A Generalization of Strassen’s Theorem on Preordered Semirings

Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.


Introduction
Motivated by the study of asymptotic tensor rank and more generally relative bilinear complexity, in [16] Strassen developed the theory of asymptotic spectra of preordered semirings (see also [17,Chapter 2] for a recent exposition). In this context, the spectrum of a (commutative, unital) semiring S with respect to a preorder is the set Δ(S, ) ofmonotone homomorphisms S → R ≥0 , i.e. maps f satisfying f (x + y) = f (x) + f (y), f (xy) = f (x)f (y), f (1) = 1 and x y =⇒ f (x) ≤ f (y). Strassen introduces the asymptotic preorder as x y iff there is a sublinear nonnegative integer sequence (k n ) n∈N such that for all n the inequality 2 k n x n y n holds (see Section 2 for precise definitions). Every element f ∈ Δ(S, ) satisfies x y =⇒ f (x) ≥ f (y). The key insight, and the content of [16,Corollary 2.6.], is that under an additional boundedness assumption the converse also holds in the sense that (∀f ∈ Δ(S, ) : f (x) ≥ f (y)) =⇒ x y (see the precise statement below).
In the main application in [16] the role of S is played by the set of (equivalence classes of) tensors over a fixed field and of fixed order but arbitrary finite dimension. The operations are the direct sum and the tensor (Kronecker) product of tensors and the preorder is given by tensor restriction (alternatively: degeneration). Asymptotic tensor rank can be characterized in terms of the resulting asymptotic restriction preorder, and as a consequence also in terms of the asymptotic spectrum.
More recently, the theory of asymptotic spectra has been applied to a range of other problems as well. In [18] Zuiddam introduced the asymptotic spectrum of graphs and found a dual characterization of the Shannon capacity [15]. Continuing this line of research, Li and Zuiddam studied the quantum Shannon capacity (defined as the regularization of the quantum independence number [12]) and entanglement-assisted quantum capacity [3] of graphs as well as the entanglement-assisted quantum capacity of noncommutative graphs [5] via the asymptotic spectra of suitable semirings of graphs and noncommutative graphs [11]. In the context of quantum information theory, tensors in Hilbert spaces represent entangled pure states, and the relevant preorder is given by local operations and classical communication (see e.g. [2]). This again gives rise to a preordered semiring [10] which refines the tensor restriction preorder and which also fits into Strassen's framework. In this application, the asymptotic spectrum provides a characterization of converse error exponents for asymptotic entantanglement transformations, a vast generalization of the results of [9].
At the same time, it became clear that the boundedness condition in Strassen's theorem is too restrictive for some purposes. For example, it does not hold in the semiring of noncommutative graphs where one could hope for a characterization of zero-error capacities similarly to [18]. To motivate a relaxation of this condition, note first that the definition of the asymptotic preorder does not make use of the additive structure and in the applications one is mainly interested in the ordered commutative monoid S \ {0} with multiplication and the preorder. Such objects have been studied in [6] as a mathematical model for resource theories [1]. In this context, asymptotic properties such as the asymptotic preorder above and more generally, rate formulas are of central interest. [6,Theorem 8.24.] implies that if an ordered commutative monoid has an element u such that for every x there is k ∈ N such that u k x and u k x 1, then u o(n) x n y n for all large n iff for all monotone homomorphisms f into the semigroup R ≥0 (with multiplication and the usual order) the inequality f (x) ≥ f (y) holds. At this point the relation to Strassen's result should be clear: if the monoid in question is the multiplicative monoid of a preordered semiring and the above condition is satisfied with u = 2, then one can restrict to monotones that preserve both operations. One may wonder if u = 2 is necessary for this stronger conclusion to hold, but simple examples show that without this assumption the set of monotone semiring homomorphisms can be too small to characterize the asymptotic preorder. In this spirit, Tobias Fritz proved a generalization of Strassen's theorem [8], listing conditions that are equivalent to ∀f ∈ Δ(S, ) : f (x) ≥ f (y) and generalize the asymptotic preorder in different ways, emphasizing also the similarity to Positivstellensätze.
Our main result is Theorem 2 below, a sufficient condition under which the spectrum does characterize the asymptotic preorder as above and which generalizes Strassen's condition. Before stating our main theorem, let us formulate Strassen's result in a way that eases comparison. We say that an element u ∈ S is power universal if u 1 and for every x ∈ S \ {0} there is a k ∈ N such that u k x 1 and u k x [8]. If S contains a power universal element then S is said to be of polynomial growth. The asymptotic preorder is defined as x y iff there is a sublinear nonnegative integer sequence (k n ) n∈N such that for all n the inequality u k n x n y n holds. Note that the asymptotic preorder does not depend on the choice of the power universal element (see Lemma 1 below). With these definitions Strassen's theorem on asymptotic spectra can be stated as follows.

Δ(S, ) is a nonempty compact Hausdorff space.
Our main result is the following (for s ∈ S, ev s : Δ(S, ) denotes the evaluation map f → f (s)).
Theorem 2 Let (S, ) be a preordered semiring of polynomial growth such that the canonical map N → S is an order embedding, and let M ⊆ S and S 0 be the subsemiring generated by M. Suppose that (M1) for all s ∈ S \ {0} there exist m ∈ M and n ∈ N such that 1 nms and ms n (M2) for all m ∈ M such that ev m : Δ(S 0 , ) → R ≥0 is bounded there is an n ∈ N such that m n.
Then for every x, y ∈ S we have

Δ(S, ) is a nonempty locally compact Hausdorff space and if u is power universal then
This result is a generalization of Theorem 1: if u = 2 is power universal then one can choose M = {1} for which both conditions are easily verified, whereas the topological part follows from the fact that ev u is a constant map in this case.
As a concrete example, we briefly describe a preordered semiring to which our theorem applies but does not satisfy the conditions of Theorem 1. This application will be more fully explored and generalized in the context of quantum information theory in a separate paper [13]. The elements of this semiring are pairs of measures on finite sets (p, q), up to bijections almost everywhere. Equivalently, one may consider pairs of equal-dimensional vectors of nonnegative real numbers. The operations are induced by the disjoint union and the tensor product, while the preorder is the relative submajorization [14], defined as (p, q) ≥ (p , q ) if there is a substochastic matrix T such that Tp ≥ p and T q ≤ q entrywise.
It may happen that (M1) is not satisfied by any subset M of S but can be satisfied after localizing at a suitable multiplicative set T . We will see that localization does not affect the asymptotic preorder or the spectrum, which leads to the following corollary, a somewhat more flexible version of our main result: Corollary 1 Let (S, ) be a preordered semiring of polynomial growth such that N → S is an order embedding. Let M ⊆ S and T ⊆ S \ {0} be a multiplicative set containing 1, and let S 0 be the subsemiring generated by M ∪ T . Suppose that (M1') for all s ∈ S \ {0} there exist m ∈ M, t 1 , t 2 ∈ T and n ∈ N such that t 2 nmt 1 s and mt 1 s nt 2 (M2') for all m ∈ M and t 1 , t 2 ∈ T such that ev m ev t 1 ev t 2 Then for every x, y ∈ S we have Verifying (M2') may require a fairly detailed knowledge of Δ(S 0 , ), therefore in practice it may be desirable to choose M and T in such a way that S 0 is as simple as possible (subject to the conditions above). We note however that having a complete classification of the elements of Δ(S 0 , ) is not a prerequisite for verifying (M2').
In particular, choosing M = {1} and T = S \ {0} (effectively replacing S with its semifield of fractions) guarantees (M1') and leads to a nontrivial equivalent condition for the characterization property (3).
Corollary 2 Let (S, ) be a preordered semiring of polynomial growth such that N → S is an order embedding. The following conditions are equivalent: Under the conditions of Theorem 1, the asymptotic spectrum has a certain uniqueness (or minimality) property [16,Corollary 2.7.]. Before stating the version that is true in the more general setting of Theorem 2, note first that the map s → ev s is a semiring homomorphism into C(Δ(S, )), the space of continuous functions on Δ(S, ), monotone with respect to the pointwise partial order. Given a preordered semiring (S, ) of polynomial growth, let us call a pair (X, Φ) an abstract asymptotic spectrum for (S, ) if X is a locally compact topological space, Φ : S → C(X) a semiring homomorphism such that Φ(S) separates the points of X, the image of every power universal element is a proper map, and ∀x, y ∈ S :

Proposition 1 Let (X, Φ) be an abstract asymptotic spectrum for (S, ). Then there is a unique homeomorphism
The proofs are organized as follows. Section 2 studies basic properties of preordered semirings, including the asymptotic preorder and its relation to localization. In Section 3 we define the spectrum as a topological space and study the continuous maps between spectra induced by monotone homomorphisms between preordered semirings. Section 4 contains the proof of our main result, Theorem 2, and of Proposition 1.

Related Work
After finishing this work we learned that in an independent work [7] Fritz has proved a different generalization of Strassen's theorem.

Asymptotic Preorder
By a semiring we mean a set equipped with binary operations + and · that are commutative and associative and have neutral elements 0 and 1 such that 0·a = a and a(b +c) = ab +ac for any elements a, b, c. A semiring homomorphism ϕ : S 1 → S 2 is a map satisfying Definition 1 A preordered semiring is a pair (S, ) where S is a semiring, is a transitive and reflexive relation on S such that 0 1 and when a, b, c ∈ S satisfy a b then a + c b + c and ac bc.
Let (S 1 , 1 ) and (S 2 , 2 ) be preordered semirings. A semiring homomorphism ϕ : Definition 2 ([8, Definition 2.8.]) Let (S, ) be a preordered semiring. An element u ∈ S is power universal if u 1 and for every x ∈ S \ {0} there is a k ∈ N such that x u k and u k x 1.
If such an element exists then S is said to be of polynomial growth.
It is clear that any element larger than a power universal one is also power universal. More generally, if u 1 and u (u ) k for some k ∈ N then u is also power universal. With the help of a power universal element we can define a generalization of the asymptotic preorder [16, (2.7)] as follows.
Definition 3 Let (S, ) be a preordered semiring and u ∈ S a power universal element. The asymptotic preorder u is defined as x u y iff there is a sequence (k n ) n∈N such that lim n→∞ k n n = 0 ( 4 ) and ∀n ∈ N : u k n x n y n .
Since u 1, we may assume whenever convenient that (k n ) n∈N is nondecreasing. By multiplying the inequalities we may replace k n with a subadditive sequence, or even require the inequality in Eq. 5 only for infinitely many n.
The asymptotic preorder is defined in terms of a power universal element. However, changing to a different power universal element does not affect the resulting preorder, as the following lemma shows.

Lemma 1
If u 1 and u 2 are power universal elements of S then u 1 = u 2 .
Proof By symmetry it is enough to prove u 1 ⊇ u 2 . Let k ∈ N such that u 2 u k 1 . Suppose that x u 2 y. This means there is a sublinear sequence (k n ) n∈N of natural numbers such that Eq. 5 holds with u 2 . Then we have u kk n 1 x n u k n 2 x n y n for all n and kk n /n → 0, therefore x u 1 y.
By Lemma 1 the asymptotic preorder is determined by the preordered semiring of polynomial growth even without specifying a power universal element. For this reason we will drop the subscript from the notation and write for the asymptotic preorder.
The following pair of lemmas show basic properties of the asymptotic preorder. Some of these are analogous to those in [17, Lemma 2.3., Lemma 2.4.], with nearly identical proofs.

Lemma 2 Let (S, ) be a preordered semiring and u a power universal element. Then
(iii) u is power universal with respect to the asymptotic preorder. (iv) Suppose that x, y ∈ S and there is an s ∈ S \ {0} such that sx sy. Then x y.
(v) Let x, y, s, t ∈ S \ {0} and suppose that for all n ∈ N the inequality sx n ty n holds.
Then x y.
Proof (i): If x y then Eq. 5 is satisfied with k n = 0. (ii): Reflexivity and 0 1 follows from (i). To prove transitivity let x y and y z. Choose sublinear sequences (k n ) n∈N and (l n ) n∈N such that u k n x n y n and u l n y n z n for all n. Then n → k n + l n is also sublinear and u k n +l n x n u l n y n z n , therefore x z. We prove compatibility with the operations. Let x y and z ∈ S, and choose (k n ) n∈N sublinear and nondecreasing such that u k n x n y n for all n. Then also u k n (xz) n (yz) n for all n, therefore xz yz. For the sum we use therefore x +z y +z. (iii) is an immediate consequence of (i). (iv): Let k ∈ N be such that u k s 1 and u k s. By induction, sx n sx n−1 y . . . sy n , and therefore u k x n y n for all n. This implies x y. (v): Let k, l ∈ N be such that u k t 1 and u l s. Then u k+l x n u k sx n u k ty n y n for all n, therefore Eq. 5 is satisfied with k n = k + l.
By (ii) and (iii) of Lemma 2 we can iterate the construction and form the asymptotic preorder that compares large powers via . However, the following lemma tells us that this does not give anything new.
Proof We have ⊆ by (i) of Lemma 2. Let x y and choose (k n ) n∈N sublinear such that u k n x n y n for all n. Let (l n,m ) n,m∈N be such that for all n lim m→∞ l n,m /m = 0 and for all n, m u l n,m +mk n x nm = u l n,m (u k n x n ) m (y n ) m = y nm (7) holds. Choose a sequence n → m n such that l n,m n ≤ m n (e.g. let m n be the first index such that l n,m n ≤ m n ). Then lim n→∞ l n,m n + m n k n nm n ≤ lim Our next goal is to understand how the asymptotic preorder interacts with localization.
. We denote the equivalence class of (s, t) by s t . There is a canonical homomorphism S → T −1 S that sends s to s 1 . The additive and multiplicative neutral elements are 0 1 and 1 1 .
Proof The roles of the primed and unprimed elements are symmetric, therefore it is enough to show that rs 1 t 2 rs 2 t 1 with r ∈ T implies ∃r ∈ T : r s 1 t 2 r s 2 t 1 . Let q 1 , q 2 ∈ T such that q 1 s 1 t 1 = q 1 s 1 t 1 and q 2 s 2 t 2 = q 2 s 2 t 2 . Let r = q 1 q 2 t 1 t 2 r. Then According to Lemma 4 we can define a relation on T −1 S as s 1 t 1 s 2 t 2 ⇐⇒ ∃r ∈ T : rs 1 t 2 rs 2 t 1 .
Lemma 5 and rs 1 t 2 s t rs 2 t 1 s t , therefore (ii): If x, y ∈ S and x y then x 1 y 1 since Eq. 10 is satisfied for r = 1. : Let u ∈ S be power universal, x = s t ∈ T −1 S \ { 0 1 }. Choose k s , k t such that the inequalities s u k s , u k s s 1, t u k t , u k t t 1 hold. Then s u k s u k s +k t t, therefore s t and t u k t u k s +k t s, therefore 1 1 It should be noted that even though we use the same symbol for the preorder on S and that induced on T −1 S, it is in general not true that x y iff x 1 y 1 . When S is of polynomial growth, we denote by the asymptotic preorder on T −1 S. Note that this could mean two different relations, depending on whether we form the asymptotic preorder on the localization or consider the preorder induced on T −1 S by the asymptotic preorder on S. However, these two relations turn out to be the same. If rs 1 t 2 rs 2 t 1 then r n u o(n) s n 1 t n 2 r n s n 2 t n 1 for all n, i.e. u 1 Conversely, if Eq. 16 holds, then there is a sequence (r n ) n∈N in T such that r n u o(n) (s 1 t 2 ) n r n (s 2 t 1 ) n , which implies s 1 t 2 s 2 t 3 by (iv) of Lemmas 2 and 3. A similar argument in the following lemma shows that the asymptotic preorder is essentially the same on S and on its image in T −1 S.
Proof Suppose that x y, i.e. u k n x n y n for some sublinear sequence k n and all n. This implies u 1 k n x 1 n y 1 n , therefore x 1 y 1 . Suppose that x 1 y 1 . This means there is a sublinear sequence k n and elements r n ∈ S \ {0} such that r n u k n x n r n y n (17) for all n ∈ N. By (iv) of Lemmas 2 and 3 this implies x y.
According to Lemma 6, we may safely identify x with x 1 for the purposes of studying the asymptotic preorder, even though the canonical homomorphism S → T −1 S is in general not injective.
We conclude this section with a construction that enlarges a preorder by forcing an ordering on certain pairs of elements.

Definition 4
Let (S, ) be a preordered semiring and R ⊆ S × S a relation. We define the relation R on S as a R b iff there is an n ∈ N and s 1 , . . . , s n , x 1 , . . . , x n , y 1 , . . . , y n ∈ S such that ∀i ∈ [n] : (x i , y i ) ∈ R and a + s 1 y 1 + · · · + s n y n b + s 1 x 1 + · · · + s n x n .
Lemma 7 Proof (i): If a b then we may choose n = 0 in the definition, therefore a R b. (ii): If (x, y) ∈ R then we can choose n = 1, s 1 = 1, x 1 = x, y 1 = y so that Eq. 19 becomes x + y y + x, therefore x R y. (iii): By (i) the relation R is reflexive. We prove transitivity. Let a R b and b R c. Then there are n, m ∈ N and elements x 1 , . . . , x n , y 1 , . . . , y n , z 1 , . . . , z m , w 1 , . . . , w m ∈ S and s 1 , . . . , s n , t 1 , . . . , t m ∈ S such that (x i , y i ) ∈ R and (z i , w i ) ∈ R for all i in [n], respectively [m], and a + s 1 y 1 + · · · + s n y n b + s 1 x 1 + · · · + s n x n (20) b + t 1 w 1 + · · · + t n w m c + t 1 z 1 + · · · + t n z m .
After adding t 1 w 1 + · · · + t n w m to both sides of the first inequality and s 1 x 1 + · · · + s n x n to both sides of the second one we can chain the inequalities and conclude a R c.
The compatibility with addition and multiplication can be seen directly by adding c to (respectively multiplying by c) both sides of Eq. 19. (iv): Expanding the definition, we see that a( R 1 ) R 2 b means that there are natural numbers n, n and elements s 1 , . . . , s n , x 1 , . . . , x n , y 1 , . . . , y n S and s 1 , . . . , s n , x 1 , . . . , x n , y 1 , . . . , y n ∈ S such that for all i (x i , y i ) ∈ R 1 and (x i , y i ) ∈ R 2 and (a+s 1 y 1 +· · ·+s n y n )+s 1 y 1 +· · ·+s n y n (b+s 1 x 1 +· · ·+s n x n )+s 1 x 1 +· · ·+s n x n , (22) which is clearly equivalent to a R 1 ∪R 2 . (v) follows from the fact that Eq. 19 involves only finitely many pairs (x i , y i ) ∈ R.

Spectrum
In this section we define and study the spectrum of a preordered semiring, i.e. the set of monotone semiring homomorphisms with the topology generated by the evaluation maps. We begin with properties related to compactness, and show that semirings of polynomial growth have locally compact spectra. Then we show that neither replacing the preorder with its asymptotic preorder nor localization affects the spectrum. Finally we prove that the inclusion of a certain type of subsemiring induces a surjective map on the spectra.

Definition 5
Let (S, ) be a preordered semiring. The spectrum Δ(S, ) is the set of monotone semiring homomorphisms S → R ≥0 equipped with the initial topology with respect to the family of evaluation maps ev s : Δ(S, ) → R ≥0 (s ∈ S). For a subset X ⊆ S we define ev X : Δ(S, ) → R X ≥0 as f → (f (x)) x∈X . Elements of the spectrum will be referred to as spectral points.
Let (S 1 , 1 ) and (S 2 , 2 ) be preordered semirings and let ϕ : S 1 → S 2 be a monotone homomorphism. We define the map Δ(ϕ) : that S is generated by a single element u. Then ev u is proper and Δ(S, ) is locally compact.
Proof (i): The evaluation map ev S clearly separates the points of Δ(S, ), thus gives an embedding into the Tychonoff space R S ≥0 . The property of being a monotone semiring homomorphism is preserved by pointwise limits, therefore the embedding is closed. (ii): every element of S can be written as a (not necessarily unique) polynomial in u with coefficients from N and ev p(u) = p(ev u ). This means that ev u already separates the points and its continuity is equivalent to the continuity of all evaluation maps. Therefore ev u : Δ(S, ) → R ≥0 is a closed embedding. This implies that ev u is proper and Δ(S, ) . C 1 is a compact subset of a Hausdorff space, therefore closed, which implies that C 2 is also closed. For all The infimum is by assumption over a nonempty set and the supremum is of a continuous function (ev x 1 ) over a compact set, therefore B x 2 is finite.
Thus ev S 2 embeds C 2 as a closed subset of the compact space

Proposition 2 Let (S, ) be a preordered semiring of polynomial growth and suppose that
To see that Δ(j ) is surjective we show that every f ∈ Δ(S, ) is also monotone under . Suppose that x y. Let (k n ) n∈N be as in Definition 3, i.e. k n /n → 0 and for all n ∈ N u k n x n ≥ y n . Then for every f ∈ Δ(S, ) we have After taking roots and letting n go to infinity we get f (x) ≥ f (y).
Thus Δ(j ) is a bijection and can be used to identify the two spectra as sets. Under this identification the evaluation maps are the same in both cases, and so are the topologies they generate. Therefore Δ(j ) is a homeomorphism.

Lemma 10 Let (S, ) be a preordered semiring of polynomial growth and T ⊆ S \ {0}
a multiplicative set containing 1. Let j : and therefore Δ(j )(f 1 ) and Δ(j )(f 2 ) are different at s. We prove that Δ(j ) is surjective. Let f ∈ Δ(S, ). Let u ∈ S be power universal. For s ∈ S \ {0} there is a k ∈ N such that 1 u k s. Applying f to both sides we get 1 ≤ f (u) k f (s), which implies f (s) > 0. If s 1 t 1 = s 2 t 2 then by definition rs 1 t 2 = rs 2 t 1 for some r ∈ T , therefore f (r)f (s 1 This means that the equalityf gives a well-defined function on T −1 S. We claim thatf ∈ Δ(T −1 S, ) and Δ(j andf This means there exists r ∈ T such that rs 1 t 2 rs 2 t 1 . Since f is monotone and f (r) > 0, this impliesf If x ∈ S then clearly Clearlyf (j (x)) =f ( x 1 ) = f (x), thusf extends f . In particular, Thus we may identify the two spectra via Δ(j ) as sets. Since for t ∈ T the map ev t vanishes nowhere, the topology generated by the maps (ev s ) s∈S and ( ev s ev t ) s∈S,t∈T is the same, therefore Δ(j ) is a homeomorphism.
Our goal in the following is to relate the spectral points of a semiring to those of a subsemiring. We will make use of a relaxed preorder whose restriction to the subsemiring is total.
We define the relation f := R f (see Definition 4) where R f is the relation Since ⊆ f by Lemma 7, we may identify Δ(S, f ) with a subset of Δ(S, ).
Applyf to both sides, divide by n and let n → ∞ to get Thereforef agrees with f on S 0 , which meansf ∈ Δ(i) −1 (f ).
Thenf is a semiring homomorphism and we need to show that it is monotone with respect to f . Let a f b. This means there are x 1 , . . . , x n , y 1 , . . . , y n ∈ S 0 and s 1 , . . . , s n ∈ S for some n ∈ N such that ∀j ∈ [n] : f (x j ) ≤ f (y j ) and a + s 1 y 1 + · · · + s n y n b + s 1 x 1 + · · · + s n x n .
Applyf to both sides and rearrange as Sincef for all j , this impliesf (b) ≥f (a), i.e.f is monotone with respect to f .

Lemma 12
Let (S, ) be a preordered semiring and S 0 ≤ S a subsemiring such that ∀s ∈ S \ {0}∃r, q ∈ S 0 such that 1 rs q. Let R ⊆ S 0 × S 0 ⊆ S × S be an arbitrary relation. Then We emphasize that R | S 0 is in general not the same preorder as ( | S 0 ) R , since the latter would only allow elements s 1 , . . . , s n ∈ S 0 in Definition 4.
Proof First note that the condition ∀s ∈ S \ {0}∃r, q ∈ S 0 such that 1 rs q implies that every f ∈ Δ(S 0 , | S 0 ) and s ∈ S 0 \ {0} satisfies f (s) = 0. To see this choose r ∈ S 0 such that 1 rs and apply f to both sides. We By (v) of Lemma 7 it is enough to prove the statement for |R| < ∞. We prove by induction on |R|. If R = ∅ then R = and there is nothing to prove. Otherwise choose (x, y) ∈ R and let Let a, b ∈ S 0 such that a R b. This means where the first inequality uses the induction hypothesis and the second one uses Eq. 38. Let r, q ∈ S 0 such that 1 rs q. Then Apply f to both sides and rearrange to get Divide by the coefficient of f (a) − f (b) (nonzero since x = 0 and r = 0). Then we use f (x) ≤ f (y) and f (q) ≥ 1 to get This inequality is true for every n ∈ N, therefore f (a) ≤ f (b).
The following proposition shows that spectral points of a subsemiring containing a power universal element can be extended to spectral points of the large semiring. Later we will use this result to relate the condition appearing in (M2) to boundednes of the evaluation function on Δ(S), but it appears to be also of independent interest. Proposition 3 Let (S, ) be a preordered semiring of polynomial growth and S 0 ≤ S a subsemiring satisfying ∀s ∈ S \ {0}∃r, q ∈ S 0 such that 1 rs q. Let i : S 0 → S be the inclusion. Then Δ(i) is surjective.
Proof We can assume that the inclusion N → S is an order embedding (otherwise both spectra are empty).
By Lemma 11 it is enough to show that for every f ∈ Δ(S 0 , ) the set Δ(S, f ) is nonempty. Let u ∈ S be power universal. By the assumption on S 0 there is an u ∈ S 0 such that u u, and any such u is also power universal in S. By (ii) of Lemma 7, therefore 2 is also power universal for (S, f ). Lemma 12 implies that Δ(S 0 , f S 0 ) = {f }. In particular, for n, m ∈ N we have n f m iff n ≤ m. Therefore we can apply Theorem 1 to the preordered semiring (S, f ) and conclude Δ(S, f ) = ∅.

Proof of Main Result and Uniqueness
Now we have all the technical tools to prove Theorem 2. In the setting of that theorem, we introduce the following notations: S − = {s ∈ S|∃n ∈ N : n s} (46) We will see that all three are subsemirings, and the definition of S b ensures that it satisfies the conditions of Theorem 1. Given a pair of elements in S, we can multiply both with the same element of M to get elements of S − , at least one of them in S b . Here Strassen's theorem ensures that the collection of monotone homomorphisms characterize the asymptotic preorder. An essential part of the proof is to show that most spectral points of S b admit an extension to S. Informally, an extension exists unless it would evaluate to ∞ on the power universal element. More precisely, by assumption (M1) we can form an "inverse up to bounded elements"ū of u, and f ∈ Δ(S b , ) extends to S iff f (1 +ū) > 1.

Lemma 13
(i) S + , S − and S b are subsemirings of S.
Proof (i): 0 and 1 are clearly contained both in S + and S − . Let s 1 , s 2 ∈ S + . This means that there are n 1 , n 2 ∈ N such that n 1 s 1 1 and n 2 s 2 1. Then n 1 (s 1 + s 2 ) n 1 s 1 1 and n 1 n 2 (s 1 s 2 ) 1 · 1 = 1, therefore s 1 + s 2 ∈ S + and s 1 s 2 ∈ S + . Let s 1 , s 2 ∈ S − . This means that there are n 1 , n 2 ∈ N such that n 1 s 1 and n 2 s 2 . Then n 1 + n 2 s 1 + s 2 and n 1 n 1 s 1 s 2 , therefore s 1 + s 2 ∈ S − and s 1 s 2 ∈ S − . S b = S + ∩ S − is the intersection of subsemirings, therefore it is also a subsemiring.
We prove thatf is monotone. Let x, y ∈ S − and suppose that x y. Then 1+x 1+y, thereforef