Basic Operations on Supertropical Quadratic Forms

In the case that a module V over a (commutative) supertropical semiring R is free, the R-module Quad(V ) of all quadratic forms on V is almost never a free module. Nevertheless, Quad(V ) has two free submodules, the module QL(V ) of quasilinear forms with base D0 and the module Rig(V ) of rigid forms with base H0, such that Quad(V ) = QL(V ) + Rig(V ) and QL(V ) ∩ Rig(V ) = {0}. In this paper we study endomorphisms of Quad(V ) for which each submodule Rq with q ∈ D0 ∪ H0 is invariant; these basic endomorphisms are determined by coefficients in R and do not depend on the base of V . We aim for a description of all basic endomorphisms of Quad(V ), or more generally of its submodules spanned by subsets of D0 ∪H0. But, due to complexity issues, this naive goal is highly nontrivial for an arbitrary supertropical semiring R. Our main stress is therefore on results valid under only mild conditions on R, while a complete solution is provided for the case that R is a tangible supersemifield. 2010 Mathematics Subject Classification: Primary 15A03, 15A09, 15A15, 16Y60; Secondary 14T05, 15A33, 20M18, 51M20


Introduction
We continue a study of quadratic forms and modules over semirings, begun in [8] and [10], where now we face a general problem over the so called supertropical semrings, as explained in §1.3 and §1.7 below.Exhibiting the contribution of the present paper, our approach is indicated in §1.8.For the reader's convenience we first recall basic terminology and results, mainly from [8] and [10], but also from other sources.
1.1.Modules over a semiring.A (commutative) semiring R is a set R equipped with addition and multiplication such that (R, +, 0) and (R, • , 1) are abelian monoids with natural elements 0 := 0 R and 1 := 1 R respectively, and multiplication distributes over addition in the standard way.In other words, R satisfies all the properties of a commutative ring except the existence of negation under addition.R is called a semifield if every nonzero element of R is invertible, i.e., R \ {0} is an abelian group.A module V over a semiring R (called also a semimodule) is an abelian monoid (V, +, 0 V ) equipped with a scalar multiplication R × V → V , (a, v) → av, such that all the customary axioms of modules over a ring are satisfied: We usually write 0 for both 0 R and 0 V , and 1 for 1 R , and often speak about elements of V as "vectors" and elements of R as "scalars".1.2.Quadratic forms on a free module.For any module V over a semiring R, a quadratic form on V is a function q : V → R with q(ax) = a 2 q(x) for any a ∈ R, x ∈ V, together with a symmetric bilinear form b : V × V → R such that (1.1) q(x + y) = q(x) + q(y) + b(x, y) for any x, y ∈ V.Here "symmetric bilinear" has the obvious meaning, butin contrast to the case where R is a ring -b is often not uniquely determined by q.We call such b a companion of q, or say that b accompanies q.
In this paper we assume throughout that V is a free R-module with base (ε i | i ∈ I), i.e., every vector x ∈ V is a linear combination with unique family of scalars (x i | i ∈ I) ⊂ R, only finitely many x i = 0, called the coordinates of x.
Then, after choosing a companion b of q, a quadratic form q : V → R can be written as (for notational convenience we choose a total order on I): where α i = q(ε i ) and α ij = b(ε i , ε j ), cf.[8, §1].

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Although the case that I is infinite is relevant for applications, we assume in this and the next introductory subsection that I = {1, . . ., n} is finite, for simplicity.Then, as customary, the presentation (1.3) of a quadratic from q is written as a triangular scheme using square brackets.A quadratic form may have presentations by different triangular schemes (cf.[8, §1]).To cope with this difficulty, we use the sign ∼ = ("equivalent") to indicate such a case.Note that the entries α i in (1.4) are uniquely determined by q, since α i = q(ε i ).
If R is embeddable as subsemiring in a ring R ′ , then also the α ij are uniquely determined by q, since by identifying R ⊂ R ′ we have However, this situation is far apart from the semirings in this paper, the so called "supertropical semirings", to be described bellow.
1.3.The problem.Assume that (ε i | i ∈ I) is a fixed base of a module V .We search for families of scalars with the following property: For any two equivalent triangular schemes Then the multiplication of the entries of a triangular scheme by the scalars µ i and µ ij yields a well defined map of the set Quad(V ) of all quadratic forms on V into itself.These maps are the "basic operations" on quadratic forms appearing in the title of the paper.Two quadratic forms q 1 , q 2 on V can be added by the rule (q 1 + q 2 )(x) = q 1 (x) + q 2 (x), and a quadratic form q can be multiplied by a scalar a ∈ R by the rule (aq)(x) = a • q(x).
In this way, the set Quad(V ) becomes an R-module.
The above presentation (1.3) of a quadratic form q translates to (1.6) with d i , h ij defined by (1.7) where as before the x i are the coordinates of x, cf (1.2).We read off from (1.6) that the d i and h ij generate the R-module Quad(V ), which gives us a linear algebraic interpretation of the basic operations as certain endomorphisms of the R-module Quad(V ), as follows.An endomorphism ϕ of Quad(V ) is called basic (w.r. to a given base (ε i | i ∈ I) of V ) if it maps the submodules Rd i (1 ≤ i ≤ n) and Rh ij (1 ≤ i < j ≤ n) to itself, and so with scalars µ i , µ ij , called the coefficients of the basic endomorphism ϕ.It is now immediate that these systems of coefficients are the same families of scalars as occurring for the basic operations (cf.(1.5)), and so basic operations are the same objects as basic endomorphisms in a different disguise.
In general the set of basic endomorphisms of Quad(V ) depends on the choice of the base (ε i | i ∈ I) of V .But, when R is a supertropical semiring (to be discussed below), the framework of the present paper, it happily turns our that any free module V has only one base up to scalar multiplication by units [8, Theorem 0.9], a phenomenon for which we use the catch-phrase "V has unique base".Actually, this property holds over a much broader class of semirings than the supertropical ones [11, §1].
If V has unique base, then the set of endomorphisms of Quad(V ) is independent of the choice of the base (ε i | i ∈ I) of V (also for infinite I).In fact, if (ε ′ i | i ∈ I) is another base, ε ′ i = u i ε i , with units u i of R, then the generators of Quad(V ) associated to this base are (1.8)d ′ i = µ −2 i d i , h ij ; = µ −1 i µ −1 j h ij , as easily verified.Thus in the presence of the unique base property, the problem of finding basic endomorphisms of Quad(V ) gains extra momentum.
1.4.Supertropical semirings.A semiring R is called supertropical ([8, Definition 0.3] and [3, §3]) if e := 1 + 1 is an idempotent (i.e., e = 1 + 1 = 1 + 1 + 1 + 1 = e + e), and the following axioms hold for all a, b ∈ R : If ea = eb, then a + b = eb.(1.10) Then the ideal eR of R is a semiring with unit element e, which is bipotent, i.e., for any u, v ∈ eR the sum u + v is either u or v.It follows that eR carries a total ordering, compatible with addition and multiplication, which is given by

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The addition in a supertropical semiring is determined by the map a → ea and the total ordering on eR as follows: If a, b ∈ R, then (1.11) a In particular (taking b = 0 in (1.11) or in (1.10)), for any a ∈ R For later use we quote another fact, true in any supertropical semiring R: When R is a supertropical semiring, the elements of T (R) := R\(eR) are called tangible elements, and those of G(R) := (eR) \ {0} are called ghost elements.
The zero of R is regarded both as tangible and ghost.The semiring R itself is called tangible if R is generated by T (R) as a semiring.Clearly, this happens iff eT (R) = G(R).If T (R) = ∅, then the set is the largest subsemiring of R which is tangible supertropical.(We have discarded the "superfluous" ghost elements.)The map ν R : R → eR is a homomorphisms of semirings, which we call the ghost map of R. When there is no ambiguity, we write T , G, ν for T (R), G(R), ν R .Sometimes we adhere to the very convenient "ν-notation" for a, b ∈ R: a ≤ ν b means that ea ≤ eb, a ∼ =ν b ("ν-equivalent") means that ea = eb, while a < ν b means that ea < eb.
We call a supertropical semiring a supersemifield if all nonzero tangible elements are invertible in R and all nonzero ghost elements are invertible in the bipotent subsemiring eR, whence both T and G are abelian groups under multiplication.Supertropical semirings have been previously introduced as a tool to refine certain aspects of tropical geometry (e.g.[13]), linear algebra [6,7], starting with [2], and tropical valuation theory [3].Up to now supertropical semifields have been prevalent in applications, but more general supertropical semirings are definitely needed for any coherent theory (cf.e.g.[3,4,5]).The relevance of quadratic forms over supertropical semirings to classical quadratic forms over rings is explained in [8, §9].
1.5.Partial orderings on R, V, and Quad(V ).Assume that V is any module over a supertropical semiring R. Then it is known from more general facts (e.g.[10]), that the binary relation defined by for any x, y ∈ V is a partial ordering on V .For the reader's convenience, we provide a direct argument giving this important fact.Reflexivity (x ≤ x) and transitivity (x ≤ y, y ≤ z ⇒ x ≤ z) are evident, but antisymmetry is subtler.Given x, y, z, w such that x + z = y, y + w = x, we need to verify that x = y.First we get x + (z + w) = x, then x + e(z + w) = x.From (1.13) we infer that ez + z = ez, and so y = x + z = x + ez + ew + z = x + ez + ew = x, as desired.
The ordering (1.15) is called the minimal ordering of V , since it is the coarsest (partial) ordering on V compatible with addition, such that 0 ≤ x for all x ∈ V .In particular we have a minimal ordering on R itself.It is immediate that scalar multiplication is compatible with both minimal orderings, i.e., for In this paper, the sign "≤" is used for both orderings on R and V .These orderings lead to a "functional ordering" on Quad(V ), again denoted by "≤", defined for q 1 , q 2 ∈ Quad(V ) as On the other hand, since Quad(V ) is an R-module, it carries a minimal ordering, denoted here by " ".Definition (1.15) now reads as: If q 1 , q 2 are quadratic forms on V , then The functional ordering refines the minimal ordering, q 1 q 2 ⇒ q 1 ≤ q 2 .The interplay between these orderings is the major theme in the second half of [10], whose results will be very useful below.
1.6.The submodules QL(V ) and Rig(V ) of Quad(V ).A quadratic form q on a module V over a semring R is called quasilinear if the zero bilinear form b = 0 is a companion of q, i.e., ( cf. (1.1)) q(x + y) = q(x) + q(y) for all x, y ∈ V , and q is called rigid if q has only one companion.It is obvious that the set QL(V ) of all quasilinear forms on V is an R-submodule of Quad(V ).Assuming that the R-module V is free with base (ε i | i ∈ I) and R is supertropical, we have the forms d i and h ij with i, j ∈ I, i < j, cf.(1.7).In consequence Documenta Mathematica 22 (2017) 1661-1707 of property (1.14) of R, every d i is quasilinear.On the other hand a quadratic form q is rigid iff q(ε i ) = 0 for all i ∈ I [8, Theorem 3.5]1 .This implies that the set Rig(V ) of all rigid forms on V is a submodule of Quad(V ) and that all forms h ij (i < j) are rigid.Having this starting point, it is an easy matter to verify that both QL(V ) and Rig(V ) are free modules with bases respectively [10,Proposition 7.2].For any κ ∈ QL(V ) and ρ ∈ Rig(V ) we have (as a special cases of (1.3)) the presentations where b is the unique companion of ρ [8, §4].From these presentations it follows that the functional ordering of Quad(V ) restricts on both QL(V ) and Rig(V ) to the minimal ordering on these free modules [10,Proposition 7.3].Every q ∈ Quad(V ) has a decomposition (1.17) q = q QL + ρ with q QL ∈ QL(V ) and ρ ∈ Rig(V ), as it is now evident from (1.3).Moreover, for any decomposition (1.17) clearly q(ε i ) = q QL (ε i ), and so we infer from (1.16) that which proves that q QL is uniquely determined by q.We call q QL the quasilinear part of q and ρ in (1.17) a rigid complement of q QL in q.Most often ρ is not unique [8, §6 and §7].
If I is finite and a triangular scheme for q is given (cf.(1.4)), then q QL is represented by the diagonal part of the scheme, while the upper triangular part gives a rigid complement of q QL in q.We have but nevertheless Quad(V ) is not a direct sum of the submodules QL(V ) and Rig(V ), as soon as Rig(V ) = {0}, i.e., |I| > 1.Indeed, then different indices i, j give us a relation

1.7.
A refinement of the problem in §1.3 for R supertropical.As before we assume that the semiring R is supertropical and V is free with base (ε i | i ∈ I).We then have the set of generators B 0 = D 0 ∪ H 0 of Quad(V ) with which up to multiplication by scalars does not depend on the choice of the base For these intersections we write D 0 (Z), H 0 (Z), B 0 (Z) respectively.Instead of the basic endomorphisms of Quad(V ) addressed in §1.3, in the present paper we search, more generally, for endomorphisms of a fixed basic submodule Z of Quad(V ) that map each submodule Rq, q ∈ B 0 (Z) into itself.Such map ϕ is called a basic endomorphism of Z.Given a fixed base (ε i | i ∈ I) of V , we denote by I [2] the set of all 2-element subsets of I, and write where K ⊂ I, M ⊂ I [2] .Then a basic endomorphisms ϕ of Z is determined by a family of scalars which we again call the coefficients of ϕ, via the formulas (i ∈ K, {i, j} ∈ M ) In the case that I = {1, . . ., n} is finite, we may use triangular schemes to present quadratic forms.Now the refined problem means that we focus on quadratic forms which are represented by a scheme as in (1.4), with zero entries at fixed places (i, j), i ≤ j, namely at (i, i), with i / ∈ K and {i, j} / ∈ M .The task is to find all systems of scalars (µ i )∪(µ ij ) such that any two equivalent schemes of this type remain equivalent after multiplication of entries by the scalars µ i and µ ij respectively.So this is indeed a very natural expansion of the problem described at the beginning of §1. 3.
In what follows we call a basic submodule of Quad(V ) simply a basic module.A basic endomorphisms ϕ of a given basic module is called a basic projector on Z, if its coefficients are all 1 or 0, and thus ϕ(z) = z or 0 for any z ∈ B 0 (Z).Then X = ϕ(Z) is a basic module with X ⊂ Z and ϕ is uniquely determined by X.We call these submodules X of Z the basic projections of Z.For example, QL(V ) is a basic projection of Quad(V ) whose associated basic projector is the endomorphism π QL of Quad(V ) which maps any q ∈ Quad(V ) to its quasilinear part q QL .But Rig(V ) is not a basic projection of Quad(V ) whenever Rig(V ) = {0}, i.e., |I| > 1.Indeed, the existence of an endomorphism of Quad(V ) with ϕ(d i ) = 0 for all i ∈ I and ϕ(h ij ) = h ij for i = j is prevented by the relations 1.8.Paper outline and main results.§3 and §4 are devoted to a study of basic projectors to obtain a classification of all basic projections of a basic module Z in combinatorial terms under the mild assumption that eR is "multiplicatively unbounded", i.e., for any x, y ∈ G there exists some z ∈ G such that y < xz (cf.Corollary 3.6 and Theorem 3.12).In particular it turns out (without the assumption of multiplicatively unboundedness) that any basic module X ⊂ Z with D 0 (X) = D 0 (Z) is a basic projection of Z.The associated projectors are constructed in §3 for Z = Quad(V ) under the name of partial quasilinearizations.For any subset Λ of the set I [2] of 2-element subsets of I we have a basic projector π Λ,QL : q −→ q Λ,QL on Quad(V ) with π Λ,QL (d i ) = d i for i ∈ I, π Λ,QL (h ij ) = h ij for {i, j} ∈ Λ, and π Λ,QL (h ij ) = 0 otherwise.This projector then restricts to a basic projector on Z for any basic module Z ⊂ Quad(V ).Its image is a basic module X ⊂ Z with and B 0 (Y ) and both X and Y are basic projections of Z (Proposition 4.2).Thus it is not surprising that the classification of the basic projections of Z in §3 in combinatorial terms leads to a description of all (possibly infinite) direct decompositions of Z, again in a combinatorial way.In particular we learn in §4 that Z has (up to permutation of summands) only one decomposition Z = α∈A Z α , such that all Z α are indecomposable basic modules, and these components Z α of Z can be described combinatorially.
In the important special case that for every h ij ∈ H 0 (Z) both d i and h ij are in Z, and so are elements of D 0 (Z), this description can be given in terms of graphs.We associate to Z a graph Γ(Z) whose sets of vertices and edges are D 0 (Z) and H 0 (Z) respectively, an edge h ij connecting the vertices d i and d j , and we call the module Z graphic.It turns out that the components Z α of Z are again graphic and the graphs Γ(Z α ) are precisely all path components of Γ(Z) (Theorem 4.18).Starting from this, we also obtain, under a mild restriction of the supertropical semring R, a description of all components of Z when Z is not graphic (Corollary 4.19).
In the last three sections §5- §8 we work on more general basic endomorphisms than basic projectors.The main result in §5 is that, under still mild conditions on R (in particular if the semiring eR is cancellative), every basic endomorphism ϕ of a basic module Z yields a basic projector p ϕ on Z by the rule p ϕ (z) = z if ϕ(z) = 0 and p ϕ (z) = 0 otherwise, for z ∈ B 0 (Z).Conversely, given a basic projector π on Z we can describe all basic endomorphisms ϕ of Z with p ϕ = π, called the satellites of π (Theorem 5.17 and Corollary 5.18).In §6 we develop other ways to obtain new basic endomorphisms from old ones.Given scalars µ, υ ∈ R, we say that υ is obedient to µ, if υ ≤ ν µ and υ is faithful to µ in the following sense: for all x, y ∈ R, if µx = µy, then υx = υy.Assume, for simplicity, that Z is graphic and ϕ is a basic endomorphism of Z with system of coefficients ( Then it turns out that every tuple of scalars (µ i | i ∈ K) ∪ (υ ij | {i, j} ∈ M ), with υ ij obedient to µ ij for all i, j, is again the coeffient system of some basic endomorphism ψ of Z (Theorem 6.9).We call such a basic endomorphism ψ an H-modification of ϕ.To give the flavor we point out what this theorem means in the case that Z = Quad(V ), ϕ = id Z , I = {1, . . ., n}.It says that for any family of scalars is a well defined basic operation on Quad(V ).The reason is that every λ ij is clearly obedient to 1.
Starting again with the system of coefficients ( While for H-modifications we obtained a best possible result, here our knowledge is less complete.We only know that a tuple (υ i ) ∪ (µ ij ) is the coefficient system of a basic endomorphism ψ if µ i ≤ υ i for all i ∈ K (minimal ordering ≤ instead of ν-dominance ≤ ν ), cf.Theorem 6.9.
In the last section §8 we determine, for a tangible supersemifield, R, all basic endomorphisms of any basic module Z.If Z is "linked", i.e., Z is graphic and Γ(Z) has no isolated vertices (the main case to be studied), it turns out that the possible coefficient systems ( are given by the condition µ 2 ij ≤ ν µ i µ j , provided that the ghost map ν R : T (R) ։ G(R) is not bijective.Otherwise there may exist more basic endomorphisms, cf.Theorem 8.5.

Partial quasilinearisation
Henceforth, R is a supertropical semiring and V is a free R-module with base (ε i | i ∈ I).Let I [2] denote the set of 2-element subsets of I. We choose a total ordering of I and often identify I [2] with the set of pairs (i, j) ∈ I × I such that i < j.If Λ is a subset of I [2] , let Λ c denote the complement I [2] \ Λ.We define a quasilinear quadratic form d i on V for every i ∈ I by (2.1) and a rigid quadratic form h ij for every i, j ∈ I with i = j by (2.2) Here, as always, the x i are the coordinates of the vector x ∈ V, x = i∈I x i ε i .
We work with the bases (d i | i ∈ I) and (h ij | (i, j) ∈ I [2] ) of the free R-modules QL(V ) and Rig(V ), respectively.For any set Λ ⊂ I [2] we introduce the free submodule Rig(Λ, V ) is a lower set in the R-module Quad(V ) both in the minimal and the functional ordering of Quad(V ).Clearly In other words, for any rigid form ρ on V we have a unique decomposition ).We call ρ 1 and ρ 2 the Λ-component and Λ c -component of ρ, respectively.
Λ-quasilinearity of q means that for every {i, j} ∈ Λ the set Proof.This is a consequence of the 1-1-correspondence between the off-diagonal companions of q and the rigid complements of q QL in q described in [8, Proposition 4.6].
Given Λ ⊂ I [2] we intend to associate to any q ∈ Quad(V ) a Λ-quasilinear form q Λ,QL ∈ Quad(V ) in a somewhat canonical way, generalizing the map q → q QL from Quad(V ) to QL(V ).The key to do this is provided by the following lemma.
Lemma 2.3.Let Λ ⊂ I [2] and q 0 ∈ QL(V ).Further let ρ, ρ ′ ∈ Rig(V ) be given, and let To prove the lemma we use part of the following notation, that shall also be helpful later.
Corollary 2.5 tells us that q Λ,QL does not depend on the choice of the rigid complement ρ in q.If Λ = I [2] , then q Λ,QL = q QL , while if Λ = ∅, then q Λ,QL = q.
Scholium 2.7.Let I = {1, 2, . . ., n}.We describe a given quadratic form q : V → R by a triangular scheme The quadratic form q Λ,QL is then given by the triangular scheme, where every entry α ij with {i, j} ∈ Λ is replaced by zero.

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Remark 2.8.(i) If q 1 , q 2 ∈ Quad(V ), then Let QL(Λ, V ) denote the R-submodule of Quad(V ) consisting of all Λquasilinear forms on V ; in other terms We have a natural map sending q ∈ Quad(V ) to its Λ-quasilinearization q Λ,QL .It is R-linear by Remark 2.8.Since π Λ,QL is additive, it is also plain that π Λ,QL respects the minimal ordering on Quad(V ), i.e., (2.7) q q ′ ⇒ q Λ,QL q ′ Λ,QL .Viewing every map π Λ,QL as an endomorphism of the R-module Quad(V ), we may state that 2) .In particular (Λ = M ), π Λ,QL ∈ End R (Quad(V )) is a projector.It can be characterized in terms of the minimal ordering of Quad(V ) as follows.
Proposition 2.9.For any q ∈ Quad(V ) the form q Λ,QL is the unique maximal form κ q, which is Λ-quasilinear.
Problem 2.10.For which supertropical semirings R, sets Λ ⊂ I [2] , and quadratic forms q ′ on R (I) , is it true that q ≤ q ′ implies q Λ,QL ≤ q ′ Λ,QL ?In addition to the cases where we know that q ≤ q ′ means the same as q q ′ (cf.[10, Corollaries 9.11 and 9.12 ]), there is one case where we can give an answer now, for any supertropical semiring R.
is the unique maximal Λ-quasilinear form κ on V (in the functional ordering of Quad(V )) with κ ≤ q. Proof.
, and so ϑ is quasilinear on Then, using Notation 2.4, we may write We have V = V J ⊕ V I\J and obtain for any x ∈ V J , y ∈ V I\J the formula q Λ,QL (x + y) = q(x) + q(y).

Basic modules and basic projections
We repeat that in the whole paper V is a free module over a supertropical semiring R and {ε i | i ∈ I} is a base (mostly fixed) of V .Let I [2] denote the set of all 2-element subsets of I.The R-module Quad(V ) has the set of generators B 0 := D 0 ∪H 0 with where d ( x) = x2 i and h ij = x i x j for x = i∈I x i ε i , as said above.Assume now that eR is multiplicatively unbounded, i.e., that for any x, y ∈ G there exists z ∈ G such that y < xz (cf.[10,Definition 6.4]). 2 Then, as proved in [10, §7], the set of generators B 0 of Quad(V ) is uniquely determined by the R-module Quad(V ) up to multiplication by units of R.More precisely, the set B := R * B 0 , consisting of all products λq with λ ∈ R * , q ∈ B 0 , coincides with the set of all "basic elements" of Quad(V ) (cf. [10, Definition 6.1]) and also with the set of all "primitive" (loc.cit.) indecomposable elements of Quad(V ) [10, Theorem 7.8, Corollary 7.9 ].Moreover, as has been shown in [10, §8], each of the sets is uniquely determined by the R-module Quad(V ), up to multiplication by units.It now makes sense to extend the notation B 0 , D 0 , H 0 as follows, and to define "basic submodules" of Quad(V ) without referring to a base of R.
Definition 3.1.Choosing sets of representatives D 0 and H 0 of the orbit sets D/R * and H/R * , we obtain a set B 0 := D 0 ∪H 0 which obviously generates the R-module Quad(V ).We call B 0 a basic set of generators of Quad(V ). 3sually we choose the sets derived from a base {ε i | i ∈ I} of the free R-module V , cf. (2.1), (2.2).Then we call B 0 := D 0 ∪ H 0 a geometric basic set of generators of Quad(V ). 4efinition 3.2.(a) We call an R-submodue Z of Quad(V ) basic if Z is spanned by a subset S of B 0 , i.e., every q ∈ Z has a presentation q = s∈S α s s with s ∈ S, α s ∈ R, almost all α s = 0. (b) If this holds, then S is uniquely determined by Z, namely S = Z ∩ B 0 , as follows immediately from the fact that every s ∈ S is primitive and indecomposable.We call S a basic set of generators of Z, and also write Z = RS (while RS just means the set of all products RS with λ ∈ R, s ∈ S).
If Z 1 and Z 2 are basic submodules of Quad(V ), then the modules Z 1 + Z 2 and Z 1 ∩ Z 2 are again basic in Quad(V ).We have and, of course, We already met preeminent basic submodules of Quad(V ).Both QL(V ) and Rig(V ) are basic in Quad(V ).If Λ is any subset of the set I [2] of two element subsets of I, then the submodule QL(Λ, V ) consisting of the Λ-quasilinear forms on V (cf.Definition 2.1) is basic.Also the submodule Rig(Λ, V ) of Rig(V ) (cf. (2.3)) is basic in Quad(V ).We are ready for the key definitions of this section.
(a) We call an endomorphism π of the R-module Z a basic projector on Z if π maps every q ∈ B 0 ∩ Z either to itself or to zero.Clearly, then π = π 2 and both We call X a basic projection of the R-module Z and Y a basic projection kernel in Z, and we also call (Z, X) a basic projection pair.(c) Whenever it is convenient, we identify the basic projector π : Z → Z with the associated projection map Z ։ X.(d) Notice that the projector π : Z → Z is uniquely determined both by the pair (Z, X) and the pair (Z, Y ).We write π = π Z,X .We call Y = π −1 (0) the kernel of the projector π, and usually write Y = ker(π).
(e) For the sake of brevity we often call a basic submodule Z of Quad(V ) simply a " basic module", suppressing the reference to the free Rmodule V , as long as V is kept fixed.
(a) If X is a basic projection of Z, then for any basic module W the intersection X ∩ W is a basic projection of Z ∩ W , and π Z∩W,X∩W is the restriction of π Z,X to W .In particular, if W ⊂ Z, then (The products are taken in End R (Z).) is also a basic projection of Z and .
(e) If X is a basic projection of Z and U is a basic projection of X, then U is a basic projection of Z and (where now π Z,X is identified with the associated projection map Z ։ X).
We strive for a combinatorial description of all basic projection pairs within the fixed R-module Quad(V ).
Notation 3.5.Given a basic module Z, we set Notice that if X and Z are basic R-modules then Corollary 3.6.Assume that Z is a basic module and N is a subset of I [2]  containing Λ(Z).Then the basic submodule X of Z with ∆(X) = ∆(Z) and Λ(X) = N is a basic projection of Z.The associated basic projector π Z,X is the restriction to Z of the N -partial quasi-linearization π N,QL (cf.(2.6)).
Proof.This follows from Remark 3.4.a,since X = Z ∩QL(N, V ) and QL(N, V ) is a basic projection of Quad(V ) with associated projector π N,QL .
Lemma 3.7.Assume that X is a basic projection of a basic module Z, and that Proof.Let π = π Z,X .In Z we have the relation (cf.[10, Eq. (7.10)]) Applying π to the relation we obtain
Proof.By Lemma 3.7 condition C Λ,∆ (i) is necessary for the existence of π.
Assuming now that C Λ,∆ (i) holds, we want to show that π exists and respects the functional ordering on Z. Without loss of generality we may assume that I is finite, I = {1, . . ., n}, and i = 1, furthermore that h 1j ∈ Z for 2 ≤ j ≤ r, but h 1j / ∈ Z for r < j ≤ n, with some r ∈ {1, . . ., n}.

Documenta Mathematica 22 (2017) 1661-1707
Basic Operations on Supertropical Quadratic Forms 1679 We claim that, given presentations of two forms q, q ′ ∈ Z with q ≤ q ′ , then (1) α 1j ≤ β 1j for 1 < j ≤ r; (2) If this is proven, then first assuming that q = q ′ , we learn that, if we omit in the presentation ( * ) the summand α 1 d 1 , we obtain a quadratic form q ∈ Z which is independent of the presentation ( * ), and thus we have a well defined basic projection π : q → q with kernel Rd 1 .(Use the claim for q ≤ q ′ and q ′ ≤ q.) Then using the claim in general, we learn that if q ≤ q ′ , then π(q) ≤ π(q ′ ), establishing (a) and (b).
We are ready to determine all basic projections X of a basic module Z.In view of Corollary 3.6 it suffices to look for those submodules X of Z where Λ(X) = Λ(Z), equivalently H 0 (X) = H 0 (Z).
Definition 3.11.We say that two elements d i and Note that the condition C Λ,∆ (i) from Theorem 3.10 means that d i is not linked in Z to any d j ∈ Z.
Theorem 3.12.Let X and Z be basic modules with X ⊂ Z and H 0 (X) = H 0 (Z).Then X is a basic projection of Z iff any two elements d i , d j ∈ D 0 (Z) which are linked in Z are both elements of D 0 (X).In more imaginative terms: we obtain all basic projections X of Z with H 0 (X) = H 0 (Z) by discarding from D 0 (Z) some elements which are not linked in Z to other elements of D 0 (Z).
Proof.(⇐): Lemma 3.7 means that, if X is a basic projection of Z, then this condition holds.
(⇒): Let Λ = Λ(Z), ∆ = ∆(Z), and Theorem 3.10 assures that for every i ∈ E there exists a basic projector ϑ i on Z with kernel Rd i .It now follows by Remark 3.4.cthat for any subset K of E the basic projection and, of course, H 0 (ϑ K (Z)) = H 0 (Z).
Corollary 3.13.Let Z be any basic module.Among the basic projections X of Z with H 0 (X) = H 0 (Z) there is a unique minimal one, X min .All basic submodules X of Z with X min ⊂ X are basic projections of Z, of course with H 0 (X) = H 0 (Z).
Theorem 3.14.With the hypotheses of Convention 3.8, let Z be any basic submodule of Quad(V ).Then any basic projector π of Z with π(q) = q for all H 0 (Z) respects the functional ordering on Z: Proof.Clear from Theorem 3.10.b,since such a projector has the shape ϑ K given in (3.2).

Direct decompositions; linked versus free modules
In this section, as well as in §5 and §6, it will be good to keep the following simple fact in mind.
Lemma 4.1.Assume that ϕ : X → Y is a linear map between R-modules X and Y (with R supertropical as always).Assume that S is a subset of Y which is convex w.r. to the minimal ordering (i.e., if s, t ∈ S, y ∈ Y and s y t, then y ∈ S).Then the preimage ϕ −1 (S) is convex in X.In particular (take Proof.This is evident, since ϕ is additive and so x x ′ implies ϕ(x) ϕ(x ′ ).
Note that a submodule S of an R-modules X is convex w.r. to the minimal ordering on X iff S is a lower set in X, since we always have 0 ∈ S.
In the sequel we assume that eR is mutiplicatively unbounded and Z is a "basic module", i.e., Z is a basic submodule of Quad(V ) for V a fixed free R-module with base {ε i | i ∈ I}, cf.Definitions 3.2 and 3.3.We want to get insight into the presentations of Z as a direct sum of submodules (which then are again basic modules).
We start with two easy facts.Proof.a): Since the elements of B 0 (Z) are indecomposable, it follows from X ∩ Y = {0} and X + Y = Z that B 0 (Z) is the disjoint union of B 0 (X) and B 0 (Y ).Thus the implication (1) ⇒ (2) is obvious.Furthermore, if X is a basic projection of Z, then Y is the associated projection kernel, i.e., Y = π −1 Z,X (0).The same holds for X, Y interchanged.This makes the implications (2) ⇔ (3) evident.
(2) ⇒ (1): Let z ∈ Z and z = x + y with x ∈ X, y ∈ Y .Then π Z,X (x) = x, π Z,X (y) = 0, π Z,Y (x) = 0, π Z,Y (y) = y, whence π Z,X (z) = x, π Z,Y (z) = y.Thus x and y are uniquely determined by z, which proves that Then each X α is a basic submodule of Z, and the basic set of generators B 0 (Z) is the disjoint union of the sets B 0 (X α ).
Proof.Since the elements of B 0 (Z) are indecomposable, it follows from X = α∈A X α and X α ∩ X β = {0} for α = β that every element of B 0 (Z) is contained in some X α , α ∈ A. Let X ′ α denote the R-submodule spanned by Documenta Mathematica 22 (2017) 1661-1707 X α ∩B 0 (Z).This is the maximal basic module of Z containing X α .Since B 0 (Z) is the union of the sets X α ∩ B 0 (Z) it is clear that Z = α∈A X ′ α .Picking any α ∈ A, we will be done by verifying that But if X and Y are convex submodules of the basic module X + Y = Z, where X ∩ Y = {0}, then it is not necessarily true that Z is the direct sum of X and Y , as the following key example shows.
As usual, we call an R-module U decomposable if there exist submodules 1 for all x 1 , x 2 ∈ R. In particular for x 1 = 1 we obtain βx 2 ≤ γ for all x 2 ∈ R, which forces β = 0 by multiplicative unboundedness.But Z is not the direct sum of the convex submodules Rd 1 and Rd 2 + Rh 12 .In fact ) Neither is Z the direct sum of Rd 2 and Rd 1 + Rh 12 .We conclude that Z is indecomposable.Remark 4.5.Hypothesis ( †) in Convention 3.8 is not needed in this example.If ( †) holds in addition to multipicativity unboundedness, then it is immediate from Theorem 3.10 that Rd 2 + Rh 12 is free.Also Rd 1 + Rh 12 is free, and so all proper basic submodules of Z are free.Definition 4.6.
(a) We call a basic module Z linked if (i) for every h ij ∈ Z both d i and d j are in Z; (ii) for each d i ∈ Z there exists some j = i such that h ij ∈ Z, and so an elementary linked module.Thus Z is linked iff Z is the sum of all elementary linked modules contained in Z.
(c) Given any basic module Z we denote the sum of all E ij ⊂ Z by Z link , which is the maximal linked submodule of Z.We call it the linked core of Z.
Example 4.4 shows that every elementary linked module is indecomposable.The formation of linked cores behaves well with respect to direct sums.
This is an immediate consequence of the fact that every elementary linked module is indecomposable, together with the following easy lemma.
Lemma 4.8.Assume that Z is a basic module and Then for any convex submodule W of Z we have Proof.We have a family (π α | α ∈ A) of basic projectors π α : Z → Z at hands with π α (Z) = X α .It follows that π α π β = π β π α = 0 for α = β and α∈A π α = id Z (which means that, given q ∈ Z, almost all values π α (q) are zero and α∈A π α (q) = q).Now π α (z) z for every z ∈ Z, α ∈ A, and thus π α (W ) ⊂ W for every α ∈ A. By restriction we obtain a family of projectors Proposition 4.9.Assume that Convention 3.8 is in force.Then Z is free iff Z link = 0.
Proof.When Z is free, all its basic submodules are free, and so Z cannot contain any elementary linked module, whence Z link = 0. (N.B.In this argument hypothesis ( †) is not needed.)Conversely, if Z link = 0, we know by Theorem 3.12 that Z ∩ Rig(V ) is a basic projection of Z with kernel Z ∩ QL(V ).(All d i ∈ D 0 (Z) may be discarded from the list B 0 (Z).)But also Z ∩ QL(V ) is a basic projection of Z, namely the image of the restriction π QL : Quad(V ) ։ QL(V ) to Z. Thus Since both QL(V ) and Rig(V ) are free, Z is also free.
Given a basic submodule X of Z, we call the unique basic module Y with Theorem 4.10.Assume that Convention 3.8 is in force.Then, the basic complement of Z link , Y , in Z is free and Proof.Y does not contain any elementary linked module, and thus is free by the preceding proposition.We will be done by verifying that both Z link and Y are basic projections of Z.
By mapping all the h ij ∈ Y to zero we obtain a basic projector Z ։ Z 1 with cf. Corollary 3.6.We have Z 1 ⊃ Z link and , and thus Theorem 3.10 gives us a basic projector Z 1 ։ Z link .Composing the two projectors we obtain a basic projector π 1 : Z ։ Z link .
On the other hand by mapping all h ij ∈ H 0 (Z link ) to zero we obtain a basic projector Z ։ Z 2 with As no d i ∈ D 0 (Z link ) is linked to any d j ∈ D 0 (Y ), we obtain a basic projector Z 2 ։ Y , again by Theorem 3.10, which together with Z ։ Z 2 yields a basic projector π 2 : Z ։ Y.The projectors π 1 and π 2 together entail Z = Z link ⊕ Y.
We denote the basic complement Y of Z link in Z by Z free , and obtain Corollary 4.11.Suppose Z is a basic module and Convention 3.8 is in force, then Z free is the unique maximal basic free submodule of Z which is a direct summand of Z.
Proof.Let Z = X ⊕ T with X free.Then (cf.Proposition 4.7) and T = T link ⊕ T free .We conclude that Z = X ⊕ T link ⊕ T free and also Z = Z link ⊕ Z free = T link ⊕ Z free , whence Z free = X ⊕ T free .
Definition 4.12.We call any indecomposable direct summand X = 0 of a basic module Z a component of Z.
We start out to determine the components of a basic R-module Z under Convention 3.8.First an easy preliminary lemma, valid over any supertropical semiring R.
Lemma 4.13.Assume that α∈A Z α is a direct decomposition of Z, where each Z α is indecomposable and = 0. Then these Z α are precisely all components of Z.
Proof.Let X be a basic nonzero submodule of Z.By Lemma 4.8 An edge h ij connects the vertices d i , d j .7 (c) A basic module Z has a unique maximal submodule which is graphic, denoted Z graph .B 0 (Z graph ) is obtained from B 0 (Z) by omitting every Note that Z is linked iff Z is graphic and Γ(Z) has no isolates vertices.When Convention 3.8 holds, clearly Z is graphic iff H 0 (Z free ) is empty.
Remark 4.15.Γ(Quad(V )) is the complete graph over the vertex set {d i | i ∈ I}, and thus may be seen as the graph (I, I [2] ).For any subgraph Γ ′ of Γ(Quad(V )) there exists a unique graphic module Z with Γ(Z) = Γ ′ .
Next we describe the components of graphic modules in graph theoretic terms.
As before we tacitly assume that G is multiplicatively unbounded.
Proposition 4.16.Assume that Z is a graphic module and The following are equivalent: (ii) Each X α is graphic and Γ(Z) is the disjoint union of the graphs Γ(X α ), for which we write When (i) and (ii) hold, the projection Z ։ X α is given, for q ∈ Z, by (cf.Notation 2.4) b) We now assume that Γ(Z) is the disjoint union of the graphs Γ(X α ), α ∈ A.
Firstly this implies that Z = α∈A X α .For each α ∈ A we define a map π α : Z → Z by formula (4.1).Let J := {i ∈ I | d i ∈ Z}, so that J is the disjoint union of the sets J α .Given i ∈ J we have (d i |V Jα ) I = d i for i ∈ J α and (d i |V Jα ) I = 0 otherwise.Given different i, j ∈ J we have a similar story: (h ij |V Jα ) I = h ij when {i, j} ⊂ J α and zero otherwise.This proves that π α is a basic projector on Z with image X α , and furthermore that π α π β = 0 if α = β and α∈A π α = id Z .It is now obvious that the sum α∈A X α is direct with associated projectors π α .
Corollary 4.17.A graphic module Z is indecomposable iff the graph Γ(Z) is connected.
Proof.We argue by contradiction.If Z = X 1 ⊕ X 2 with X 1 = 0, X 2 = 0, then by Proposition 4.16 both X 1 , X 2 are graphic and Γ(Z) = Γ(X 1 ) ⊔ Γ(X 2 ); so Γ(Z) is not connected.Conversely, assume Γ(Z) = Γ 1 ⊔Γ 2 and let X i denote the graphic submodule of Z with Γ(X i ) = Γ i (i = 1, 2).Then Z = X 1 ⊕ X 2 by Proposition 4.16, and thus Z is decomposable.Proof.We know by Proposition 4.16, that Z is the direct sum of the Z γ and by Corollary 4.17 that the Z γ are indecomposable.It follows from Lemma 4.8 that the Z γ are all components of Z.
We emphasize that in our study of the components of a basic module Z up to now hypothesis ( †) has not been needed.But if ( †) holds, then we know that Documenta Mathematica 22 (2017) 1661-1707 Z = Z link ⊕ Z free .Applying Theorem 4.18 to Z link , we obtain the following corollary.Furthermore, Z link is the direct sum of those components Z γ , which are not free, while Z free is the direct sums of all others.They are free of rank one.

Basic endomorphisms and their associated projectors
As before R is a supertropical semiring.Let Z be a basic module over R, i.e., a basic submodule of Quad(V ), for V a fixed free R-module V , cf.Definitions 3.2 and 3.3.While in §3 and §4 we studied basic projectors on Z, we now proceed to the more general "basic endomorphisms" of Z.We denote the set of endomorphisms of Z by End R (Z) or simply End(Z).This is an R-algebra in the obvious sense.As before we work with the set of generators of Quad(V ) derived from a fixed base of V , cf. (3.1).By intersection with Z it gives a "basic set of generators" B 0 (Z) = D 0 (Z) ∪ H 0 (Z), cf.Notation 3.5.
Definition 5.1.An endomorphism ϕ of Z is called basic, if ϕ(Rq) ⊂ Rq for every q ∈ B 0 , whence for all d i ∈ Z, h ij ∈ Z, with elements µ i , µ ij in R, which we name the coefficients of ϕ.The set of all basic endomorphisms of Z, denoted by End b (Z), is a commutative subalgebra of End(Z).
We remark that for any basic submodule Z ′ of Z, every ϕ ∈ End b (Z) restricts to a basic endomorphism ϕ|Z ′ of Z ′ .Note also that the basic projectors on Z are precisely the basic endomorphisms of Z with all coefficients in {0, 1}.They are idempotents of the R-algebra End b (Z). 8xample 5.2.If ρ is an endomorphism of the free R-module V , then for every quadratic form q : V → R, the composite q • ρ : V → R is again a quadratic form on V , and so we obtain an endomorphism of the R-module Quad(V ).We call these ρ * the geometric endomorphisms of Quad(V ).If ρ itself is "basic", i.e., ρ(ε i ) = µ i ε i for every i ∈ I with some µ i ∈ R, then an easy computation shows that whence ρ * is a basic endomorphism of Quad(V ).We denote this endomorphism by γ µ , where µ := (µ i | i ∈ I), and call the γ µ the geometric basic endomorphisms of Quad(V ).
These endomorphisms γ µ are the "easy" basic endomorphisms of Quad(V ).
Every tuple µ = (µ i | i ∈ I) ∈ R I gives such an endomorphism with system of coefficients (µ ).By restriction we obtain for γ µ a basic endomorphism γ Z,µ := γ µ |Z, and then have the following upshot of Example 5.2.
Proposition 5.3.Let Z be a basic module and set We call the γ Z,µ the geometric basic endomorphisms of Z.They form a subset of End b (Z), closed under multiplication.Note that this set does not depend on the choice of the base In particular we have the geometric basic projectors of Z at hands.These are the endomorphisms γ Z,µ with µ i ∈ {0, 1} for all i ∈ I(Z), and thus they correspond uniquely to the subsets J = {i ∈ I(Z) | µ i = 1} of I(Z).
Notations 5.4.We denote the basic projection coming from such a set J by π Z,J .In the most important case, namely Z = Quad(V ), we write π J instead of π Quad(V ),J and so Proposition 5.5.(a) For any J ⊂ I the geometric basic projector π J on Quad(V ) can be also described by the formula (q ∈ Quad(V )) (5.1) π J (q) = (q|V J ) I =: q J , cf.Notation 2.4.More generally, for any J ⊂ I and ϕ ∈ End b (Quad(V )), q ∈ Quad(V ), we have the formula Proof.(a): (5.1) is the formula (5.2) in the special case that ϕ is the identity map.In order to verify (5.2) it is suffices to check this formula for every (ϕ(q)|V J ) I has exactly the same values for all q ∈ B 0 .
(b): Obvious by considering the coefficients of the occurring projectors, or (better) the endomorphisms of V including these geometric endomorphisms of Quad(V ).
In the case of |J| ≤ 2 we simplify the notation by writing (5.3) These projectors will play a very helpful role later.All basic endomorphisms of Quad(V ) can be built from basic endomorphisms of the elementary linked submodules of Quad(V ) (cf.Definition 4.6) as follows.
Assume that Assume further that ϕ ij |Rd i = ϕ ik |Rd i for any three distinct i, j, k ∈ I. Then there is a unique ϕ ∈ End b (Z) with ϕ|E ij = ϕ ij , for all {i, j} ∈ I [2] .
Proof.We have elements for all {i, j} ∈ I [2] .The basic endomorphism ϕ of Z with system of coefficients (µ i ) ∪ (µ ij ) has the required properties, and clearly is the unique one, provided that ϕ exists.
For notational convenience we choose a total ordering on I.If q ∈ Quad(V ), and two presentations of q are given, (of course with only finitely many scalars α i , α ij , β ij = 0 ), we need to verify that For any pair i < j in I we apply the projector π ij to (A), viewed as a map onto E ij , and then the map ϕ ij .We obtain Since this holds for all i < j, (B) is now evident.From B we conclude that there is a well defined map ϕ : Theorem 5.8 (Extension Theorem).Let ϕ be a basic endomorphism of a graphic module Z ⊂ Quad(V ), and for every Proof.We choose a family ( We define ψ ij (d j ) by the same rule, setting ψ ij (h ij ) = 0.This map ψ ij is the composite of the quasilinear projector π QL |E ij and an endomorphism of the free module Rd i + Rd j with prescribed values for d i , d j , and thus is well defined.By construction it is clear that for any three different indices i, j, k we have ψ ij (d i ) = ψ ik (d i ).Thus the pasting Lemma 5.6 applies and yields a basic endomorphism ψ of Quad(V ), which clearly extends ϕ.
In particular, we can choose in Theorem 5.8 all v i = 0 to obtain an extension ψ = ϕ of ϕ to Quad(V ) with ϕ(q) = 0 for all q ∈ B 0 \ B 0 (Z).We call ϕ the extension of ϕ by zero.Convention 5.9.Up to end of §5 we assume, usually without explicitly stating it, that for R and Z one of the two following conditions holds.
Hypothesis A: G is multiplicatively unbounded and Z is a graphic module.Hypothesis B : G is multiplicatively unbounded and has property ( †), cf.Convention 3.8.Here Z can be any basic module.
Recall that, since G is assumed to be multiplicatively unbounded, the set R\{0} is closed under multiplication ([10, Remark 6.5]).
We are ready for a central result of this section.
Theorem 5.10.Given a basic endomorphism ϕ of Z there exists a unique basic projector on Z, denoted by p ϕ , such that any q ∈ B 0 (Z) has the image p ϕ (q) = q if ϕ(q) = 0 and p ϕ (q) = 0 if ϕ(q) = 0.
Proof.a) We assume Hypothesis A. We extend ϕ to a basic endomorphism ψ of Quad(V ) in some way, which is possible by Theorem 5.8.It suffices to prove the theorem for ψ instead of ϕ.Restricting p ψ to Z we then obtain the desired p ϕ .Thus we furthermore assume that Z = Quad(V ).We employ the Pasting Lemma 5.6.For any two indices i = j in I let There remains the case that exactly one of the vectors ϕ Inserting the vector x = v i +cv j for any c ∈ R we obtain that µ i + cµ ij = µ i for all c ∈ R. Since G is multiplicatively unbounded, this forces µ ij = 0, i.e., ϕ(h ij ) = 0. Now define p ij as the composite of π QL |E ij with the obvious projection from Rd i + Rd j to Rd i .Then As just proved there exists a unique basic projector π 1 = p ϕ1 on Y 1 with π 1 (q) = q ⇔ ϕ 1 (q) = 0 for all q ∈ B 0 (Y 1 ).Since Y 2 is free we trivially also have a unique basic projector π 2 on Y 2 such that for all q ∈ B 0 (Y 2 ) π 2 (q) = q ⇔ ϕ 2 (q) = 0.The projector p ϕ := π 1 ⊕ π 2 on Z has the required property addressed in the theorem.
We call p ϕ the basic projector associated to ϕ.We now strive for a characterization of the image and the kernel of p ϕ in terms of the image and kernel of ϕ.It is obvious that but determining B 0 (M b ) can be difficult.For example, if M = Rq with q ∈ Quad(V ), then D 0 (M b ) consists of all d i showing up in q QL , while H 0 (M b ) is the minimal subset of H 0 appearing in a rigid complement of q QL , in other terms, the minimal subset Λ ⊂ I [2] such that q is Λ c -quasilinear.But there is an extended class of submodules M of Quad(V ) at hands, for which the determination of B 0 (M b ) is trivial.
Definition 5.12.We call a submodule M of Quad(V ) diagonal if We read off from (5.4) that which proves that p ϕ (Z) = ϕ(Z) b .In the same way with q i ∈ B 0 , λ i = 0, then n 1 λ i ϕ(q i ) = 0, whence λ i ϕ(q i ) = 0, and so all ϕ(q i ) = 0, since R has no zero-divisors.Therefore, p ϕ = π can also be characterized by the property that, for all q ∈Z, π(q) = 0 iff ϕ(q) = 0.It will be helpful to work in End b (Z) with a partial ordering finer than the minimal ordering, analogous to our setting for Z itself.Given ϕ, ψ ∈ End b (Z), we define ϕ ≤ ψ ⇔ ∀z ∈ Z : ϕ(z) ≤ ψ(z), where on the right hand side " ≤ ′′ stands for the function ordering on Z ⊂ Quad(V ).We call this finer relation ≤ the function ordering on End b (Z), in contrast to the minimal ordering which is denoted by and defined as for all i with d i ∈ Z and all {i, j} with h ij ∈ Z.
The finiteness assumption in Theorem 5.17 is not essential; it can be easily extended to the case in which V has an infinite basis {ε i | i ∈ I}, as follows: Corollary 5.18.Assume that either Hypothesis A or B holds, and that there is some ϑ < e in G. Let π ∈ P b (Z) and ϕ ∈ End b (Z).Then ϕ is a satellite of π iff for every finite subset K of I there exist α where Z K := (Z ∩ V K ) I = π K (Z) for V K = i∈K Rε i , and π K is the (geometric) basic projector (5.1) on Quad(V ) with image Quad(V K ) I (cf.Proposition 5.5).
Scholium 5.20.Suppose π 1 , π 2 ∈ P b (Z).Then (a) It turns out that each fiber of τ Z is closed under addition and multiplication.More generally the following holds.

Modifications of basic endomorphisms
Up to now the only explicit examples of basic endomorphisms, which we have met, are the basic projectors ( §3, §4) and the geometric basic endomorphisms (cf.§5, Example 5.2 and Proposition 5.3).We now look for procedures to obtain new basic endomorphisms from old ones.We start with a definition and a lemma valid in any supertropical semiring R. We intensely use the ν-notation, cf.§1.4.Definition 6.1.Given µ, υ ∈ R, we say that υ is obedient to µ if where µ ∈ T .(f) If υ is obedient to µ and υ ∈ T , then also eυ is obedient to µ, but most often υ is disobedient to eυ.Lemma 6.3.Assume that υ ∈ R is obedient to µ ∈ R and that a + µb = a + µc for a, b, c ∈ R. Then a + υb = a + υc.
We run through three subcases.
2.a: Suppose a ∈ G. Then ( * ) implies directly that a + υb = a = a + υc.We have proved that in all cases a + υb = a + υc.
The lemma ensures the following fact about quadratic forms, valid for any module V over a supertropical semiring.
In the following we assume as before that V is a free quadratic R-module with base {ε i | i ∈ I}, and that |I| > 1, discarding a trivial case.

Documenta Mathematica 22 (2017) 1661-1707
Basic Operations on Supertropical Quadratic Forms 1697 Theorem 6.5.Assume that Z is a graphic submodule of Quad(V ).As before, we write with K ⊂ I, M ⊂ I [2] .Assume that ϕ is a basic endomorphism of Z, having the system of coefficients and furthermore that a family (υ ij | {i, j} ∈ M ) of elements of R is given such that υ ij is obedient to µ ij for every {i, j} ∈ M .Then there exists a basic endomorphism ψ of Z having the coefficient system Proof.a) We first prove the theorem for the special case that I = {1, 2} and In this case ϕ ∈ End b (Z) has three coefficients µ 1 , µ 2 , µ, where µ = µ 12 , and we are given an element υ = υ 12 of R obedient to µ.Then and we claim that there is a basic endomorphism ψ of E 12 with ψ(d q 1 = αh 12 , q ′ 1 = βh 12 , we see that indeed (2) is a consequence of (3).b) We employ the Pasting Lemma 5.6 to prove the theorem for Z = Quad(V ) in general.Given {i, j} ∈ I [2] , let having the coefficients µ i , µ j , µ ij .By step a), for every {i, j} ∈ I [2] , there exists Thus Lemma 5.6 applies and gives us a basic endomorphism ψ of Quad(V ) with ψ|E ij = ψ ij for every {i, j} ∈ I [2] .This basic endomorphism has the desired system of coefficients (µ i | i ∈ I) ∪ (υ ij | {i, j} ∈ I [2] ).
c) Finally, we prove the theorem in its full generality.Given an endomorphism ϕ ∈ End b (Z), we extend it to a basic endomorphism ϕ ∈ End b (Z) with coefficients ( µ i | i ∈ I) ∪ ( µ ij | {i, j} ∈ I [2] ), where µ i = µ i when i ∈ K, otherwise µ i = 0, and µ ij = µ ij for {i, j} ∈ M , otherwise µ ij = 0 (extension by zero, cf.Theorem 5.8).We further extend the family (υ ij | {i, j} ∈ M ) to a family ( υ ij | {i, j} ∈ M ) by setting υ ij = υ ij when {i, j} ∈ M and υ ij = 0 otherwise.It is evident that υ ij is obedient to µ ij for any {i, j} ∈ I [2] .By step b) there exists a basic endomorphism ψ of Quad(V ) with coefficients ( µ i | i ∈ I) ∪ ( υ ij | {i, j} ∈ I [2] ).The restriction ψ := ψ|Z has the desired system of coefficients (µ i | i ∈ K) ∪ (υ ij | {i, j} ∈ M ).Comment.We may interpret the pair (Z, ϕ) as a weighted graph by assigning weights to the edges and to the vertices of the graph Γ(Z) of Z, namely weight µ i to the vertex d i ∈ VerΓ(Z) and weight µ ij to the edge h ij ∈ EdgΓ(Z).For example, for Z = Quad(V ) and I = {1, 2, 3} we have the weighted triangle If, say, R is cancellative, we obtain all H-modifications of ϕ by lowering the ν-values of the edges.Definition 6.7.Let Z be a graphic submodule of Quad(V ).Let K ⊂ I, and let ϕ be a basic endomorphism of Z, having the system of coefficients (µ i | i ∈ K) ∪ (µ ij | {i, j} ∈ M ), according to the notation in Theorem 6.5.We say, that a basic endomorphism ψ of Z with coefficients (υ i | i ∈ K) ∪ (υ ij | {i, j} ∈ M ) is a D-modification of ϕ, if µ ij = υ ij for {i, j} ∈ M , but µ i ≤ ν υ i for i ∈ K, in other terms ψ(h ij ) = ϕ(h ij ) for all h ij ∈ Z, while ψ(d i ) ≥ ν ϕ(d i ) for all d i ∈ Z.
Open Problem 6.8.In contrast to the situation of H-modifications we do not know whether for every list of coefficients (υ i | i ∈ K) ∪ (µ ij | {i, j} ∈ M ), where υ i ≥ ν µ i for all i ∈ K, a D-modification of ϕ exists .
In general we only have the following weaker result.Theorem 6.9.Given a basic endomorphism ϕ on a graphic submodule Z with coefficients (µ i | i ∈ K) ∪ (µ ij | {i, j} ∈ M ), as in the notation of Theorem 6.5, and a family (υ i | i ∈ K) in R such that µ i ≤ υ i (minimal ordering instead of ν-dominance), there exists a basic endomorphism ψ of Z with coefficients
At the moment R may be any supertropical semiring.Given a triple (µ 1 , µ 2 , µ) ∈ R 3 , we enquire whether a basic endomorphism ϕ of Z with coefficients µ 1 , µ While in the last sections, starting from §3, we have been eager to impose only mild restrictions on the supertropical semiring R, we now solve this problem for the class of tangible supersemifields.We start with a general lemma, implementing a previous argument in different context.
We now obtain a partial answer, to Problem 8.1 as follows.
in R serves as the system of coefficients of a basic endomorphism of Quad(V ) iff (8.3) µ 2 ij ≤ ν µ i µ j for all i, j ∈ I with i < j.The same holds when T e = {1} and R is dense.
(b) When R is discrete and T e = {1}, Condition (8.3) has to be replaced by the more complicated condition, that for all i < j either µ 2 ij ≤ ν µ i µ j or µ 2 ij ∼ =ν π −1 µ i µ j , (µ i , µ j , µ ij ) ∈ G 3 .(c) Mutatis mutandis, all this remains true if we replace Quad(V ) by any linked submodule Z of Quad(V ).
Proof.This follows from Theorem 8.3 by the same line of thought as in parts (b) and (c) of the proof of Theorem 6.5, using the Pasting Lemma 5.6.
When Z is a basic R-module, and R is a tangible supersemifield, then Z = Z link ⊕ Y with Y a free R-module (cf.Theorem 4.10).Every basic endomorphism ϕ of Z is a direct sum ϕ = ϕ 1 ⊕ ϕ 2 , with ϕ 1 and ϕ 2 basic endomorphisms of Z link and Y respectively.Since Y is free there are no restrictions on the coefficients system of ϕ 2 , and therefore we know all the endomorphisms of Z.We omit the details.
Remark 8.6.As a consequence of Theorems 8.5 and 6.9, Problem 6.8 about Dmodifications has a positive answer when R is a tangible supersemifield.Every change of coefficients µ i → υ i (i ∈ K), prescribed in Definition 6.7 for a given basic endomorphism ϕ is realized by a D-modification of ϕ.Here we do not need to bother about the case that T e = {1}, since then Problem 6.8 vanishes: If υ i ≤ ν µ i , then υ i ≤ µ i .

Convention 3 . 8 .Proposition 3 . 9 .
Up to the end of this section, we assume that the supertropical semiring, R, besides multiplicative unboundedness, satisfies the following condition.(†) If a, b ∈ R and there exists somec 0 ∈ R such that ac ≤ bc for all c ≥ c 0 , then a ≤ b.This condition is a rather mild hypothesis, as the following proposition reveals.We introduce the setcanc(G) := {c ∈ G | ∀a, b ∈ G : ac = bc ⇒ a = b},consisting of the "cancellative" elements of G. Assume that the set canc(G) is unbounded in G, and furthermore that T is closed under multiplication and eT is unbounded in G. Then R has Property ( †).Proof.Let a, b ∈ R and assume that ac ≤ bc for all c ≥ c 0 in R, for some c 0 ∈ R. We want to verify that a ≤ b.If a < ν b then a < b, and we are done.Henceforth assume that a ≥ ν b.Then ac ≥ ν bc for all c ∈ R, and we conclude that ac ∼ =ν bc for all c ≥ c 0 .Since canc(G) is unbounded, this implies a ∼ =ν b.If a, b ∈ eR this means that a = b; and if a ∈ T and b ∈ G, then a ∼ =ν b implies a < b.Assume finally that b ∈ T and pick u ∈ T with u ≥ c 0 , which is possible since eT is unbounded in G. Now au ≤ bu ∈ T and au ∼ =ν bu.This forces au = bu.Thus au ∈ T .which implies a ∈ T .From a ∼ =ν b and a, b ∈ T we conclude again that a = b.Thus a ≤ b in all cases.Theorem 3.10.Let Z be a basic module, Λ := Λ(Z), ∆ := ∆(Z).Assume that i ∈ I is an index with d i ∈ Z.

Proposition 4 . 2 . 2 ) 3 )
Assume that X and Y are two basic submodules of a basic module Z with X ∩ Y = {0} and X + Y = Z.(a) The following are equivalent (1) Z = X ⊕ Y ; (Both X and Y are basic projections of Z; (Both X and Y are basic projection kernels of Z.(b) If (1)-(3) hold the both X and Y are convex submodules of Z.

Definition 4 .
14. (a) We call a basic module Z graphic if for any h ij ∈ H 0 (Z) both d i d j are in Z.(b) When Z is graphic, we define the graph Γ(Z) (simple, undirected, without loops) as follows.Γ(Z) has the sets of vertices and edges Ver(Γ(Z)) = D 0 (Z) and Edg(Γ(Z)) = H 0 (Z).

Theorem 4 . 18 .
Assume that Z is a graphic module.Let (Γ γ | γ ∈ C) denote the set of components of the graph Γ(Z), arbitrarily indexed, and for every γ ∈ C let Z γ denote the graphic module with Γ(Z γ ) = Γ γ .Then Z = γ∈C Z γ and the Z γ are precisely all components of the basic module Z.

Corollary 4 .
19. Assume that Convention 3.8 is in force.Let Z be a basic module, and let (Z γ | γ ∈ C) denote the set of components of Z indexed in some way.Then Z = γ∈C Z γ .
0. By construction p ij (d i ) = p ik (d i ) for any three different indices i, j, k ∈ I. Thus there exists a basic endomorphism p of Quad(V ) with p|E ij = p ij for any i = j.It is the projector p ϕ we were looking for.b) We assume now Hypothesis B. By Theorem 4.10 there is a decomposition Z = Y 1 ⊕ Y 2 with Y 1 = Z link and Y 2 a free module.Given ϕ ∈ End b (Z), we have ϕ(Y i ) ⊂ Y i for i = 1, 2, and thus ϕ = ϕ 1 ⊕ ϕ 2 with ϕ i ∈ End b (Y i ).

Definition 5 . 11 .
Given an R-submodule M of Quad(V ), let M b denote the unique maximal basic submodule of Quad(V ) contained in M , and let M b denote the unique minimal basic submodule of Quad(V ) containing M .M b and M b are respectively called the basic core and basic hull of M .

Examples 5 .Proposition 5 . 14 .
13. (a) Every basic module Z is diagonal.(b) Every submodule of Quad(V ) that is convex in the minimal ordering of Quad(V ) is easily seen to be diagonal.(c) If M is a diagonal submodule of a basic module Z, then for any ϕ ∈ End b (Z) also ϕ(M ) is diagonal.(d) Intersections and sums of families of diagonal submodules of Quad(V ) are diagonal.(e) R(d i + d j ) is not diagonal if i = j.(This module has the basic hull Rd i + Rd j .)If ϕ is a basic endomorphism of a basic module Z then ϕ(Z) is a diagonal module and p ϕ (Z) = ϕ(Z) b .The kernel ϕ −1 (0) is itself basic and so p −1 ϕ (0) = ϕ −1 (0) = ϕ −1 (0) b .Proof.It is evident that the modules ϕ(Z) and ϕ −1 (0) are diagonal by Examples 5.13.(b) and (c).By definition of p ϕ we have

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Mathematica 22 (2017) 1661-1707 Example 5.15.Let µ = (µ i | i ∈ I) ∈ R I .Then the geometric basic endomorphism γ µ of Quad(V ) (cf.Example 5.2) has the associated basic projector π J (cf.Notation 5.4) with J = {i ∈ I | µ i = 0}.We turn to the inverse problem of analyzing the set of basic endomorphisms ϕ with p ϕ = π for a given basic projector π.Definition 5.16.Let Z be a basic module.(a) We denote the set of all basic projectors on Z by P b (Z).(b) If π ∈ P b (Z), then we call an endomorphism ϕ ∈ End b (Z) with p ϕ = π a satellite of π; the set of these ϕ is denoted by Stl(π).