition 10.2.
We define on a D -ordered R -module V the binary relation (cid:22) D as x (cid:22) D y ⇔ y + D ⊂ x + D. This relation is again a partial ordering on the set V . But, without more assumptions on R ,there is no reason that (cid:22) D is compatible with scalar multiplication.We search for minimal D -cosets, i.e., maximal vectors with respect to (cid:22) D , which show upin connection with a pair ( A, T ) of submodules of V with amalgamation, where A ∩ T = D , A is an SA-extension of D in V , and T is a D -complement of A in V . More specifically wepick an SA-extension D ⊂ A in the D -ordered R -module V . Thus we know that A \ D isclosed under addition , and that ( A \ D ) + D = A \ D . We define D ↓ := { x ∈ V | ∃ d ∈ D : x ≤ D d } . (10.2)Thus D ↓ is the set of all x ∈ V with x + d = d ′ for some d, d ′ ∈ D . In other words, D ↓ is thesubtractive hull of D in V . We verify directly (without involving §
9) that[( A \ D ) + D ↓ ] ∩ D ↓ = ∅ . (10.3)Indeed, suppose there exist x ∈ A \ D , y ∈ D ↓ with x + y ∈ D ↓ . Then y + d = d , x + y + d = d for some d i ∈ D. This implies x + y + d + d = d + d , and then x + d + d = d + d , in contradiction to ( A \ D ) + D ↓ ⊂ A \ D . Thus D ↓ is complementaryto A over D (Definition 7.4) and we infer from Theorem 7.5 that B := [( A \ D ) + D ↓ ] ∪ D ↓ (10.4)is an SA-extension of D containing A (the saturation of A by the complementary module D ↓ ),and A + D ↓ = B + D ↓ . Theorem 7.7 tells us that the pair ( B, D ↓ ) has amalgamation in V .We arrive at the diagram B SA A + D ↓ = B + D ↓ SA A ❣❣❣❣❣❣❣❣❣❣❣❣❣ SA D ↓ D ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (10.5) for any SA-extension D ⊂ A in V .Before focusing on minimal D -cosets in A + D ↓ = B + D ↓ , we study D -cosets in anarbitrary submodule U ⊃ D of V . Proposition 10.3.
Let U ⊃ D be any submodule of V containing D . The set of maximalelements in U with respect to the ordering (cid:22) D is the set Fix D ( U ) := { x ∈ U | ∀ d ∈ D : x + d = x } of fix points in U under the family of maps z z + d , U → U with d ∈ D . Thus all minimal D -cosets in U are singletons.Proof. Of course, if x + D is a singleton, then x + D is a minimal D -coset. Conversely, d + D ⊂ D for any d ∈ D . Let x ∈ U , whence x + d + D ⊂ x + D . If x + D is minimal, thisforces x + d + D = x + D . We conclude by Remark 10.1 that x + d = x for any d ∈ D . (cid:3) Example 10.4 (The case of U = D ) . If D has a maximal vector d max in the D -orderingof V , then { d max } is the unique minimal D -coset in D . Otherwise D does not contain anyminimal D -coset. We may study D -cosets in more general subsets of V than submodules. Definition 10.5.
A (nonempty) set X ⊂ V is stable under D , if X + D ⊂ X , i.e., x + d ∈ X for every x ∈ X and d ∈ D . If this holds, we define Fix D ( X ) := { x ∈ X | ∀ d ∈ D : x + d = x } . Proposition 10.6.
If a set X ⊂ V is stable under D , then { x } is stable under D ↓ for every x ∈ Fix D ( X ) , i.e., x + D ↓ = { x } .Proof. Let x ∈ Fix D ( X ) and a ∈ D ↓ . There exist d , d ∈ D with a + d = d . Thus x + a = x + d + a = x + d = x . (cid:3) Proposition 10.7.
Fix D ( V ) is an “ideal” of the additive monoid ( V, +) , i.e., V +Fix D ( V ) ⊂ Fix D ( V ) . Proof.
Let v ∈ V and x ∈ Fix D ( V ), then ( x + v ) + d = ( x + d ) + v = x + v for any d ∈ D , (cid:3) Using this chain of propositions we determine the minimal D -cosets in B + D ↓ , and alsoget hold on some minimal D ↓ -cosets in A + D ↓ = B + D ↓ . Theorem 10.8.
Assume that D is a submodule of an R -module V , R any semiring, onwhich ≤ D is an ordering. Let D ⊂ A be any SA-extension in V and B = [( A \ D ) + D ↓ ] ∪ D the saturation of A by the complementary module D ↓ over D . Recall that ( B, D ↓ ) has amalgamation in V , B ∩ D ↓ = D , and A + D ↓ = B + D ↓ (Theorem 7.7).The minimal D -cosets in B are the singletons { x } with x ∈ Fix D ( A \ D ) and, in the casethat D has a maximal element, also x = d max . We have x + D ↓ = { x } for all these vectors,and so the singletons { x } are also minimal D ↓ -cosets in A + D ↓ = B + D ↓ .If x ∈ Fix D ( A ) = Fix D ( B ) , then x + v ∈ Fix D ( B ) for every v ∈ V . Thus Fix D ( B ) is anupper set under the quasiordering ≤ V , restricted to B .Proof. a) Let v ∈ B + D ↓ . Assume that v + D is a minimal D -coset, and write v = x + t with x ∈ B , t ∈ D ↓ . Then v + d ⊂ v + d + D = x + t + d + D = x + d + D. MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 23
Due to the minimality of v + D this forces v + D = x + d + D ⊂ B + D. More elaborately v + D ∈ ( A \ D ) + D = A \ D or v + D ⊂ D ↓ . In the latter case v + D is aminimal D -coset in D ↓ . This can only happen if D has a maximal element d max , and then v + D = { d max } (cf. Example 10.4). In the former case v + D = { x } with x ∈ A \ D . Thusthe minimal D -cosets in A + D ↓ = B + D ↓ are the singletons { x } with x ∈ Fix D ( A \ D ) andthe singleton { d max } , if d max exists.b) Assuming now that v + D ↓ is a minimal D ↓ -coset in A + D ↓ = B + D ↓ , we choose aminimal D -coset u + D in the D -stable set A + D ↓ . From u + D ⊂ v + D ↓ we obtain u + D ↓ = u + D + D ↓ ⊂ v + D ↓ , whence u + D ↓ = v + D ↓ , by minimality of v + D ↓ . By a) weknow that u ∈ Fix D ( A ) = Fix D ( B ), and conclude that u + D ↓ = { u } by Proposition 10.6.c) The last assertion in Theorem 10.8 is now clear by Proposition 10.6. (cid:3) The question remains, which minimal D ↓ -cosets v + D ↓ contain D -cosets and how many.This question can be answered in a very general context. We assume that V is an R -moduleover a semiring R , and consider for any submodule E of V the minimal E -coset v + E inthe set theoretic sense, not assuming that V is E -ordered. Lemma 10.9. If v + E is minimal, then v + E = u + E for every u ∈ v + E .Proof. u + E ⊂ v + E . This forces u + E = v + E , since v + E is minimal. (cid:3) Proposition 10.10.
Let D be any R -submodule of V . Assume that v + D ↓ is a minimal D ↓ -coset in V . There exists at most one minimal D -coset u + D ⊂ v + D , and then u + D ↓ = v + D ↓ .Proof. Suppose that u + D and u + D are minimal D -cosets contained in v + D ↓ . Wehave vectors t , t ∈ D ↓ for which u = v + t , u = v + t , and vectors d , d ′ , d , d ′ in D with t + d = d ′ , t + d = d ′ . Then u + d = v + d ′ , u + d = v + d ′ , and so u + d + d ′ = v + d ′ + d ′ = u + d ′ + d . We conclude by Lemma 10.9 that u + D = u + d + d ′ + D = u + d ′ + d + D = u + D . Furthermore u + D ↓ = v + D ↓ , again byLemma 10.9. (cid:3) D -isolated vectors Given a module V over a semiring R and a submodule D of V , we call a vector v ∈ VD -isolated , if there exists neither a vector d ∈ D with v + d = v nor vectors x = v in V and d ∈ D such that x + d = v . In other terms, v + D = { v } , and v / ∈ x + D for every x ∈ V \ D .We are interested in cases where D -isolated vectors show up in connection with a pair( A, T ) of submodules of V with amalgamation, where A ∩ T = D , A an SA-extension of D in V , and T a D -complement of A in V . First a simple but basic example. Example 11.1.
Let R = N ≥ and let V be a totaly ordered set with a smallest element .We introduce the addition λ + λ = max { λ , λ } , λ , λ ∈ V, on V and regard V as R -module in the obvious way. Assume that D is a subset of V containing , and that D \ { } is convex in V .Using the notations from §
9, the set of vectors in V without D -access is N = Nac D ( D ) = { λ ∈ V | λ > D } , and the complement of N in V is N c = D ↓ = { x ∈ V | ∃ d ∈ D : x ≤ d } .A := N ∪ D is an SA-extension of D , since obviously A \ D is closed under addition and ( A \ D ) + D = A \ D (cf. Proposition 7.3). T := D ↓ is a D -complement of A in V = A + D ↓ and B := [( A \ D ) + T ] ∪ D = A. Thus A = B is a saturated SA-extension of D , and so we know by Theorem 7.7 that ( A, D ↓ ) has amalgamation in V .We enquire, which elements v of V are D -isolated. Note that this is only of interest ifthere exist nonzero elements µ < D . (i) Assume first D \ { } is not a singleton. (a) If v < D , then v + D = D , and thus v is not D -isolated. (b) Let v ∈ D . If D has a maximal element d max , then v + d max = v , and v + d max = d max for all v < d max . If D has no maximum, then v + D = { v } for any v ∈ D .Thus D is void of D -isolated vectors. (c) Let v > D . Then v + D = { v } , and if x < v then x + d = max( x, d ) < v . Thus v is D -isolated.We conclude: The D -isolated vectors of V are the vectors v > D . (ii) If D \ { } = { d } , then beside the vectors v > d also v = d is D -isolated. We introduce a special class of additive monoids, which will play a central role below.
Definition 11.2.
An additive monoid ( X, +) is bipotent (also called selective), if for any x , x ∈ X x + x ∈ { x , x } . (11.1)For bipotent monoids we define a binary relation ≤ by x ≤ x ⇔ x + x = x . (11.2)Clearly x ≤ x for any x ∈ X , and x ≤ x , x ≤ x implies x = x . If x ≤ x and x ≤ x ,then x + x = x + x + x = x + x = x . Furthermore 0 ≤ x for all x ∈ X . Thus, therelation (11.2) is a total ordering on the set X with smallest element 0 = 0 X .We compare this ordering ≤ with the quasiordering ≤ X of ( X, +). If x ≤ x , then x + x = x , and so x ≤ X x . Conversely, if x ≤ X x , then there exists y ∈ X such that x + y = x , whence x ≤ x . Thus ≤ coincides with the ordering ≤ X of ( x, +). This proves Proposition 11.3.
Every bipotent monoid ( X, +) is an upper bound monoid with totalordering ≤ X and smallest element X . This fact has a strong converse. Let ( X, ≤ ) be a totally ordered set with smallest element 0.We define a composition X × X + −→ X by the rule x + x = max( x , x ) (11.3)(as done for ( V, +) in Example 11.1). The composition + is clearly commutative, and0 + x = x for any x ∈ X . It can be proved by an easy straightforward way that thecomposition + is associative { e.g. check that ( x + x ) + x = x + ( x + x ) by goingthrough the four cases x ≤ x ≤ x , x ≤ x ≥ x , . . . } . Thus ( X, +) is an additive MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 25 monoid, whose zero element is the minimal element of ( X, ≤ ) . This monoid is bipotent dueto (11.3).Finally notice that the a map ϕ : X → X ′ form X to a second bipotent monoid ( X ′ , +) isa monoid homomorphism iff it respects the total orderings ≤ X and ≤ X ′ (in the weak sense, x ≤ x ⇒ ϕ ( x ) ≤ ϕ ( x )), and maps 0 X to 0 X ′ . Thus we may state Proposition 11.4.
The category of bipotent monoids is canonically equivalent to the categoryof totally ordered sets with minimal element.
In what follows we view bipotent monoids as the same objects as totally ordered sets withminimal element.
Definition 11.5. (a) A bipotent retraction of an additive monoid ( V, +) is a monoid homomorphism ϕ : ( V, +) → ( X, +) to a bipotent submonoid X of V with ϕ ( x ) = x for any x ∈ X . (b) We call such a retraction special , if for v , v ∈ V the following holds: v + v ∈{ v , v } if ϕ ( v ) = ϕ ( v ) , while v + v = ϕ ( v ) if ϕ ( v ) = ϕ ( v ) . Below we will use special retractions of V to exhibit D -isolated vectors for suitable sub-monoids D of V . Remark 11.6. If ϕ : ( V, +) → ( X, +) is a special bipotent retraction, then for any v , v ∈ Vv + v = v if ϕ ( v ) < ϕ ( v ) ,v if ϕ ( v ) > ϕ ( v ) ,ϕ ( v ) if ϕ ( v ) = ϕ ( v ) , (11.4) as follows from the fact that ϕ respects addition, and so respects the quaiordering ≤ V on V and is restriction to the ordering ≤ X on X .It will turn out that for ϕ : ( V, +) → ( X, +) a special bipotent retraction the monoid ( V, +) is upper bound, but bipotent retractions of ( V, +) in general retains sense if V is not upperbound and will be useful also then, cf. Theorem 11.10 below. We obtain all special bipotent retractions by the following construction. Let V be a set and ϕ : V → X a map to a subset X of V with ϕ ( x ) = x for every x ∈ X (a “set theoreticretraction”). We choose a total ordering on X with a minimal element 0 = 0 X , and view X as a bipotent monoid ( X, +), as explained above. Then we define a composition by therule (11.4) above. Theorem 11.7. (i) ( V, +) is an upper bound additive monoid and ϕ : V → X is a special bipotentretraction. (ii) Every fiber ϕ − ( x ) of x is convex with respect to the partial ordering ≤ V , in particular ϕ − (0) = { } . (iii) Let v ∈ V , x ∈ X . If ϕ ( v ) ≤ ϕ ( x ) = x , then v + x = x . If ϕ ( v ) > ϕ ( x ) , then v + x = v . (iv) v + v = ϕ ( v ) for every v ∈ V . Thus the bipotent submonoid X of ( V, +) is uniquelydetermined by ( V, +) . If v + v = v + v , then v + v = v + v . We do not demand here that ( V, +) is upper bound, but in the construction below this will be the case. Proof. a) It is obvious from rule (11.4) that the composition V × V + −→ V is commutativeand v + 0 = v for all v ∈ V , where 0 = 0 X ∈ X . For any v , v , v ∈ V it can be verifiedin a straightforward way that ( v + v ) + v = v + ( v + v ) { e.g. run through all cases ϕ ( v ) (cid:3) ϕ ( v ) (cid:3) ϕ ( v ) with (cid:3) ∈ { <, = , > } ; some cases can be settled simultaneously byinterchanging v and v } . Thus ( V, +) is an additive monoid. Furthermore ( X, +) is asubmonoid of ( V, +) by (11.4), and so ϕ : V → X is a special bipotent retraction of V .b) We verify that ( V, +) is upper bound. Let v, w , w ∈ V and v + w + w = v. ( ∗ )We need to prove that v + w = v . Applying ϕ to ( ∗ ) gives ϕ ( v ) + ϕ ( w ) + ϕ ( w ) = ϕ ( v ) . Since X is bipotent, we conclude that ϕ ( w ) ≤ ϕ ( v ) and ϕ ( w ) ≤ ϕ ( v ). If ϕ ( w ) < ϕ ( v ),then v + w = v . If ϕ ( w ) < ϕ ( v ), then v + w = v , and then by ( ∗ ) again v + w = v . Iffinally ϕ ( w ) = ϕ ( w ) = ϕ ( v ), then by ( ∗ ) v + w = ϕ ( v ), v + w + w = ϕ ( v ), and we readoff from ( ∗ ) that ϕ ( v ) = v , v + w = v. Thus v + w = v in all cases.c) Claims (iii) and (iv) of Theorem 11.7 now follow by easy observations. (iii) is clearby (11.4). If v ≤ v ≤ v , then ϕ ( v ) ≤ ϕ ( v ) ≤ ϕ ( v ), and so, if ϕ ( v ) = ϕ ( v ) also ϕ ( v ) = ϕ ( v ). Thus ϕ − ( λ ) is convex for any λ ∈ X . (We use here in an essential waythat V is upper bound.) ϕ − (0) = { } is obvious directly from (11.4). By (11.4) also ϕ ( v ) = v + v . Finally, if v + v = v + v , then ϕ ( v ) = ϕ ( v ), and so v + v = v + v ,again by (11.4). (cid:3) Example 11.8.
Let R = ( R, G , ν ) be a supertropical semiring [14] , or more generally a ν -semiring [4] , where G is a bipotent subsemiring of R and ν is a projection R → G , i.e., ν ( a ) = a for every a ∈ G , satisfying a + b = ν ( a ) if ν ( a ) = ν ( b ) . Then the projection ν : R → G is a bipotent retraction (Definition 11.5).When R is a supertrpical semifield [3, 14] , i.e., T := R \ G is an abelian group and therestriction ν | T : T → G is onto, the subsemiring G is totally ordered and a + b ∈ { a, b } whenever ν ( a ) = ν ( b ) . So for this case, the projection ν : R → G is a special bipotentretraction.The familiar tropical (max-plus) semifield T = ( R ∪{−∞} , max , +) is a biopotent semifield.It extends to a supertropical semifield F = ( R ∪ {−∞} ∪ R ν , + , · ) , where R ν is a second copyof R , in which T embeds in F as its ghost ideal G . Then the projection ν : F → T is a specialbipotent retraction. We now assume only that ϕ : ( V, +) → ( X, +) is a bipotent retraction, V not necessarilyupper bound. We choose a subset D of X that contains 0, for which D \ { } is convex in X .Then D := ϕ − ( D ) (11.5)is a submonoid if V , for which the following holds:If d ≤ V v ≤ V d where ϕ ( d ) and ϕ ( d ) are in D \ { } , then v ∈ D . (11.6)Let D ↓ := { v ∈ V | v ≤ V d for all d ∈ D } . Lemma 11.9. D ↓ = ϕ − ( D ↓ ) . MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 27
Proof. ( ⊆ ): Let v ∈ D ↓ , i.e., v + w = d with w ∈ V , d ∈ D . Then ϕ ( v ) + ϕ ( w ) = ¯ d with¯ d := ϕ ( d ) ∈ D . Thus ϕ ( v ) ∈ D ↓ .( ⊇ ): Let v ∈ V , ϕ ( v ) ≤ ¯ d . There exists w ∈ V with ϕ ( v + w ) = ϕ ( v ) + ϕ ( w ) = ¯ d , and so v + w ∈ ϕ − ( D ) = D . (cid:3) Theorem 11.10.
Assume that λ ∈ X is D -isolated. Then every v ∈ ϕ − ( λ ) is D -isolated.Proof. Let ϕ ( v ) = λ . Then for any d ∈ D ϕ ( v + d ) = ϕ ( v ) + ϕ ( d ) = ϕ ( v ) = λ . If u + d = v ,then ϕ ( u ) + ϕ ( d ) = ϕ ( v ) = λ . Thus ϕ ( u ) = λ , and so u + d = u , as proved. We concludethat u = v . (cid:3) Thus, what had been observed about D -isolated vectors in Example 11.1, remains validin the present more general setting mutatis mutandis.12. SA-submodules induced by actions on upper bound modules
Let ( V, +) and ( X, +) be additive monoids. An action of V on X is a map α : V × X → X having the following properties: Write α ( u, x ) = u + α xu + α ( x + x ) = ( u + α x ) + x (and so also = x + ( u + α x )) , ( u + u ) + α x = u + α ( u + α x ) , α x = x. In this situation, for any x ∈ S define C α ( s ) := { u ∈ V | u + α s = s } . (12.1)It is immediate that C α ( s ) is a submonoid of V : If u + α s = s and u + α s = s , then( u + u ) + α s = u + α ( u + α s ) = u + α s = s, and 0 + α s = s. We define for u ∈ V : e u := u + α X ∈ X. Note that for u + u ∈ V we have ^ u + u = ( u + u ) + α u + α ( u + α u + α e u = u + α (0 + e u ) = ( u + α
0) + e u = e u + e u . Furthermore, e α X = 0 X . Thus the map u e u is a homomorphism of the monoid V onto a submonoid of X .From now on we assume that the monoid ( X, +) is upper bound . Remark 12.1.
For any u ∈ V , x ∈ X , u + α x = u + α (0 + x ) = ( u + α
0) + x = e u + x, and so u + α x ≥ X x. Proposition 12.2. C α ( s ) is an SA-submonoid of V for any s ∈ X .Proof. Let u + u ∈ C α ( s ). Then by Remark 12.1, s = ( u + u ) + α s = u + α ( u + α s ) ≥ X u + α s ≥ X s, and thus u + α s = s (and so also u + α s = s ). (cid:3) Proposition 12.3. If s , s ∈ X , then C α ( s ) + C α ( s ) ⊂ C α ( s + s ) . Thus s ≤ X t ⇒ C α ( s ) ⊂ C α ( t ) . Proof.
Let u ∈ C α ( s ). Then u + ( s + s ) = ( u + s ) + s = s + s . Thus u ∈ C α ( s + s ), implying that C α ( s ) ⊂ C α ( s + s ). Similarly, C α ( s ) ⊂ C α ( s + s ),and so C α ( s ) + C α ( s ) ⊂ C α ( s + s ). (cid:3) We further define C α ( S ) := [ s ∈ S C α ( s ) = { u ∈ V | ∃ s ∈ S : u + α s = s } . (12.2)for any (nonempty) subset S of X . Proposition 12.4. (a) If S is closed under addition then C α ( S ) is an SA-submodule of V . (b) If T is a second subset of X , closed under addition, then C α ( S ) + C α ( T ) ⊂ C α ( S + T ) . (c) If S and T are cofinal subsets of X , i.e., for every s ∈ S there is some t ∈ T with s ≤ X t , and vice versa, then C α ( S ) = C α ( T ) .Proof. We infer from the definition (12.1) of C α ( S ) and Proposition 12.2 that C α ( S ) is asubmodule of V . By this proposition also claims (b) and (c) are evident. Given u , u ∈ V with u + u ∈ C α ( S ), there is some s ∈ S for which u + u ∈ C α ( s ). By Proposition 12.3we conclude that u , u ∈ C α ( s ) ⊂ C α ( S ) . Thus the submodule C α ( S ) is SA in V . (cid:3) Example 12.5.
Given x ∈ V \ { } , i.e., x > , let S = { nx | n ∈ N } . We have x ∈ C ( nx ) iff nx = ( n + 1) x , and thus meet the following dichotomy: If thereexists n ∈ N with nx = ( n + 1) x , then C ( S ) = C ( nx ) for the smallest such number n .Otherwise C ( S ) is the union of the submonoids C ( x ) $ C (2 x ) $ C (3 x ) $ · · · . In this casewe write C ( S ) = C ω ( x ) . We turn to the primordial example for an action of V on X . Here X = V and α is givenby v + α x := v + x . We denote the monoids C α ( x ) simply by C ( x ), and then for any x ∈ V have C ( x ) = { v ∈ V | v + x = x } . (12.3)Recall that each of these sets C ( x ) is an SA-submodule of V , and for any x, y ∈ V thesubmonoid C ( x + y ) contains C ( x ) and C ( y ), whence x ≤ V y ⇒ C ( x ) ⊂ C ( y ) . (12.4)Turning to the SA-submonoids C ( S ) := S s ∈ S C ( s ), for S a subset of V , closed under ad-dition, the following case deserves special interest. Writing ≤ instead of ≤ V , for short, anarchimedean feature comes into sight. Definition 12.6.
The archimedean class of any x ∈ V is Arch( x ) := { y | ∃ n : y ≤ nx, ∃ m : x ≤ my } . The following is easily seen and certainly very well known.
MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 29
Remark 12.7.
Let x, y ∈ V \ { } be in the same archimedean class. Then either both se-quences { mx | m ∈ N } , { ny | n ∈ N } become constant of values m x , n y , and Arch( m x ) =Arch( n y ) , or C ω ( x ) = C ω ( y ) . Of course,
Arch(0) = { } . In any upper bound monoid ( X, +) we define archimedean classes as above in Definition12.6, here with the ordering ≤ X on X .Assume now that an action α : V × X → X is given, where both V and X are modulesover a semiring R = { } , and, as before, X is upper bound. Then, for any x = 0 in X , theset R x := { λ ∈ R | λ · C ( x ) ⊂ C ( x ) } (12.5)is a subsemiring of R , as is immediate from the fact that C ( x ) is a submonoid of V . Conse-quently, N · R ⊂ R x , (12.6)since N · R is the unique smallest subsemiring of R .A subset Y of R is called convex in R (with respect to the quasiordering ≤ R ) if for λ ≤ λ in Y every λ ∈ R with λ ≤ λ ≤ λ is also in R . Proposition 12.8.
For every x ∈ X the semiring R x is convex in R .Proof. Let λ , λ ∈ R x with λ ≤ R λ ≤ R λ , i.e., λ + µ = λ , λ + ν = λ for some µ, ν ∈ R .Given u ∈ C ( x ), we have λ u + µu = λu , λu + νu = λ u . Thus x = ( λ u ) + α x ≤ ( λ u ) + α x + ( µu ) + α x = ( λu ) + α x ≤ ( λu ) + α x + ( νu ) + α x = ( λ u ) + α x = x. This implies that ( λu ) + α x = x , and proves that λu ∈ C ( x ). (cid:3) Definition 12.9.
Let o R denote the convex hull of N · R in the semiring R with respectto ≤ R . It is the smallest SA-subsemiring of R , cf. [11, § . Corollary 12.10.
For any subset S of X , closed under addition, C α ( x ) is an o R -submoduleof V .Proof. For any s ∈ S R s is a convex subsemiring of R containing N · R , whence contain-ing o R . Thus C α ( s ) is an o R -submodule of V . The same holds for the union (=sum) C α ( S )of the C α ( s ), s ∈ S , cf. (12.2). (cid:3) Amalgamation in the category of upper bound monoids
The amalgamation theory for submodules of an additive monoid ( V, +), as developed in § §
5, can be amended in a natural way to an amalgamation of upper-bound monoids, sincethere is a canonical reflection V → V from the former to the latter category (cf. [9, § V , +) arises from ( V, +) by dividing out the natural congruence relation ≡ V , which turnsthe quasiordering ≤ V on V to a (partial) ordering ≤ V on V = V / ≡ V . This congruence isgiven by x ≡ V y ⇔ x ≤ V y and y ≤ V x. We denote the congruence class of a vector x ∈ V by ¯ x , and then have for x, y ∈ V theexplicit description ¯ x = ¯ y ⇔ ∃ z, w ∈ V : x + z = y, y + w = x. (13.1) We name V the upper bound monoid associated to V .If V is a module over a semiring R , then it is immediate that V is a module over the upperbound semiring R = R/ ≡ , with scalar mutiplication given by¯ a ¯ v = av (13.2)for a ∈ R , v ∈ V . Then consequently, we say that the V is the upper bound R - moduleassociated to V .Below we most of the time work in the category of R -modules for R an upper boundsemiring. Monoids can be subsumed here by taking R = R = N .As common, we say that a subset S ⊂ V is convex , if s ∈ S for any s , s ∈ S , s ∈ V with s ≤ V s ≤ V s . We cite the following useful fact, valid in any monoid ( V, +). Proposition 13.1 ([9, Proposition 5.7]) . A submodule S of V is SA in V iff S is a unionof congruence classes and S = S/ ≡ V is SA in V . In §
12 we started a study of the SA-submodule C ( x ) = { v ∈ V | v + x } of V for every x ∈ V in the case that V is an upper bound monoid. We now continue this study for theupper bound monoid V = V / ≡ V associated to any additive monoid ( V, +), but instead ofarguing in V and then passing to V by the use of Proposition 13.1, we work directly in V by using the quasiordering ≤ V . For a given x ∈ V we define C ( x ) := { u ∈ V | u + x ≤ V x } . (13.3)Since always x ≤ V u + x , this means that C ( x ) = { u ∈ V | u + x ≡ V x } , (13.4)and thus, if V happens to be upper bound, C ( x ) = C ( x ). More generally we may assumethat V is an R -module, R any semiring. Then V is an R -module for R = R/ ≡ R . Let o R = conv( N · R )be the convex hull of N · R in R with respect to ≤ R . Then we conclude by Corollary 12.10that C ( x ) is an o R -submodule of V . We often write ≤ , ≡ , . . . , for ≤ V , ≡ V , . . . , when theambient monoid V is clear from the context. Proposition 13.2. C ( x ) is an SA-submonoid of V .Proof. If u + x ≤ x , u + x ≤ x , then u + u + x ≤ u + x ≤ x . Thus C ( x ) is a submonoid.Conversely, if u + u + x ≤ x , then u + x ≤ u + u + x ≤ x , and also u + x ≤ x . Thus C ( x ) is SA in V . (cid:3) Proposition 13.3. If x ≤ V x ′ , then C ( x ) ⊂ C ( x ′ ) . Consequently, if x ≡ V x ′ , then C ( x ) = C ( x ′ ) .Proof. If x ≤ V x ′ , then x ′ = x + y for some y ∈ V . Thus u + x ≤ V x implies u + x + y ≤ V x + y ,i.e., u + x ′ ≤ V x ′ . (cid:3) Definition 13.4.
For any x ∈ V we introduce the subset C ω ( x ) := [ n ∈ N C ( nx ) of V . In the special case V = R already defined in § MALGAMATION AND EXTENSIONS OF SUMMAND ABSORBING MODULES 31
Since by Proposition 13.2 C ( nx ) ⊂ C ( nx + x ) = C (( n + 1) x ) (13.5)it is clear, that C ω ( x ) is again an SA-submodule of V . Consequently, in the case that V isupper bound we define C ω ( x ) := [ n ∈ N C ( nx ) , which extends the notation in Example 12.5 to all x ∈ V . Definition 13.5.
The archimedean class
Arch V ( x ) of an element x ∈ V is the set of all y ∈ V such that x ≤ V ny and y ≤ V mx for some n, m ∈ N . Lemma 13.6.
Let x, y ∈ V , and assume that x ≤ V my for some m ∈ N . Then C ( x ) ⊂ C ( my ) and C ω ( x ) ⊂ C ω ( y ) .Proof. For every k ∈ N we have kx ≤ V kmy , whence C ( kx ) ⊂ C ( kmy ) by Proposition 13.3.This gives both claims. (cid:3) The following in now evident.
Proposition 13.7. If Arch V ( x ) = Arch V ( y ) , then C ω ( x ) = C ω ( y ) . More generally, given a subset S of V with S + S ⊂ S , the subset C ( S ) := [ s ∈ S C ( s ) (13.6)of V is an SA-submodule, since for x ∈ C ( s ), y ∈ C ( t ) we have x + y ∈ C ( s ) + C ( t ) ⊂ C ( s + t )(cf. Proposition 13.3), and C ( s + t ) is an SA-submodule of V . Again C ( S ) is an o R -submoduleof V in the case that V is an R -module. For any x ∈ VC ω ( x ) = C ( N x ) . (13.7)Given two subsets S and T of V , closed under addition, suppose that for any t ∈ T thereis some s ∈ S with t ≤ s . Then C ( T ) ⊂ C ( S ). It follows that C ( T ) = C ( S ), if S and T arecofinal under ≤ V .We are ready to construct an additive monoid which is the amalgamation of submonoids A , . . . , A r , in which for any tuple ( x , . . . , x r ) ∈ A ×· · ·× A r the family ( C ω ( x ) , . . . , C ω ( x r ))has amalgamation in V and C ω ( x ) + · · · + C ω ( x r ) is SA in V .Starting with finitely many submonoids A , . . . , A r of an additive monoid V , we introducethe amalgamation (cf. § V = A ∞ V · · · ∞ V A r . (13.8)We identify each A k with the submonoid j k ( A k ) of V , as explained in §
4. Then V = A + · · · + A r . (13.9)Here two tuples ( a , . . . , a r ), ( b , . . . , b r ) in A × · · · × A r with the same sum a + · · · + a r = b + · · · + b r are exchange equivalent in V . Given x = x + · · · + x r , x k ∈ A k , we define thesubmonoid W = W ( x ) := { y ∈ V | ∃ n ∈ N : y ≤ V nx } . (13.10) It is the convex hull of N x in V (with respect to ≤ V ). For any z ∈ WC ω ( x ) = { u ∈ V | ∃ n ∈ N : u ≤ V nz } = { u ∈ W | ∃ n ∈ N : u ≤ V nz } , since W is convex (=SA) in V , and so C ω ( z ) is SA in W . In particular, each C ω ( x k ) is SAin W , and C ω ( x ) = C ω ( x ) ∞ V · · · ∞ V C ω ( x r ) ⊂ SA W, (13.11)as follows from Theorem 4.6.Furthermore, if x ′ = x ′ + · · · + x ′ r is a vector with x ′ k ∈ A k , and x ≤ x ′ , then we inferfrom (13.1) that W ( x ) ⊂ W ( x ′ ) . Thus we meet a hierarchy of amalgamated SA-submodulesof V , C ω ( x ) ∞ · · · ∞ C ω ( x r ) ⊂ C ω ( x ′ ) ∞ · · · ∞ C ω ( x ′ r ) . (13.12)This construction can be enlarged by choosing instead of the vectors x k ∈ A k subsets S k of A k which are closed under addition. In a complete analogy to arguments above we obtainthe following. Theorem 13.8.
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Institute of Mathematics, University of Aberdeen, AB24 3UE, Aberdeen, UK.
E-mail address : [email protected] Department of Mathematics, NWF-I Mathematik, Universit¨at Regensburg 93040 Regens-burg, Germany
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