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Convexity Theories 0 cont. Foundations

Bodo Pareigis, Dieter Pumplün, and Helmut Röhrl

Abstract. The main issue of this paper is an axiomatization of the notion of absolutely convergent series involving a set of summands of fixed (but unrestricted) infinite cardinality N. This notion is used to define the category N_{R}pnSmod^{1} of R-prenormed R-semimodules with N-summation whose homomorphisms are contractive. Based on this we introduce left N-convexity theories Γ and the category ΓC of left Γ-convex modules. We show that the closed unit ball functor N_{R}pnSmod^{1} ⟶ Set, the forgetful functor ΓC ⟶ Set, and the associated Γ-convex module functor N_{R}pnSmod^{1} ⟶ ΓC have left adjoints.

1991 Mathematics Subject Classification: 16Y60, 52A01.

Introduction

The basic definitions of this paper are contained in §1. They concern an axiomatic approach to the notion of “absolutely convergent series” in prenormed semirings and prenormed semimodules . The type of series we are interested in have a set of summands of a fixed, but arbitrary, infinite cardinality N. Semimodules equipped with a family of such “summable” series are said to have N-summation. The section concludes with few statements directly related to the basic definitions. §2 consists of several results involving semimodules with N-summation. In addition we introduce maps from a fixed, but arbitrary, set to a semimodule with N-summation that are “summable”. These maps are used in §3 to define, for an arbitrary set A, the free semimodule L^{N}(A) with N-summation on the set A. L^{N}(A) is a generalization of the well known functional-analytic concept of ℓ_{1}-space on the set A. The functor Set ∋ A ↦ L^{N}(A) ∈ N_{R}pnSmod^{1} is the left adjoint of the forgetful functor N_{R}pnSmod^{1} ⟶ Set; here, N_{R}pnSmod^{1} stands for the category of R-prenormed R-semimodules with N-summation and their contractive homomorphisms. In section 4 we deal with N-convexity theories Γ over prenormed semirings with N-summation (generalizing the corresponding concept in [5], §3 and §7), derive few of the properties of Γ-convex modules, and give an explicit construction of the free Γ-convex modules. This construction differs from and is more perspicuous than the one given in [3], §5. In §5 we introduce the functor O_{Γ} : N_{R}pnSmod^{1} ⟶ ΓC, where for a given N-convexity theory Γ the category ΓC is the category of Γ-convex modules and their homomorphisms. O_{Γ}(M) is the closed unit ball of M equipped with the obvious operation of Γ on the closed unit ball of M. In addition we exhibit the left adjoint S^{Γ} of O_{Γ}. The paper ends with a section presenting several examples for the previously introduced concepts.

1. The basic definitions

Let R be a semiring (in the sense of [5], §1). Let furthermore N be a fixed set of cardinality ≥ ℵ_{0}. Maps from N to R, that is elements of R^{N}, will be denoted by lower case greek letters with a lower placeholder symbol, e.g. α_{∗} or α_{□}; occasionally we will write: {α_{n} : n ∈ N} or {α(n) : n ∈ N} instead of α_{∗}. If we define α_{∗}+·β_{∗}

as the map N ∋ n ⟼ α_{n}+·β_{n} ∈ R then R^{N} becomes a semiring. If r ∈ R and if

we denote the constant map N ⟶ R with value r by r_{∗} then R ∋ r ↦ r_{∗} ∈ R^{N} is a homomorphism of semirings.

Let M be a semimodule (in the sense of [5], §1) over the semiring R. Then we can again form M^{N}. Let α_{∗} ∈ R^{N} and μ′_{∗}, μ_{∗} ∈ M^{N}, and define α_{∗}μ_{∗} and μ′_{∗} + μ_{∗} pointwise — as in the case of R — then M^{N} becomes a R^{N}-semimodule. As before we let m_{∗}, m ∈ M, be the constant map with value m. Then M ∋ m ↦ m_{∗} ∈ M^{N} is a homomorphism of R-semimodules.

If R is a partially ordered semiring and M is a partially ordered R-semimodule (in the sense of [5], §1) then we define μ′_{∗} ≤ μ_{∗}, where μ′_{∗}, μ_{∗} ∈ M^{N}, as μ′_{n} ≤ μ_{n} for all n ∈ N. This makes M^{N} a partially ordered semimodule over the partially ordered semiring R^{N}. Obviously, R ∋ r ↦ r_{∗} ∈ R^{N} and M ∋ m ↦ m_{∗} ∈ M^{N} are order preserving. μ_{∗} ∈ M^{N} is said to be bounded if there are m′, m′′ ∈ M with m′_{∗} ≤ μ_{∗} ≤ m′′_{∗}, and a family {μ_{∗}^{i} : i ∈ I} is called uniformly bounded if there are m′, m′′ ∈ M with m′_{∗} ≤ μ_{∗}^{i} ≤ m′′_{∗} for all i ∈ I.

If M is an R-prenormed R-semimodule over the prenormed semiring R and if ∥ ∥ denotes the prenorm with value cone C (see [5], §2) then, with μ_{∗} ∈ M^{N}, we denote by ∥μ_{∗}∥ the map N ∋ n ↦ ∥μ_{n}∥ ∈ C; hence ∥μ_{∗}∥ is in C^{N}. This makes M^{N} a R^{N}-prenormed R^{N}-semimodule over the prenormed semiring R^{N} with value cone C^{N}.

Finally a construction that will be used later. Let A be some set and χ : N ⟶ A a set map. Let furthermore M be an R-semimodule and μ_{∗} ∈ M^{N}. Given a ∈ A we denote by μ_{∗}^{χ^{−1}(a)} the map N ⟶ M given by

N ∋ n ↦ \{ \begin{matrix}μ_{n} & , if χ(n) = a; \\ 0 & , otherwise. \end{matrix}

In other words, μ_{∗}^{χ^{−1}(a)} is given by the formulae

μ_{∗}^{χ^{−1}(a)}|χ^{−1}(a) = μ_{∗}|χ^{−1}(a)  and  μ_{∗}^{χ^{−1}(a)}|N ∖ χ^{−1}(a) = 0_{∗}|N ∖ χ^{−1}(a).

Note that any subset of N can be obtained as χ^{−1}(a), provided that A contains at least two elements, and that each partition of N can be written as {χ^{−1}(a) : a ∈ A′} for some subset A′ of A, provided that A is large enough.

In the following definition we refer to the concept of positive semiring. According to [5], §1, a positive semiring is a partially ordered semiring with 0 as its smallest element.

(1.1) Definition. Let C be a positive semiring. By a left N-summation for C is meant a pair (S_{C}, Σ_{C}) consisting of a C-subsemimodule S_{C} of C^{N} and a Chomomorphism Σ_{C} : S_{C} ⟶ C such that

(i) C^{(N)} := {α_{∗} ∈ C^{N} : supp α_{∗} is finite} is contained in S_{C} and for all α_{∗} ∈ C^{(N)}, Σ_{C}(α_{∗}) = ∑′{α_{n} : n ∈ supp α_{∗}}, where ∑′ stands for the usual sum of finitely many elements in C;

(ii) for all α_{∗} ∈ S_{C} and β_{∗} ∈ C^{N} with β_{∗} ≤ α_{∗}, β_{∗} is in S_{C} and Σ_{C}(β_{∗}) ≤

Σ_{C}(α_{∗}); (iii) for every α_{∗} ∈ S_{C} and every map φ : N ⟶ N, α_{∗}^{φ^{−1}(n)} is in S_{C} for all

n ∈ N, and the map α_{∗}^{φ^{−1}} given by N ∋ n ↦ Σ_{C}(α_{∗}^{φ^{−1}(n)}) ∈ C is in S_{C} and satisfies Σ_{C}(α_{∗}^{φ^{−1}}) = Σ_{C}(α_{∗});

(iv) if α_{∗} is in C^{N} and there exists a map φ : N ⟶ N such that α_{∗}^{φ^{−1}(n)} is in S_{C} for all n ∈ N and that α_{∗}^{φ^{−1}} is in S_{C} then α_{∗} is in S_{C}.

(1.2) Lemma. Let C be a positive semiring with left N-summation (S_{C}, Σ_{C}). Then the conditions (1.1), (i)-(iii), imply that for every α_{∗} ∈ S_{C} the inequalities α_{n} ≤ Σ_{C}(α_{∗}), n ∈ N, hold; in particular, if sup{α_{n} : n ∈ N} exists, sup{α_{n} : n ∈ N} ≤ Σ_{C}(α_{∗}) is satisfied. ⊓ ⊔

The next definition uses the notion of prenormed semiring. Due to [5], (2.1), this is a semiring R together with a map ∥ ∥ : R ⟶ C, where C is a positive and complete (with respect to the partial order) semiring, such that

(o) ∥0∥ = 0 and ∥1∥ = 1;

(i) ∥r_{1} + r_{2}∥ ≤ ∥r_{1}∥ + ∥r_{2}∥ , for all r_{1}, r_{2} ∈ R; (ii) ∥r_{1}r_{2}∥ ≤ ∥r_{1}∥∥r_{2}∥ , for all r_{1}, r_{2} ∈ R.

The semiring C is called the value cone of R and ∥ ∥ is said to be the prenorm of R.

(1.3) Definition. Let R be a prenormed semiring with prenorm ∥ ∥ : R ⟶ C. By a left N-summation for R is meant a left N-summation (S_{C}, Σ_{C}) for C together with a pair (S_{R}, Σ_{R}) consisting of an R-subsemimodule S_{R} of R^{N} and an Rhomomorphism Σ_{R} : S_{R} ⟶ R such that

(o) α_{∗} ∈ R^{N} is in S_{R} if and only if ∥α_{∗}∥ is in S_{C};

(i) R^{(N)} := {α_{∗} ∈ R^{N} : supp α_{∗} is finite} is contained in S_{R} and for all α_{∗} ∈ R^{(N)}, Σ_{R}(α_{∗}) = ∑′{α_{n} : n ∈ supp α_{∗}}, where ∑′ stands for the usual sum of finitely many elements in R;

(ii) for all α_{∗} ∈ S_{R} and β_{∗} ∈ R^{N} with ∥β_{∗}∥ ≤ ∥α_{∗}∥, β_{∗} is in S_{R} and ∥Σ_{R}β_{∗}∥ ≤ Σ_{C}∥α_{∗}∥; (iii) for every α_{∗} ∈ S_{R} and every φ : N ⟶ N, α_{∗}^{φ^{−1}(n)} is in S_{R} for all n ∈ N,

and the map α_{∗}^{φ^{−1}} given by N ∋ n ↦ Σ_{R}(α_{∗}^{φ^{−1}(n)}) ∈ R is in S_{R} and satisfies Σ_{R}(α_{∗}^{φ^{−1}}) = Σ_{R}(α_{∗}).

Definition (1.4) requires the notion of R-prenormed left R-semimodule over the prenormed semiring R with prenorm ∥ ∥ : R ⟶ C. By this is meant a left R-semimodule M together with a map ∥ ∥ : M ⟶ C such that

(o) ∥0∥ = 0;

(i) ∥m_{1} + m_{2}∥ ≤ ∥m_{1}∥ + ∥m_{2}∥ , for all m_{1}, m_{2} ∈ M; (ii) ∥rm∥ ≤ ∥r∥∥m∥ , for all r ∈ R, m ∈ M.

(1.4) Definition. Let R be a prenormed semiring with prenorm ∥ ∥ : R ⟶ C and N-summation (S_{R}, Σ_{R}) for R and (S_{C}, Σ_{C}) for C. An R-prenormed left R-semimodule M with left N-summation is an R-prenormed left R-semimodule together with a pair (S_{M}, Σ_{M}) consisting of an R-subsemimodule S_{M} of M^{N} and an R-homomorphism Σ_{M} : S_{M} ⟶ M such that

(o) μ_{∗} ∈ M^{N} is in S_{M} if and only if ∥μ_{∗}∥ is in S_{C};

(i) M^{(N)} := {μ_{∗} ∈ M^{N} : supp μ_{∗} is finite} is contained in S_{M} and for all μ_{∗} ∈ M^{(N)}, Σ_{M}(μ_{∗}) = ∑′{μ_{n} : n ∈ supp μ_{∗}}, where ∑′ stands for the usual sum of finitely many elements in M;

(ii) for all μ_{∗} ∈ S_{M} and ν_{∗} ∈ M^{N} with ∥ν_{∗}∥ ≤ ∥μ_{∗}∥, ν_{∗} is in S_{M} and ∥Σ_{M}ν_{∗}∥ ≤

Σ_{C}∥μ_{∗}∥; (iii) for every μ_{∗} ∈ S_{M} and every φ : N ⟶ N, μ_{∗}^{φ^{−1}(n)} is in S_{M} for all n ∈ N,

and the map μ_{∗}^{φ^{−1}} given by N ∋ n ↦ Σ_{M}(μ_{∗}^{φ^{−1}(n)}) ∈ M is in S_{M} and satisfies Σ_{M}(μ_{∗}^{φ^{−1}}) = Σ_{M}(μ_{∗}).

Note that (1.3) is a special case of (1.4) and that (1.1) can be viewed as a special case of (1.3) by setting R = C, ∥ ∥ = id_{C}, S_{R} = S_{C}, and Σ_{R} = Σ_{C}. Hence the statements following (1.5) concerning semimodules apply to prenormed semirings and positive semirings as well.

(1.5) Definition. Let M and M′ be R-prenormed R-semimodule with left Nsummation (S_{M}, Σ_{M}) resp. (S_{M′}, Σ_{M′}). Then a homomorphism from M to M′ is a map f : M ⟶ M′ such that

(i) f is a homomorphism of left R-semimodules from M to M′ satisfying

f^{N}(S_{M}) ⊆ S_{M′} and Σ_{M′}(f^{N}(μ_{∗})) = f(Σ_{M}(μ_{∗})) , for all μ_{∗} ∈ S_{M};

(ii) there is a c ∈ C (depending on f) with

∥Σ_{M′}f^{N}(μ_{∗})∥ ≤ (Σ_{C}(∥μ_{∗}∥))c  , for all μ_{∗} ∈ S_{M}.

It is clear from (1.18) that the totality of R-prenormed R-semimodules with left N-summation together with their homomorphisms and the set-theoretical composition of these forms a category N_{R}pnSmod.

A homomorphism of R-prenormed R-semimodules is called contractive (or a contraction) if c = 1 can be chosen in (1.5), (ii). Again it is easy to see that the totality of R-prenormed R-semimodules together with their contractive homomorphisms forms a subcategory N_{R}pnSmod^{1} of N_{R}pnSmod.

We close with three statements directly related to the above definitions.

(1.6) Lemma. (1.4), (o), implies that, whenever μ_{∗} ∈ S_{M} and ν_{∗} ∈ M^{N} with ∥ν_{∗}∥ ≤ ∥μ_{∗}∥ holds then ν_{∗} ∈ S_{M} is satisfied.

Proof. Due to (1.4), (o), ∥μ_{∗}∥ is in S_{C}. Hence (1.1), (ii), shows that ∥ν_{∗}∥ is in S_{C} and thus, by (1.4), (o), ν_{∗} is in S_{M}. ⊓ ⊔

(1.7) Lemma. (1.4), (o) and (ii), imply that, whenever μ_{∗} ∈ S_{M} and φ : N ⟶ N is a map, μ_{∗}^{φ^{−1}(n)} is in S_{M}, for all n ∈ N, and μ_{∗}^{φ^{−1}} is in S_{M}.

Proof. Again ∥μ_{∗}∥ is in S_{C}. Since ∥μ_{∗}^{φ^{−1}(n)}∥ ≤ ∥μ_{∗}∥, ∥μ_{∗}^{φ^{−1}(n)}∥ is in S_{C} and thus μ_{∗}^{φ^{−1}(n)} is in S_{M}. By (1.4), (ii),

∥Σ_{M}(μ_{∗}^{φ^{−1}(n)})∥ ≤ Σ_{C}(∥μ_{∗}^{φ^{−1}(n)}∥) = Σ_{C}(∥μ_{∗}∥^{φ^{−1}(n)}).

Due to (1.1), (iii), N ∋ n ↦ Σ_{C}(μ_{∗}^{φ^{−1}(n)}) ∈ C is in S_{C} as ∥μ_{∗}∥ is in S_{C}. Therefore the map N ∋ n ↦ ∥Σ_{M}(μ_{∗}^{φ^{−1}(n)})∥ ∈ C is in S_{C} and hence the map μ_{∗}^{φ^{−1}}, that is N ∋ n ↦ Σ_{M}(μ_{∗}^{φ^{−1}(n)}), is in S_{M}. ⊓ ⊔

(1.8) Lemma. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}) and denote the N-summation of R by (S_{R}, Σ_{R}). If α_{∗} ∈ S_{R} and μ^{∗} ∈ S_{M} then α_{∗}μ^{∗}, defined as the map N ∋ n ↦ α_{n}μ^{n} ∈ M, is in S_{M}.

Proof. Since α_{∗} is in S_{R}, it follows from (1.3), (o), that ∥α_{∗}∥ is in S_{C}, whence ∥α_{n}∥ ≤ Σ_{C}∥α_{∗}∥, n ∈ N, on account of (1.2). Therefore ∥α_{∗}μ^{∗}∥ ≤ (Σ_{C}∥α_{∗}∥)∥μ^{∗}∥, and (1.1) and (1.4), (o), show that the right hand side of this inequality is in S_{C}. Thus ∥α_{∗}μ^{∗}∥ is in S_{C} due to (1.1), (ii), whence α_{∗}μ^{∗} is in S_{M} by (1.4), (o).

2. Elementary results

(2.1) Lemma. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}). Let furthermore μ_{∗} ∈ S_{M} and ν_{∗} ∈ M^{N} be such that there is a bijection \overline{φ} : supp μ_{∗} ⟶ supp ν_{∗} with μ_{n} = ν_{\overline{φ}(n)}, for all n ∈ supp μ_{∗}. Then ν_{∗} is in S_{M} and Σ_{M}(ν_{∗}) = Σ_{M}(μ_{∗}).

Proof. Extend \overline{φ} to some map φ : N ⟶ N. Then one checks quickly that ν_{∗} = μ_{∗}^{φ^{−1}} holds. Hence (1.1), (iii), shows that ν_{∗} is in S_{M} and that Σ_{M}(ν_{∗}) = Σ_{M}(μ_{∗}) is valid. ⊓ ⊔ (2.2) Corollary. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}). Let furthermore f : A ⟶ M be a map with card(supp f) ≤ cardN. Then there is a map χ : N ⟶ A, a μ_{∗} ∈ M^{N}, and sets A′ and N′ with supp f ⊆ A′ ⊆ A, supp μ_{∗} ⊆ N′ ⊆ N, and cardA′ ≤ cardN such that

a) A′ ⊆ χ(N) and A′|χ|N′ is a bijection

b) μ_{n} = f(χ(n)) , for all n ∈ N′. Moreover, if \overline{χ}, \overline{μ}_{∗}, \overline{A}′ and \overline{N}′ is another set of data satisfying the above conditions, for all n ∈ N′, then μ_{∗} is in S_{M} if and only if \overline{μ}_{∗} is in S_{M}, in which case Σ_{M}(μ_{∗}) = Σ_{M}(\overline{μ}_{∗}) holds.

Proof. The existence of χ, μ_{∗}, A′, and N′ is obvious. The balance of (2.2) is an immediate consequence of (2.1). ⊓ ⊔

(2.3) Remark. Suppose we are in the situation of (2.2) but are dealing with a family {f_{n} : n ∈ Ñ} of maps A ⟶ N, whose index set satisfies cardÑ ≤ cardN, instead of looking at a single such map. If card(supp f_{n}) ≤ cardN, for all n ∈ Ñ, then, as cardN^{2} = cardN, card(⋃{supp f_{n} : n ∈ Ñ}) ≤ cardN holds. Therefore, in (2.2), χ, A′, and N′ can be chosen such that for every f_{n}, n ∈ Ñ, and the (now uniquely determined) μ_{∗n}, n ∈ Ñ , the conditions in (2.2) are satisfied. ⊓ ⊔

Due to (2.2) we can formulate

(2.4) Definition. Let M be an R-prenormed R-semimodule with left N-summa-tion (S_{M}, Σ_{M}). Let furthermore A be any set. Then we define S_{M,A} as the set of maps f : A ⟶ M such that card(supp f) ≤ cardN and for some data χ, μ_{∗}, A′, and N′ in (2.2), μ_{∗} ∈ S_{M} holds. Moreover, we define Σ_{M,A}(f) := Σ_{M}(μ_{∗}), for all f ∈ S_{M,A}, to obtain a map Σ_{M,A} : S_{M.A} ⟶ M.

Note that S_{M,N} = S_{M} and Σ_{M,N} = Σ_{M} hold. We will write occasionally Σ_{M,A}{f(a) : a ∈ A} instead of Σ_{M,A}(f).

For f : A ⟶ M we denote by ∥f∥ the map A ∋ a ↦ ∥f(a)∥ ∈ C.

(2.5) Lemma. For every set A and every R-prenormed R-semimodule M with left N-summation, S_{M,A} is an R-subsemimodule of M^{A} and Σ_{M,A} : S_{M,A} ⟶ M is an R-homomorphism of R-prenormed semimodules; in particular, ∥Σ_{M,A}(f)∥ ≤ Σ_{C,A}(∥f∥).

Proof. Let f and g be in S_{M,A}. Choose the data in (2.2) to serve both f and g (see (2.3)). If μ_{∗} corresponds to f and ν_{∗} corresponds to g then, obviously, μ_{∗} + ν_{∗} corresponds to f + g. Hence f + g belongs to S_{M,A}. Similarly one shows that rf, r ∈ R, f ∈ S_{M,A}, also belongs to S_{M,A}. Moreover,

Σ_{M,A}(f + g) = Σ_{M}(μ_{∗} + ν_{∗}) = Σ_{M}(μ_{∗}) + Σ_{M}(ν_{∗}) = Σ_{M,A}(f) + Σ_{M,A}(g)

and

Σ_{M,A}(rf) = Σ_{M}(r_{∗}μ_{∗}) = rΣ_{M}(μ_{∗}) = rΣ_{M,A}(f).

Finally,

∥Σ_{M,A}(f)∥ = ∥Σ_{M}(μ_{∗})∥ ≤ Σ_{C}(∥μ_{∗}∥) = Σ_{C,A}(∥f∥).  □

(2.6) Lemma. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}). Furthermore let A be any set. Then f ∈ M^{A} belongs to S_{M,A} if and only if ∥f∥ ∈ C^{A} belongs to S_{C,A}.

Proof. (1.4),(o), and (2.3). ⊓ ⊔

(2.7) Lemma. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}). Then, for any set A, M^{(A)} := {f ∈ M^{A} : supp f is finite} is contained in S_{M,A} and for all f ∈ M^{(A)}, Σ_{M,A}(f) = ∑′{f(a) : a ∈ supp f}, where ∑′ stands for the usual sum of finitely many elements in M.

Proof. (1.4), (i), and (2.2). ⊓ ⊔

(2.8) Lemma. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}). Furthermore let A be any set. If f is in S_{M,A} and g ∈ M^{A} with ∥g∥ ≤ ∥f∥ then g is in S_{M,A} and ∥Σ_{M,A}(g)∥ ≤ Σ_{C,A}(∥f∥).

Proof. (1.4),(ii), and (2.2). ⊓ ⊔

(2.9) Lemma. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}). Furthermore let A be any set. If f is in S_{M,A} and ψ : A ⟶ A is any map then, for any a ∈ A, the map f^{ψ^{−1}(a)} given by

A ∋ b ↦ \{ \begin{matrix}f(b) & , if ψ(b) = a; \\ 0 & , otherwise. \end{matrix}

is in S_{M,A}; moreover, the map f^{ψ^{−1}} given by A ∋ a ↦ Σ_{M,A}(f^{ψ^{−1}(a)}) ∈ M is in S_{M,A} and Σ_{M,A}(f^{ψ^{−1}}) = Σ_{M,A}(f).

Proof. (1.4),(iii), and (2.2). ⊓ ⊔

(2.10) Corollary. Let M be an R-prenormed R-semimodule with left N-summa-tion (S_{M}, Σ_{M}). Furthermore let A be any set and π : A ⟶ A be any bijection.

Then f ∈ M^{A} is in S_{M,A} if and only if f^{π} := f ∘ π is in S_{M,A}, in which case Σ_{M,A}(f^{π}) = Σ_{M,A}(f).

Proof. Immediate consequence of (2.9) ⊓ ⊔

(2.11) Lemma. Let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}). Furthermore let A and B be any sets. For f ∈ S_{M,A×B} define f(a, -) : B ⟶ M resp. f(-, b) : A ⟶ M as the maps

B ∋ b ↦ f(a, b) ∈ M  resp.  A ∋ a ↦ f(a, b) ∈ M.

Then f(a, -) in in S_{M,B}, for all a ∈ A, and f(-, b) is in S_{M,A}, for all b ∈ B, and the maps

A ∋ a ↦ Σ_{M,B}(f(a, -)) ∈ M  resp.  B ∋ b ↦ Σ_{M,A}(f(-, b)) ∈ M

are in S_{M,A} resp. S_{M,B} and satisfy

\begin{aligned}Σ_{M,A}\{Σ_{M,B}\{f(a, b) : b ∈ B\} : a ∈ A\} & = Σ_{M,B}\{Σ_{M,A}\{f(a, b) : a ∈ A\} : b ∈ B\} \\ & = Σ_{M,A×B}(f). \end{aligned}

Proof. By (2.2), (2.11) can be reduced to the case A = B = N. In this situation choose a bijection N^{2} ⟶ N, use (2.1), and apply (1.4), (iii), twice. ⊓ ⊔

As a special case of (2.11) we obtain

(2.12) Corollary. Let M be an R-prenormed R-semimodule with left N-summa-tion (S_{M}, Σ_{M}). If α_{∗} ∈ S_{R} and m ∈ M then the map α_{∗}m given by N ∋ n ↦ α_{n}m ∈ M is in S_{M} and Σ_{M}(α_{∗}m) = (Σ_{R}α_{∗})m. Similarly if r ∈ R and μ_{∗} ∈ S_{M} then the map rμ_{∗} given by N ∋ n ↦ rμ_{n} ∈ M is in S_{M} and Σ_{M}(rμ_{∗}) = r(Σ_{M}μ_{∗}). ⊓ ⊔

(2.13) Corollary. Let f ∈ M^{A}, φ : A ⟶ B a map. For b ∈ B let ∥f∥^{φ^{−1}(b)} be the map

A ∋ a ↦ \{ \begin{matrix}∥f(a)∥ & , if φ(a) = b; \\ 0 & , otherwise. \end{matrix}

Suppose that ∥f∥^{φ^{−1}(b)} is in S_{C,A} for every b ∈ B and that the map

B ∋ b ↦ \{ \begin{matrix}Σ_{C,A}(∥f∥^{φ^{−1}(b)}) & , if b ∈ φ(A); \\ 0 & , otherwise; \end{matrix}

is in S_{C,B}. Then f is in S_{M,A}.

Proof. This is an immediate consequence of (1.1),(iii’),(1.4), (o), and (2.2). ⊓ ⊔

If should be pointed out that, with all index sets assumed to have cardinality ≤ cardN, there is a correspondence between certain axioms in [7], §6, and some of the results obtained here. This correspondence is as follows:

Equivalent Families Axiom ≡ (2.9),
Unary Sum Axiom ≡ (1.4), (i),
Generalized Partition Axiom ≡ (1.4), (iii),
Weak Double Sum Axiom ≡ (2.11).

3. The closed unit ball functor

As in [5] one defines the closed unit ball functor B_{N} : N_{R}pnSmod^{1} ⟶ Set. Its value on the object M of N_{R}pnSmod^{1} is

B_{N}(M) := {m ∈ M : ∥m∥ ≤ 1}.

(3.1) Theorem. B_{N} : N_{R}pnSmod^{1} ⟶ Set has a left adjoint L^{N}.

Proof. We put L^{N}(∅) := {0}. If A ≠ ∅ is any nonempty set, we put – as a set –

L^{N}(A) := S_{R,A} ⊆ R^{A}.

Due to (2.5), L^{N}(A) is an R-subsemimodule of R^{A}.

Next we define a prenorm ||| ||| : L^{N}(A) ⟶ C by putting

(3.2)  |||f||| := Σ_{C,A}(∥f∥) = Σ_{C,A}{∥f(a)∥ : a ∈ A}  , f ∈ L^{N}(A).

Since ∥f + g∥ ≤ ∥f∥ + ∥g∥, f and g in L^{N}(A), we obtain from (2.5) and (2.8) |||f + g||| ≤ |||f||| + |||g|||. |||rf||| ≤ ∥r∥ · |||f|||, r ∈ R, f ∈ L^{N}(A), follows similarly. This means that L^{N}(A) is an R-prenormed R-semimodule.

It remains to define (S_{L^{N} (A)}, Σ_{L^{N} (A)}). Let F_{∗} ∈ (R^{A})^{N}. Given a ∈ A we denote the map N ∋ n ↦ F_{n}(a) ∈ R by F_{∗}(a). Due to (1.3), (o), F_{∗}(a) is in S_{R} if and only if ∥F_{∗}(a)∥, that is the map N ∋ n ↦ ∥F_{n}(a)∥ ∈ C, is in S_{C}. In this case we can form the map Σ_{C}∥F_{∗}∥ : A ⟶ C that is given by

(Σ_{C}∥F_{∗}∥)(a) := Σ_{C}(∥F_{∗}(a)∥)  , a ∈ A.

With these notations we have

S : = S_{L^{N} (A)}

= {F_{∗} ∈ L^{N}(A)^{N} : ∥F_{∗}(a)∥ ∈ S_{C}, for all a ∈ A, and Σ_{C}∥F_{∗}∥ ∈ S_{C,A}}.

In order to define Σ := Σ_{L^{N} (A)}, let F_{∗} be in S. Since ∥F_{∗}(a)∥ is in S_{C} we have that F_{∗}(a), that is the map N ∋ n ↦ F_{n}(a) ∈ R, is in S_{R}. Due to (1.3), (ii), we obtain ∥Σ_{R}F_{∗}(a)∥ ≤ Σ_{C}∥F_{∗}(a)∥. If Σ_{R}F_{∗} denotes the map A ∋ a ↦ Σ_{R}F_{∗}(a) ∈ R then we have ∥Σ_{R}F_{∗}∥ ≤ Σ_{C}∥F_{∗}∥. Since the latter function is in S_{C,A}, (2.8) shows that ∥Σ_{R}F_{∗}∥ is in S_{C,A}, whence Σ_{R}F_{∗} is in S_{R,A} due to (2.6). In other words, Σ_{R}F_{∗} is in L^{N}(A), and we put ΣF_{∗} := Σ_{R}F_{∗}.

At this point we have to show that (S, Σ) is a left N-summation for L^{N}(A). So, let F_{∗} and G_{∗} be in S. Then F_{∗} + G_{∗} is the map N ∋ n ↦ F_{n} + G_{n} ∈ L^{N}(A) and hence F_{∗} + G_{∗} ∈ L^{N}(A)^{N}. Moreover, (F_{n} + G_{n})(a) = F_{n}(a) + G_{n}(a) for all a ∈ A. Thus, ∥(F_{∗} + G_{∗})(a)∥ ≤ ∥F_{∗}(a)∥ + ∥G_{∗}(a)∥. Since both ∥F_{∗}(a)∥ and ∥G_{∗}(a)∥ are in S_{C}, (1.1) shows that ∥F_{∗}(a)∥ + ∥G_{∗}(a)∥ is in S_{C}. Therefore (1.1), (ii), implies that ∥(F_{∗} + G_{∗})(a)∥ is also in S_{C} and that

Σ_{C}(∥(F_{∗} + G_{∗})(a)∥) ≤ Σ_{C}(∥F_{∗}(a)∥ + ∥G_{∗}(a)∥) = Σ_{C}(∥F_{∗}(a)∥) + Σ_{C}(∥G_{∗}(a)∥)

holds for all a ∈ A. This means that Σ_{C}∥F_{∗} + G_{∗}∥ ≤ Σ_{C}∥F_{∗}∥ + Σ_{C}∥G_{∗}∥ is valid. By assumption both Σ_{C}∥F_{∗}∥ and Σ_{C}∥G_{∗}∥ are in S_{C}, whence Σ_{C}∥F_{∗}∥ + Σ_{C}∥G_{∗}∥ is in S_{C}. Thus (1.1), (ii), shows that Σ_{C}∥F_{∗} +G_{∗}∥ is in S_{C}. Therefore F_{∗} +G_{∗} is in S. Similarly, but more simply, one shows that F_{∗} ∈ S and r ∈ R implies rF_{∗} ∈ S. Thus we have shown that S is an R-subsemimodule of L^{N}(A)^{N}.

Next we need to prove that Σ is a homomorphism of R-semimodules. Again let F_{∗} and G_{∗} be in S. Then Σ_{R}(F_{∗}(a) + G_{∗}(a)) = Σ_{R}F_{∗}(a) + Σ_{R}G_{∗}(a) and thus Σ_{R}(F_{∗} + G_{∗}) = Σ_{R}F_{∗} + Σ_{R}G_{∗}. Moreover, (Σ_{R}(F_{∗} + G_{∗}))_{∗} = (Σ_{R}F_{∗})_{∗} + (Σ_{R}G_{∗})_{∗}, whence

Σ(F_{∗} + G_{∗}) = Σ_{R}((Σ_{R}(F_{∗} + G_{∗}))_{∗}) = Σ_{R}((Σ_{R}F_{∗})_{∗} + (Σ_{R}G_{∗})_{∗})
= Σ_{R}((Σ_{R}F_{∗})_{∗}) + Σ_{R}((Σ_{R}G_{∗})_{∗}) = ΣF_{∗} + ΣG_{∗}.

Similarly one obtains ΣrF_{∗} = rΣF_{∗}, and Σ is recognized as a homomorphism of R-semimodules.

Now we wish to verify (1.4), (o). For this, let F_{∗} be in L^{N}(A)^{N}. We have to show that F_{∗} ∈ S is equivalent with |||F_{∗}||| ∈ S_{C}, where |||F_{∗}||| is the map N ∋ n ↦ |||F_{n}||| ∈ C. Let (F_{∗}) denote the map A × N ∋ (a, n) ↦ ∥F_{n}(a)∥ ∈ C. Suppose that F_{∗} ∈ S holds. Let φ : A × N ⟶ A be the projection onto the first factor. Then, for every a ∈ A,

(F_{∗})^{φ^{−1}(a)}(b, n) = \{ \begin{matrix}∥F_{n}(a)∥ & , if b = a, \\ 0 & , otherwise. \end{matrix}

Since ∥F_{∗}(a)∥ is in S_{C}, (F_{∗})^{φ^{−1}(a)} belongs to S_{C,A×N} due to (2.1). Moreover, Σ_{C}∥F_{∗}(a)∥ = Σ_{C,A×N}(F_{∗})^{φ^{−1}(a)}. Since Σ_{C,A}∥F_{∗}∥ is in S_{C,A}, (2.12) shows that (F_{∗}) is in S_{C,A×N}. Thus (2.11) implies that, with ψ : A × N ⟶ N the projection onto the second factor, (F_{∗})^{ψ^{−1}(n)} is in S_{C,A×N} for every n ∈ N. But due to (2.1)

Σ_{C,A×N}(F_{∗})^{ψ^{−1}(n)} = Σ_{C,A}{∥F_{n}(a)∥ : a ∈ A} = |||F_{n}|||  , n ∈ N.

Hence (2.11) shows that |||F_{∗}||| is in S_{C}. The same type of argument shows that |||F_{∗}||| ∈ S_{C} implies F_{∗} ∈ S.

On to (1.4), (i). Here we are dealing with F_{∗} ∈ L^{N}(A)^{N} with finite support. Hence ∥F_{∗}(a)∥ has finite support and is therefore in S_{C}, for all a ∈ A. In particular,

Σ_{C}∥F_{∗}(a)∥ = ∑′{∥F_{n}(a)∥ : n ∈ supp F_{∗}}

and hence

Σ_{C}∥F_{∗}∥ = ∑′{∥F_{n}∥ : n ∈ supp F_{∗}}.

Since S_{C,A} is an R-semimodule, the right hand side of the last equation, and thus Σ_{C}∥F_{∗}∥, is in S_{C,A}. Therefore F_{∗} is in S. Moreover, with F_{∗} having finite support, so does F_{∗}(a), for all a ∈ A, and

Σ_{R}F_{∗}(a) = ∑′{F_{n}(a) : n ∈ supp F_{∗}}  , a ∈ A.

Consequently,

ΣF_{∗} = Σ_{R}F_{∗} = ∑′{F_{n} : n ∈ supp F_{∗}},

and (1.4), (i), is satisfied.

Next comes (1.4), (ii). Let F_{∗} ∈ S and G_{∗} ∈ L^{N}(A)^{N} with |||G_{∗}||| ≤ |||F_{∗}|||. By (1.4), (o), |||F_{∗}||| ∈ S_{C} holds. Due to (1.1), (ii), |||G_{∗}||| is in S_{C}. By (1.4),(o), G_{∗} is in S. Since (Σ_{R}G_{∗})(a) = Σ_{R}(G_{∗}(a)) = Σ_{R}{G_{n}(a) : n ∈ N}, it follows from (1.3), (ii), that

∥(Σ_{R}G_{∗})(a)∥ ≤ Σ_{C}{∥G_{n}(a)∥ : n ∈ N} = Σ_{C}∥G_{∗}(a)∥.

Therefore we obtain, as in the proof of (1.4), (o),

\begin{aligned}|||ΣG_{∗}||| & = Σ_{C,A}\{∥(Σ_{R}G_{∗})(a)∥ : a ∈ A\} ≤ Σ_{C,A}\{Σ_{C}\{∥G_{n}(a)∥ : n ∈ N\} : a ∈ A\} \\ & = Σ_{C,A×N}((G_{∗})) = Σ_{C}|||G_{∗}||| \end{aligned}

and, through (1.1), (iii),

|||ΣG_{∗}||| ≤ Σ_{C}|||G_{∗}||| ≤ Σ_{C}|||F_{∗}|||.

There remains (1.4), (iii). Let F_{∗} ∈ S and let φ : N ⟶ N be a map. By (1.7), F_{∗}^{φ^{−1}(n)} is in S, for all n ∈ N, and F_{∗}^{φ^{−1}} is in S. Moreover, for every a ∈ A, due to (1.3), (iii),

(ΣF_{∗})(a) = Σ_{R}(F_{∗}(a)) = Σ_{R}(F_{∗}(a)^{φ^{−1}}) = Σ_{R}{(Σ_{R}(F_{∗}(a)^{φ^{−1}(n)})) : n ∈ N}
= Σ_{R}{(Σ_{R}(F_{∗}^{φ^{−1}(n)})(a)) : n ∈ N} = Σ_{R}{(ΣF_{∗}^{φ^{−1}(n)})(a) : n ∈ N}
= (Σ{ΣF_{∗}^{φ^{−1}(n)} : n ∈ N})(a) = (ΣF_{∗}^{φ^{−1}})(a),

and thus Σ(F_{∗}^{φ^{−1}}) = Σ(F_{∗}), as had to be shown.

At this point we know that L^{N}(A) is an R-prenormed R-semimodule with left N-summation (S_{L^{N} (A)}, Σ_{L^{N} (A)}).

For a ∈ A let δ^{a} be the Dirac function at a on A, that is the map A ⟶ R with δ^{a}(a) = 1 and δ^{a}(b) = 0, for all a ≠ b ∈ A. By (1.3), (i), δ^{a} is in L^{N}(A) and |||δ^{a}||| = 1 holds. The map A ∋ a ↦ δ^{a} ∈ B_{N}(L^{N}(A)) is denoted by δ and is called the Dirac map on A. We claim that δ : A ⟶ B_{N}(L^{N}(A)) is universal with respect to N_{R}pnSmod^{1}. For this let M be an R-prenormed R-semimodule with left N-summation (S_{M}, Σ_{M}) and let h : A ⟶ B_{N}(M) be any set map. Let furthermore f ∈ L^{N}(A) and consider the map fh given by A ∋ a ↦ f(a)h(a) ∈ M.

Then ∥fh∥(a) = ∥f(a)h(a)∥ ≤ ∥f(a)∥∥h(a)∥ ≤ ∥f(a)∥, for every a ∈ A. By (1.3), (o), f ∈ S_{R,A} implies ∥f∥ ∈ S_{C,A}. The last inequalities together with (1.1), (ii), show ∥fh∥ ∈ S_{C,A}. Hence (1.4), (o), implies fh ∈ S_{M,A}. Hence we have \overline{h}(f) := Σ_{M,A}(fh) ∈ M. Thus we obtain the map \overline{h} : L^{N}(A) ⟶ M. By (1.4), (i), \overline{h}(δ^{a}) = h(a), for all a ∈ A. Since Σ_{M,A} is a homomorphism of R-semimodules, the same is true for \overline{h}. We want to show that \overline{h} is a homomorphism of R-prenormed Rsemimodules with left N-summation. Let F_{∗} ∈ S. Then |||F_{∗}||| ∈ S_{C} due to (1.4), (o). Since, for any f ∈ L^{N}(A), |||\overline{h}(f)||| ≤ |||f||| holds, we have |||\overline{h}^{N}(F_{∗})||| ≤ |||F_{∗}||| and thus |||\overline{h}^{N}(F_{∗})||| ∈ S_{C} by (1.1), (ii), and hence \overline{h}^{N}(F_{∗}) ∈ S_{C} by (1.4), (o). That is, \overline{h}^{N}(S) ⊆ S_{M}. Now consider F_{∗} as a map A × N ⟶ R. Since |||F_{∗}||| is in S_{C}, it follows from (2.12) that F_{∗} is in S_{R,A×N}. Let p : A × N ⟶ A be the first projection and put h̃ := h ∘ p. Then F_{∗} · h̃ is in M^{A×N}, and it follows from (2.8) that F_{∗} · h̃ is in S_{M,A×N}. Hence (2.11) and (2.12) lead to

Σ_{M,A×N}(F_{∗} · h̃) = Σ_{M,A}{Σ_{M}{F_{n}(a) · h(a) : n ∈ N} : a ∈ A}
= Σ_{M,A}{(Σ_{R}{F_{n}(a) : n ∈ N}) · h(a) : a ∈ A}

= Σ_{M,A}\{(Σ_{R}F_{∗}(a)) · h(a) : a ∈ A\} = Σ_{M,A}((ΣF_{∗}) · h) = \overline{h}(ΣF_{∗}).

On the other hand

\begin{aligned}M_{M,A×N}(F_{∗} · h̃) & = Σ_{M}\{Σ_{M,A}\{F_{n}(a)h(a) : a ∈ A\} : n ∈ N\} \\ & = Σ_{M}\{\overline{h}(F_{n}) : n ∈ N\} = Σ_{M}(\overline{h}^{N}(F_{∗})). \end{aligned}

This means that Σ_{M}(\overline{h}^{N}(F_{∗})) = \overline{h}(ΣF_{∗}), which is the formula in (1.5), (i). Finally, as |||\overline{h}^{N}(F_{∗})||| ≤ |||F_{∗}||| has been shown before, (1.4), (ii), implies

∥Σ_{M} \overline{h}^{N}(F_{∗})∥ ≤ Σ_{C}|||\overline{h}^{N}(F_{∗})||| ≤ (Σ_{C}|||F_{∗}|||) · 1,

showing (1.5), (ii), as well as proving that \overline{h} is a contraction.

The very final step is now to show that \overline{h} is unique. So, let h′ : L^{N}(A) ⟶ M be a contractive homomorphism of R-prenormed R-semimodules with left Nsummation such that h = h′ ∘ δ. Let f ∈ L^{N}(A). As in (2.2) we have supp f ⊆ A′ ⊆ A, N′ ⊆ N, with cardA′ ≤ cardN, and χ : N ⟶ A satisfying (2.2), a). Let f_{∗} be the map

N ∋ n ↦ \{ \begin{matrix}f(χ(n))δ^{χ(n)} & , if n ∈ N′ \\ 0 & , otherwise. \end{matrix}

Obviously, f_{∗} is in (R^{A})^{N}. Since f is in S_{R,A}, f_{∗} is in S and Σf_{∗} = f. Since h^{′N}(f_{∗}) is the map

N ∋ n ↦ \{ \begin{matrix}h′(f(χ(n))δ^{χ(n)}) = f(χ(n))h′(δ^{χ(n)}) & , if n ∈ N′; \\ 0 & , otherwise; \end{matrix}

we have

\begin{aligned}h′(f) & = h′(Σf_{∗}) = Σ_{M}(h^{′N}(f_{∗})) = Σ_{M,A}\{f(a)h′(δ^{a}) : a ∈ A\} \\ & = Σ_{M,A}\{f(a)h(a) : a ∈ A\} = \overline{h}(f), \end{aligned}

proving the required uniqueness. ⊓ ⊔

4. N-convexity theories

(4.1) Definition. Let R be a prenormed semiring with left N-summation (S_{R}, Σ_{R}). By a left N-convexity theory over R is meant a subset Γ of S_{R} such that

(o) ∥α_{∗}∥ ∈ S_{C} and Σ_{C}∥α_{∗}∥ ≤ 1 , for all α_{∗} ∈ Γ,

(i) δ_{∗}^{n} ∈ Γ , for all n ∈ N, (ii) for all α_{∗}, β_{∗}^{n}, n ∈ N, in Γ the map ⟨α_{□}, β_{∗}^{□}⟩ given by N ∋ n ↦ Σ_{R}α_{∗}β_{n}^{∗} ∈ R

is in Γ.

It is a simple consequence of (1.1), (ii) – (iv), that ⟨α_{□}, β_{∗}^{□}⟩ satisfies (4.1), (o). Let X be any set and denote the elements of X^{N} by lower case letters with an

upper placeholder symbol, e. g. x^{∗} or x^{□}.

(4.2) Definition. Let Γ be a left N-convexity theory over R. By a left Γ-convex module is meant a set X ≠ ∅ together with a map

Γ × X^{N} ∋ (α_{∗}, x^{∗}) ↦ ⟨α_{∗}, x^{∗}⟩ ∈ X

such that (i) ⟨δ_{∗}^{n}, x^{∗}⟩ = x^{n} , for all n ∈ N, α_{∗} ∈ Γ, x^{∗} ∈ X^{N}, (ii) ⟨α_{□}, ⟨β_{∗}^{□}, x^{∗}⟩⟩ = ⟨⟨α_{□}, β_{∗}^{□}⟩, x^{∗}⟩ , for all α_{∗} ∈ Γ, β_{∗}^{□} ∈ Γ^{N}, x^{∗} ∈ X^{N}.

(4.3) Definition. Let Γ be a left N-convexity theory. By a homomorphism of left Γ-convex modules X ⟶ X′ is meant a map f : X ⟶ X′ such that

f(⟨α_{∗}, x^{∗}⟩) = ⟨α_{∗}, f^{N}(x^{∗})⟩  , for all α_{∗} ∈ Γ, x^{∗} ∈ X^{N}.

Let Γ be an N-convexity theory. Then the totality of left Γ-convex modules and their homomorphisms, with composition the set-theoretical one, form a category ΓC, the category of left Γ-convex modules. Clearly, ΓC is an algebraic category. Since it has a rank ([2], p.56), it has free objects on any set. However, we want to construct such free objects explicitly. First, three technical statements about Γ-convex modules. They correspond to [4], (2.4), (iii), (iv), and (viii), and the proofs there carry over to the current situation with nominal changes only.

(4.4) Lemma. Let X be a left Γ-convex module, let α_{∗} ∈ Γ with supp α_{∗} ⊆ N_{0} ⊆ N, and let y^{∗}, z^{∗} ∈ X^{N} be such that y^{n} = z^{n}, for all n ∈ N_{0}. Then ⟨α_{∗}, y^{∗}⟩ = ⟨α_{∗}, z^{∗}⟩. ⊓ ⊔

(4.5) Lemma. Let X be a left Γ-convex module, let α_{∗} ∈ Γ and x^{∗} ∈ X. For any bijection σ : N ⟶ N define ^{σ}α_{∗} resp. ^{σ}x^{∗} as the maps

N ∋ n ↦ α_{σ(n)} ∈ R  resp.  N ∋ n ↦ x^{σ(n)} ∈ X.

Then ⟨α_{∗}, x^{∗}⟩ = ⟨^{σ}α_{∗},^{σ}x^{∗}⟩. ⊓ ⊔

(4.6) Lemma. Let X be a left Γ-convex module, α_{∗} and β_{∗} in Γ, and x^{∗} and y^{∗} in X. Let furthermore φ : N ⟶ N be an injection and assume

\begin{matrix}β_{n} = α_{φ^{−1}(n)} , n ∈ φ(N),  and  β_{n} = 0 , n ∉ φ(N); \\ x^{n} = y^{φ^{−1}(n)} , n ∈ φ(N). \end{matrix}

Then ⟨α_{∗}, y^{∗}⟩ = ⟨β_{∗}, x^{∗}⟩. ⊓ ⊔

Next we have

(4.7) Theorem. (see [3], 5.4) The forgetful functor V_{Γ} : ΓC ⟶ Set has a left adjoint F^{Γ}.

Proof. Let R be the prenormed semiring with left N-summation (S_{R}, Σ_{R}) that appears in the definition (4.1) of Γ. Given any set A we define F^{Γ}(A) as the set of maps f : A ⟶ R such that card(supp f) ≤ cardN and for some data χ, β_{∗}, A′, and N′ in (2.2), β_{∗} ∈ Γ holds. In order to make F^{Γ}(A) a left Γ-convex module we have to define ⟨α_{∗}, f^{∗}⟩, for all α_{∗} ∈ Γ and f^{∗} ∈ F^{Γ}(A)^{N}. According to (2.3) we can choose χ, A′ and N′ such that for every f^{n} ∈ F^{Γ}(A) ⊆ S_{R,A}, n ∈ N, the conditions in (2.2) are satisfied. Suppose that β_{∗}^{n} is associated to f^{n} via the data χ, A′ and N′. Then β_{∗}^{n} ∈ Γ, for all n ∈ N. Due to (4.1), (ii), ⟨α_{□}, β_{∗}^{□}⟩ is in Γ and, as is seen easily, supp ⟨α_{□}, β_{∗}^{□}⟩ ⊆ ⋃{supp β_{∗}^{n} : n ∈ N} ⊆ N′. Hence there is a unique f ∈ F^{Γ}(A) with supp f ⊆ A′ and f(χ(n)) = ⟨α_{□}, β_{∗}^{□}⟩, for all n ∈ N′. Denote this f by ⟨α_{∗}, f^{∗}⟩. By (2.2), f does not depend on the data chosen. It is clear from the construction that (4.2), (i), is satisfied. In order to verify (4.2), (ii), it suffices to show the existence and equality of the two terms

⟨α_{□}, ⟨β_{∗}^{□}, γ_{Δ}^{∗}⟩⟩  and  ⟨⟨α_{□}, β_{∗}^{□}⟩, γ_{Δ}^{∗}⟩

for all α_{□} ∈ Γ, β_{∗}^{□} ∈ Γ^{N}, γ_{Δ}^{∗} ∈ Γ^{N}. However, the existence of these expressions is an immediate consequence of (4.1), (ii). As for equality, consider the map α_{□}β_{∗}^{□}γ_{t}^{∗} given by

N × N ∋ (n, p) ↦ α_{n}β_{p}^{n}γ_{t}^{p} ∈ R.

Since by (1.2), ∥α_{n}β_{p}^{n}γ_{t}^{p}∥ ≤ ∥α_{n}∥, it follows from (1.1), (o) and (ii), and (1.4), (o), that the map N ∋ n ↦ α_{n}β_{p}^{n}γ_{t}^{p} ∈ R is in S_{R}, whence Σ_{R}{α_{n}β_{p}^{n}γ_{t}^{p} : n ∈ N} is defined. Since, due to (1.2) and (1.3), (ii),

\begin{aligned}∥Σ_{R}\{α_{n}β_{p}^{n}γ_{t}^{p} : n ∈ N\}∥ & ≤ Σ_{C}\{∥α_{n}∥∥β_{p}^{n}∥∥γ_{t}^{p}∥ : n ∈ N\} \\ & ≤ Σ_{C}\{∥α_{n}∥∥β_{p}^{n}∥ : n ∈ N\}, \end{aligned}

and since by (1.1), (iv), the right hand side of this inequality (as a function of p) is in S_{C}, it follows from (1.3), (o), that

N ∋ p ↦ Σ_{R}{α_{n}β_{p}^{n}γ_{t}^{p} : n ∈ N} ∈ R

is in S_{R}. Hence (1.1), (iii’), and the use of a bijection N^{2} ⟶ N show that α_{□}β_{∗}^{□}γ_{t}^{∗} is in S_{R,N×N}. Therefore (2.11) leads to the desired equality. Which means, that F^{Γ}(A) is a left Γ-convex module. The Dirac map δ : A ⟶ F^{Γ}(A) assigns to each a ∈ A the Dirac function δ^{a} at a. We claim that δ : A ⟶ F^{Γ}(A) is a universal arrow. Let X be a left Γ-convex module and let h : A ⟶ X be a set map. We want to define an appropriate map h′ : F^{Γ}(A) ⟶ X. Let f ∈ F^{Γ}(A). Choose for f the data χ, β_{∗}, A′, and N′ and denote by h^{∗} the map

N ∋ n ↦ \{ \begin{matrix}h(χ(n)) & , if n ∈ N′; \\ x_{0} & , if n ∉ N′; \end{matrix}

where x_{0} is some element of X. It follows from (4.4) – (4.6) that ⟨β_{∗}, h^{∗}⟩ is independent of x_{0} and of the data chosen, and we put h′(f) := ⟨β_{∗}, h^{∗}⟩. Obviously, h′(δ^{a}) = h(a), a ∈ A, whence h′ ∘ δ = h holds. Next we check that h′ is a homomorphism of left Γ-convex modules. Let α_{∗} ∈ Γ and f^{∗} ∈ F^{Γ}(A)^{N}. By (2.3) we can choose data so that they serve for all f^{n}, n ∈ N. If β_{∗}^{n} is the element of Γ associated with f^{n}, n ∈ N, via such data then

h′(⟨α_{∗}, f^{∗}⟩) = ⟨⟨α_{□}, β_{∗}^{□}⟩, h^{∗}⟩,
while  ⟨α_{∗}, h^{′N}(f^{∗})⟩ = ⟨α_{□}, ⟨β_{∗}^{□}, h^{∗}⟩⟩,

whence the equality of the left hand sides of the last two equations follows from (4.2), (ii), showing that h′ is a homomorphism of left Γ-convex modules. As for uniqueness of the required factorization, let h̃ : F^{Γ}(A) ⟶ X be a homomorphism of left Γ-convex modules with h̃ ∘ δ = h. For f ∈ F^{Γ}(A) choose the data as above to obtain β_{∗} ∈ Γ associated with f. Denote furthermore by δ^{χ(∗)} the map N ∋ n ↦ δ^{χ(n)} ∈ F^{Γ}(A). An easy computation shows that f = ⟨β_{∗}, δ^{χ(∗)}⟩. Hence

h̃(f) = h̃(⟨β_{∗}, δ^{χ(∗)}⟩) = ⟨β_{∗}, h̃^{N}(δ^{χ(∗)})⟩ = ⟨β_{∗}, h^{∗}⟩ = h′(f).  □

5. The associated Γ-convex module functor

(5.1) Proposition. Let Γ be a left N-convexity theory over R. Then there is a functor O_{Γ} : N_{R}pnSmod^{1} ⟶ ΓC whose object function is the following:

if M is an R-prenormed R-semimodule with N-summation then the set underlying O_{Γ}(M) is B_{N}(M) and the Γ-convex module structure on O_{Γ}(M) is given by

Γ × O_{Γ}(M)^{N} ∋ (α_{∗}, μ^{∗}) ↦ Σ_{M}(α_{∗}μ^{∗}) =: ⟨α_{∗}, μ^{∗}⟩ ∈ O_{Γ}(M).

Proof. Since ∥α_{∗}μ^{∗}∥ ≤ ∥α_{∗}∥ and since ∥α_{∗}∥ ∈ S_{C} by (4.1), (o), it follows from (1.1), (ii), that ∥α_{∗}μ^{∗}∥ is in S_{C} and thus α_{∗}μ^{∗} is in S_{M}, due to (1.4), (o). Hence Σ_{M}(α_{∗}μ^{∗}) is well defined and

∥Σ_{M}(α_{∗}μ^{∗})∥ ≤ Σ_{C}(∥α_{∗}μ^{∗}∥) ≤ Σ_{C}(∥α_{∗}∥∥μ^{∗}∥) ≤ Σ_{C}(∥α_{∗}∥) ≤ 1,

whence ⟨α_{∗}, μ^{∗}⟩ = Σ_{M}(α_{∗}μ^{∗}) is in O_{Γ}(M). By (1.4), (i),

⟨δ_{∗}^{n}, μ^{∗}⟩ = Σ_{M}{δ_{p}^{n}μ^{p} : p ∈ N} = μ^{n}  , for all n ∈ N.

This verifies (4.2), (i). Now denote by α_{□}β_{∗}^{□}μ^{∗} the map N × N ∋ (n, p) ↦ α_{n}β_{p}^{n}μ^{p} ∈ M. Then ∥α_{n}β_{p}^{n}μ^{p}∥ ≤ ∥α_{n}∥∥β_{p}^{n}∥∥μ^{p}∥ ≤ ∥α_{n}∥∥β_{p}^{n}∥, for all n, p ∈ N. By (1.1), (iv), the right hand side of these inequalities, as a map N × N ⟶ C, is in S_{C,N×N}. Therefore ∥α_{□}β_{∗}^{□}μ^{∗}∥ is in S_{C,N×N} by (1.1), (ii), and α_{□}β_{∗}^{□}μ^{∗} is in S_{M,N×N} by (1.4), (o). Hence (2.11) and (2.12) imply

Σ_{M,N×N}(α_{□}β_{∗}^{□}μ^{∗}) = Σ_{M}{Σ_{M}{α_{n}β_{p}^{n}μ^{p} : n ∈ N} : p ∈ N}
= Σ_{M}{(Σ_{M}{α_{n}β_{p}^{n} : n ∈ N})μ^{p} : p ∈ N}
= ⟨⟨α_{□}, β_{∗}^{□}⟩, μ^{∗}⟩

and

Σ_{M,N×N}(α_{□}β_{∗}^{□}μ^{∗}) = Σ_{M}{Σ_{M}{α_{n}β_{p}^{n}μ^{p} : p ∈ N} : n ∈ N}
= Σ_{M}{α_{n}(Σ_{M}{β_{p}^{n}μ^{p} : p ∈ N}) : n ∈ N}
= ⟨α_{□}, ⟨β_{∗}^{□}, μ^{∗}⟩⟩,

which is (4.2), (ii). Thus the object function described in (5.1) is indeed in ΓC. Finally, given f : M ⟶ M′ in N_{R}pnSmod^{1}, we obtain from (1.5), (i),

\begin{aligned}f(⟨α_{∗}, μ^{∗}⟩) & = f(Σ_{M}α_{∗}μ^{∗}) = Σ_{M′}(f^{N}(α_{∗}μ^{∗})) \\ & = Σ_{M′}(α_{∗}f^{N}(μ^{∗})) = ⟨α_{∗}, f^{N}(μ^{∗})⟩, \end{aligned}

showing that f induces a homomorphism of Γ-convex modules. ⊓ ⊔

Since the value cone C of a prenormed semiring is partially ordered, we obtain an induced partial order on C^{N} (it was described at the beginning of §1). If T is a subset of C^{N}, then inf T refers to this partial order on C^{N}.

(5.2) Theorem. Let R be a prenormed semiring with left N-summation. Suppose that the value cone C of R satisfies the following conditions

CO: C is complete (in the sense of [5], §1); IS: for every T ⊆ S_{C}, inf{Σ_{C}(t_{∗}) : t_{∗} ∈ T } = Σ_{C}(inf{t_{∗} : t_{∗} ∈ T }); LD: for every ∅ ≠ U ⊆ C, u_{0} := inf{u : u ∈ U}, and every t ∈ C with

u_{0} < u_{0} + t there is a u_{1} ∈ U with u_{0} ≤ u_{1} ≤ u_{0} + t; LIM: for every U ⊆ C and every c ∈ C, inf{cu : u ∈ U} = c inf{u : u ∈ U}; OP: for every c_{∗} ∈ S_{C} there is a d_{∗} ∈ C^{N} such that

(o) c_{∗} + d_{∗} ∈ S_{C};

(i) if c_{n} is not a maximal element of C (with respect to the partial

order of C) then c_{n} < c_{n} + d_{n}.

Then for every left N-convexity theory Γ over R, O_{Γ} has a left adjoint S^{Γ}.

Proof. Let X be a Γ-convex module and denote the set underlying X again by X. Form, as in the proof of (3.1), the R-semimodule L^{N}(X) together with the Dirac map δ : X ⟶ L^{N}(X) and consider the set

S := {(δ^{⟨α_{∗},x^{∗}⟩}, ⟨α_{∗}, δ^{x^{∗}}⟩) : α_{∗} ∈ Γ, x^{∗} ∈ X^{N}} ⊆ L^{N}(X) × L^{N}(X),

where δ^{x^{∗}} is the map N ∋ n ↦ δ^{x^{n}} ∈ L^{N}(X). If h : L^{N}(X) ⟶ M is a contracting homomorphism of R-prenormed R-semimodules with N-summation we say that h is S-compatible if S_{h} := {(f, f′) : h(f) = h(f′)} ⊆ L^{N}(X) × L^{N}(X) contains S. Clearly, there are such contracting homomorphisms, e.g. the zero homomorphism. Let ∼ be the intersection of all these S_{h}. Then ∼ is an equivalence relation, S^{Γ}(X) := L^{N}(X)/ ∼ is an R-semimodule, and the quotient map q : L^{N}(X) ⟶ S^{Γ}(X) is a homomorphism of R-semimodules satisfying

q(δ^{⟨α_{∗},x^{∗}⟩}) = q(⟨α_{∗}, δ^{x^{∗}}⟩)  , for all α_{∗} ∈ Γ, x^{∗} ∈ X^{N}.

Now define ||| ||| : S^{Γ}(X) ⟶ C by

|||s||| := inf{|||f||| : q(f) = s}  , s ∈ S^{Γ}(X).

Since C is complete (in the sense of [5], §1), the above infimum exists, and it follows easily from [5], (2.10), (IA), – which is a simple consequence of (IS) – and (LIM) that ||| ||| : S^{Γ}(X) ⟶ C is a prenorm.

Next we put S_{S^{Γ}(X)} := q^{N}(S_{L^{N} (X)}). Suppose now that f_{∗}, f′_{∗} are in S_{L^{N} (X)} and satisfy q^{N}(f_{∗}) = q^{N}(f′_{∗}), that is q(f_{n}) = q(f′_{n}) for all n ∈ N. Then (f_{n}, f′_{n}) ∈ ∼ and hence (f_{n}, f′_{n}) ∈ S_{h} for all contracting homomorphisms h : L^{N}(X) ⟶ M that are S-compatible, for all n ∈ N. This shows that h^{N}(f_{∗}) = h^{N}(f′_{∗}) and therefore

h(Σ_{L^{N} (X)}f_{∗}) = Σ_{M}(h^{N}(f_{∗})) = Σ_{M}(h^{N}(f′_{∗})) = h(Σ_{L^{N} (X)}f′_{∗}),

that is we obtain the relation (Σ_{L^{N} (X)}f_{∗}, Σ_{L^{N} (X)}f′_{∗}) ∈ S_{h} for all such h, whence (Σ_{L^{N} (X)}f_{∗}, Σ_{L^{N} (X)}f′_{∗}) ∈∼ holds. Thus we can define

Σ_{S^{Γ}(X)}(s_{∗}) := q(Σ_{L^{N} (X)}f_{∗})  , s_{∗} ∈ S_{S^{Γ}(X)},

where f_{∗} ∈ S_{L}^{N}(X) is chosen such that g^{N}(f_{∗}) = s_{∗}. In particular, q(Σ_{L^{N} (X)}(f_{∗})) = Σ_{S^{Γ}(X)}(q^{N}(f_{∗})) for all f_{∗} ∈ S_{L^{N} (X)}. One checks easily that S_{S}^{Γ}(X) is an Rsubsemimodule of S^{Γ}(X)^{N} and that Σ_{S^{Γ}(X)} : S_{S^{Γ}(X)} ⟶ S^{Γ}(X) is a homomorphism of R-semimodules.

Next we verify (1.4), (o), for S^{Γ}(X). Let s_{∗} ∈ S_{S^{Γ}(X)} and choose f_{∗} ∈ S_{L^{N} (X)} with q^{N}(f_{∗}) = s_{∗}. Then |||s_{n}||| ≤ |||f_{n}|||, n ∈ N. Since |||f_{∗}||| is in S_{C}, so is |||s_{∗}||| due to (1.1), (ii). Conversely, assume s_{∗} ∈ S^{Γ}(X)^{N} and |||s_{∗}||| ∈ S_{C}. We apply (OP) to c_{∗} := |||s_{∗}||| and obtain d_{∗} with the properties stated there. If |||s_{n}||| is a maximal element of C, choose f_{∗} ∈ L^{N}(X) such that q(f_{n}) = s_{n}. Then |||f_{n}||| = |||s_{n}|||. If |||s_{n}||| is not a maximal element of C, (LD) implies the existence of an f_{n} ∈ L^{N}(X) with q(f_{n}) = s_{n} and |||s_{n}||| ≤ |||f_{n}||| ≤ |||s_{n}||| + d_{n}. Hence f_{∗} is in L^{N}(X) and |||f_{∗}||| ≤ |||s_{∗}||| + d_{∗} ∈ S_{C}, whence |||f_{∗}||| is in S_{C} due to (1.1), (ii), and thus f_{∗} is in S_{L^{N} (X)} by (1.4), (o). Therefore s_{∗} = q^{N}(f_{∗}) is in S_{S^{Γ}(X)}.

(1.4), (i), is trivially satisfied in the current situation.

On to (1.4), (ii). It follows from (1.5) that s_{∗} ∈ S_{S^{Γ}(X)} and t_{∗} ∈ S^{Γ}(X)^{N} with |||t_{∗}||| ≤ |||s_{∗}||| implies t_{∗} ∈ S_{S^{Γ}(X)}. Hence there are g_{∗} ∈ S_{L^{N} (X)} with t_{∗} = q^{N}(g_{∗}) and

|||Σ_{S^{Γ}(X)}t_{∗}||| = |||q(Σ_{L^{N} (X)}g_{∗})||| ≤ |||Σ_{L^{N} (X)}g_{∗}||| ≤ Σ_{C}|||g_{∗}|||,

due to (1.4), (ii), for L^{N}(X). Hence we have

|||Σ_{S^{Γ}(X)}t_{∗}||| ≤ inf{Σ_{C}|||g_{∗}||| : q^{N}(g_{∗}) = t_{∗}}
= Σ_{C}(inf{|||g_{∗}||| : q^{N}(g_{∗}) = t_{∗}}) = Σ_{C}|||t_{∗}|||.

Finally (1.4), (iii). Let φ : N ⟶ N be a map and let s_{∗} ∈ S_{S^{Γ}(X)}. Then there is

an f_{∗} ∈ S_{L^{N} (X)} with s_{∗} = q^{N}(f_{∗}). Since s_{∗}^{φ^{−1}(n)} = q^{N}(f_{∗}^{φ^{−1}(n)}) and since f_{∗}^{φ^{−1}(n)}

is in S_{L^{N} (X)}, due to (1.4), (iii), applied to L^{N}(X), we have s_{∗}^{φ^{−1}(n)} ∈ S_{S^{Γ}(X)} for

every n ∈ N. Moreover, by definition, Σ_{S^{Γ}(X)}s_{∗}^{φ^{−1}(n)} = q(Σ_{L^{N} (X)}f_{∗}^{φ^{−1}(n)}). Since

f_{∗}^{φ^{−1}}, that is the map N ∋ n ↦ Σ_{L^{N} (X)}f_{∗}^{φ^{−1}(n)} ∈ L^{N}(X), is in S_{L^{N} (X)} it follows

that s_{∗}^{φ^{−1}} is in S_{S^{Γ}(X)}. Finally

Σ_{S^{Γ}(X)}s_{∗}^{φ^{−1}} = q(Σ_{L^{N} (X)}f_{∗}^{φ^{−1}}) = q(Σ_{L^{N} (X)}f_{∗}) = Σ_{S^{Γ}(X)}s_{∗}.

Thus we have shown that S^{Γ}(X) is an R-prenormed R-semimodule with left Nsummation (S_{S^{Γ}(X)}, Σ_{S^{Γ}(X)}).

Additionally we claim that q : L^{N}(X) ⟶ S^{Γ}(X) is a contracting homomorphism of R-prenormed R-semimodules with left N-summation. (1.5), (i), is obvious from the construction of S^{Γ}(X), while (1.5), (ii), – with c = 1 – was established in the above verification of (1.4), (ii).

What remains to be done is to show that B_{N}(q) ∘ δ : X ⟶ O_{Γ}(S^{Γ}(X)) is a universal arrow. Since q is a homomorphism of R-prenormed R-semimodules with N-summation, B_{N}(q) ∘ δ is a homomorphism of left Γ-convex modules. Let h : X ⟶ O_{Γ}(M) be such a homomorphism. Due to (3.1) there is a contractive homomorphism h′ : L^{N}(X) ⟶ M of R-prenormed R-semimodules with N-summation with B_{N}(h′) ∘ δ = h. Since h is a homomorphism of left Γ-convex modules, h′ is S-compatible and hence gives rise to a factorization h′ = h′′ ∘ q, where h′′ : S^{Γ}(X) ⟶ M is a contractive homomorphism of R-prenormed Rsemimodules with N-summation. Hence h = B_{N}(h′′ ∘ q) ∘ δ, which is the required factorization. We claim that h determines \overline{h} uniquely. So, let h̃ ∘ q ∘ δ = h be another factorization. Each s ∈ S^{Γ}(X) can be written as q(f), with f ∈ L^{N}(X). Due to (2.4), f equals Σ_{L^{N} (X)}(α_{∗}δ^{χ(∗)}), where χ : N ⟶ A is a suitable map and α_{∗} ∈ S_{R} is chosen appropriately. Hence we have

\begin{aligned}h̃(s) & = h̃(q(f)) = h̃(q(Σ_{L^{N} (X)}(α_{∗}δ^{χ(∗)}))) \\ & = Σ_{M}(α_{∗}(h̃ ∘ q)^{N}(δ^{χ(∗)})) = Σ_{M}(α_{∗}h̃^{χ(∗)}), \end{aligned}

where h̃^{χ(∗)} is the map N ∋ n ↦ h(χ(n)) ∈ O_{Γ}(M). Thus h determines \overline{h} uniquely. ⊓ ⊔

6. Examples

Clearly, every positive semiring C has (C^{(N)}, ∑′_{C}), with ∑′_{C} the usual sum in C, as a left N-summation. Similarly, every prenormed semiring R has (R^{(N)}, ∑′_{R}) as a left N-summation, just as (M^{(N)}, ∑′_{N}) is a left N-summation for every Rprenormed R-semimodule. This means that the positive (resp. prenormed) semirings, the R-prenormed R-semimodules, and the finitary convexity theories discussed in [5], §1 - §3, are special cases of the notions investigated in the present paper.

One checks quite easily that the Banach semirings R discussed in [5], §6, are another instance of the concepts treated here. The cone of such a Banach semiring R has to satisfy suitable properties (see [5], 4.14 and 4.15); S_{C} is then the subset of C^{ℕ} consisting of all those α_{∗} for which Σα_{∗} has a limit (in the sense of [5], §4), while Σ_{C}α_{∗} := sup{α_{n} : n ∈ ℕ}. In addition, S_{R} is the set of those β_{∗} ∈ R^{ℕ} for which Σβ_{∗} is an absolute Cauchy sum (in the sense of [5], §4), while Σ_{R}β_{∗} is the limit (in the sense of [5], §4) of the infinite sum ∑ β_{∗}. Analogously one obtains (S_{M}, Σ_{M}) for each Banach R-semimodule M; in particular, each Banach R-semimodule is an R-prenormed R-semimodule with ℕ-summation in current terminology, while the converse in general fails to be correct. However, if C := ℝ_{+} = {r ∈ ℝ : r ≥ 0}, R := ℝ, ∥ ∥ : ℝ ⟶ ℝ_{+} is the usual absolute value, and S_{C} is the set of all α_{∗} ∈ ℝ_{+}^{ℕ} for which ∑ α_{∗} converges and Σ_{C}α_{∗} = ∑ α_{∗} then the Banach spaces over ℝ (in the sense of functional analysis) are precisely the ℝ-prenormed ℝ-semimodules with ⋉-summation as follows from a well known characterization of Banach spaces ([6], 3.1.2).

Now we want to characterize explicitly the concepts of the present paper in the case where the semiring involved is the smallest semiring that is not a ring. Define on the two-element set {0, 1}

addition by  0 + 0 = 0, 0 + 1 = 1 + 0 = 1 + 1 = 1,
multiplication by 0 · 0 = 0 · 1 = 1 · 0 = 0, 1 · 1 = 1,
partial order by 0 < 1.

Then {0, 1} equipped with this structure is a complete, commutative, and unital semiring 𝔻. Define an N-summation on 𝔻 by putting S_{𝔻} := 𝔻^{N} and Σ_{𝔻}α_{∗} := max{α_{n} : n ∈ N}. One checks easily that these data make 𝔻 a positive semiring with N-summation.

Put R := 𝔻 and define ∥ ∥ : R ⟶ 𝔻 by ∥r∥ := r, r ∈ R. Let furthermore S_{R} := 𝔻^{N} and Σ_{R} := Σ_{𝔻}. Then R = 𝔻 is a prenormed (even normed) semiring with N-summation. A 𝔻-prenormed 𝔻-semimodule M is commutative, idempotent (i.e. m+m = m for all m ∈ M) monoid together with a submonoid M_{0} (corresponding to {m ∈ M : ∥m∥ = 0}); one check easily that M_{0} can be an arbitrary submonoid of M. The monoid M has the additional property that 0 is the only element of M that possesses an (additive) inverse. Next we define “m_{1} ≤ m_{2}” as “there is an m ∈ M with m_{1} + m = m_{2}”, for all m_{1}, m_{2} ∈ M. One checks easily that this is a partial order relation on M (in particular, m_{1} ≤ m_{2} and m_{2} ≤ m_{1} imply m_{1} = m_{2}) that is compatible with the additive monoid structure on M. With respect to this order relation, M has finite suprema and sup{m_{1}, …, m_{n}} = m_{1} +…+m_{n}. Hence M_{0} is closed under finite suprema. If the 𝔻-prenormed 𝔻-semimodule M has Nsummation (S_{M}, Σ_{M}) then (1.4), (o), implies S_{M} = M^{N}. If μ_{∗} ∈ M^{N} and T ⊆ N we denote by μ_{∗}^{T} the map given by μ_{∗}^{T} |T = μ_{∗}|T and μ_{∗}^{T} |N ∖ T = 0_{∗}|N ∖ T. Then (1.4), (iii), shows that Σ_{M}μ_{∗}^{T} ≤ Σ_{M}μ_{∗} holds for all μ_{∗} ∈ M^{N} and T ⊆ N. It follows from (3.1) that for every m ∈ M, m = Σ_{M}m_{∗} is valid. An immediate consequence of this is Σ_{M}μ_{∗} = sup{μ_{n} : n ∈ N}. This means that M has N-suprema, that is suprema of all subsets of M of cardinality ≤ cardN.

Conversely one checks easily that any commutative, idempotent, partially ordered monoid M with a distinguished submonoid M_{0} such that the partial order is compatible with the monoid structure and has N-suprema is in fact a 𝔻-prenormed 𝔻-semimodule with N-summation (M^{N}, Σ_{M}), where Σ_{M}μ_{∗} = sup{μ_{n} : n ∈ N}.

Let Γ_{𝔻} := 𝔻^{N}. It is easy to see that Γ_{𝔻} is an N-convexity theory over 𝔻. Let X be a Γ-convex module. Then we say that for x_{1}, x_{2} ∈ X the relation x_{1} ≤ x_{2} is valid precisely when there are α_{∗} ∈ Γ and x^{∗} ∈ X^{N} such that

(i) there is an i ∈ supp α_{∗} with x^{i} = x_{1}, (ii) ⟨α_{∗}, x^{∗}⟩ = x_{2}.

One checks easily that this defines a partial order on X and that (with respect to this partial order) X has N-suprema. In fact, ⟨α_{∗}, x^{∗}⟩ = sup{x^{n} : n ∈ supp α_{∗}}. Moreover, if Y ⊆ X is a subset of cardinality ≤ cardN, let φ : Y ⟶ N be an injective map, and define α_{∗} resp. x^{∗} as the maps (with y_{0} ∈ X chosen arbitrarily)

N ∋ n ↦ \{ \begin{matrix}1 & , if n ∈ imφ; \\ 0 & , otherwise; \end{matrix}  resp.  N ∋ n ↦ \{ \begin{matrix}y & , if n = φ(y); \\ y_{0} & , otherwise. \end{matrix}

Then sup(Y) = ⟨α_{∗}, x^{∗}⟩.

Conversely, if X is a partially ordered set that has N-suprema, define

⟨α_{∗}, x^{∗}⟩ := sup{x^{n} : n ∈ supp α_{∗}}  , α_{∗} ∈ Γ and x^{∗} ∈ X^{N}.

A simple computation shows that this makes X a Γ_{𝔻}-convex module. Finally one concludes from (4.3) that a map f : X ⟶ X′ between Γ_{𝔻}-convex modules is a homomorphism of Γ_{𝔻}-convex modules precisely when for each subset Y ⊆ X of cardinality ≤ cardN, sup(f(Y)) = f(sup(Y)) gilt. Hence the category Γ_{𝔻}C is isomorphic to the category of partially ordered sets with N-suprema and Nsuprema preserving maps.

Instead of Γ_{𝔻} one could take the set Γ_{𝔻,} of all α_{∗} ∈ Γ_{𝔻} with card(supp α_{∗}) < cardN. A simple computation shows that Γ_{𝔻,} is an N-convexity theory over 𝔻. Then one obtains the same results as in the case of Γ_{𝔻} except that the requirement “existence of the supremum of every subset Y with cardY ≤ cardN” has to be replaced by the requirement “existence of the supremum of every subset Y with cardY < cardN”.

Instead of Γ_{𝔻} one could take the set Γ_{𝔻} ∖ {0_{∗}}. Again it is easy to see that Γ_{𝔻,sc} := Γ_{𝔻} ∖ {0_{∗}} is an N-convexity theory over 𝔻. Again, as before, the same results remain in force, except that the subsets Y in question now have to be nonempty. It should be pointed out, that Γ_{𝔻,sc} is the 𝔻-analog to the superconvexity theory Ω_{sc} := {α_{∗} ∈ ℝ^{ℕ} : α_{n} ≥ 0, for all n ∈ ℕ, and Σ_{ℕ}α_{n} = 1}. The 𝔻-analog to the classical convexity theory Ω_{c} := {α_{∗} ∈ Ω_{sc} : supp α_{∗} is finite} is then Γ_{𝔻,c} := {α_{∗} ∈ Γ_{𝔻,sc} : supp α_{∗} is finite}; results similar to the above hold for Γ_{𝔻,c}.

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Bodo Pareigis Math. Inst. Univ. München Theresienstr. 39 80333 München Germany Dieter Pumplün Fachb. Math. Fernuniv. Hagen 58084 Hagen Germany Helmut Röhrl 9322 La Jolla Farms Road. La Jolla, CA 92037 USA and Math. Inst. TU München Arcisstr. 21 80333 München

Germany