Salvatore Tringali
email: salvatore.tringali@upmc.fr

A UNIFIED APPROACH TO THE THEORY OF NORMED STRUCTURES -PART I: THE SINGLE-SORTED CASE

Keywords: 2010 Mathematics Subject Classification. 06F99, 18C10, 18C30, 46B99, 46H99

We introduce the concept of a prenormed model of a particular kind of singlesorted finitary first-order theories, interpreted over a category with finite products. These are referred to as prealgebraic theories, for the fact that their signature comprises, together with arbitrary function symbols (of finite ariety), only relation symbols whose interpretation, in any possible model, is a reflexive and transitive binary relation, namely a preorder. The result is an abstract approach to the very notion of norm and, consequently, to the theory of normed structures.

Introduction

There is no doubt that norms, along with diverse analog concepts such as valuations and seminorms, occupy a central place in mathematics, not only in relation to the notion of distance and metric spaces, but also in their own right, as for instance in the theory of Banach spaces [START_REF]Handbook of the Geometry of Banach Spaces -Volumes I and II[END_REF], valuated rings [START_REF] Bosch | Non-Archimedean Analysis[END_REF] and normed groups [START_REF] Bingham | Normed groups: dichotomy and duality[END_REF] (differently from other authors, we use here the term "valuation" with the meaning of "absolute value"). In fact, the present article is intended as one-half of a two-part work in a series of papers devoted to norms and normed structures. Since the "many-sorted case" is an essentially technical complication of the "one-sorted case" and adds no significant insights to the theory (at least in its basic aspects), we will concentrate here on the latter and consider the former only in the second part 1 . The long-term goal, as well as our original motivation, is the development of a framework suitable to carry out computations relevant to the a priori convergence theory of approximation schemes in numerical analysis, with a special focus on reduced basis methods [START_REF] Buffa | A Priori Convergence of the Greedy Algorithm for the Parametrized Reduced Basis[END_REF] (a standard technique used by several authors in applied mathematics to provide effective solutions of numerical problems depending on a large number of parameters). The link is the spectral theory of linear operators and Banach algebras [START_REF] Maday | A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations[END_REF], but we are not really going to dig into this in the sequel.

One of the main achievements here is, instead, the introduction of an abstract notion of norm for models of a special kind of finitary first-order theories interpreted over a category K with (all) finite products. Such theories will be called subalgebraic, due to the fact that their signature includes, together with arbitrary function symbols, only relation symbols whose interpretation is always a partial order (algebraic structures are covered as a special case). The corresponding models will be referred to as K-models, to stress the role of K in the picture. In particular, it will be proved that all the K-models of a given collection of prealgebraic theories form themselves a category whose morphisms can be eventually understood as "norms". This is used, in turn, to build up another category (over a fixed K-model), whose objects are ultimately an abstraction of normed spaces and whose morphisms are, in a generalized sense, "short maps" between them.

Upon these premises, we show by a number of examples how to recover down-to-earth constructions of common use in the everyday practice, for a variety of applications ranging from functional analysis to linear algebra and number theory, such as normed groups and valuated rings (Part I), normed spaces and normed algebras (Part II), and variants, generalizations or specializations thereof including seminormed semigroups, non-Archimedean pseudo-semivaluated fields, normed modules over valuated rings, etc. In this respect, we will face, in the next pages, the following (somewhat vague) questions: What is abstractly a norm? And what are the essential features that one should retain in order to give a purely algebraic definition of a norm, to the degree that normed structures can be ultimately identified with the objects of an appropriate category and norms with the arrows between these objects? The answers provided in the sequel are certainly far from being exhaustive and definite, but we are confident that the subject may be worth the effort and our hope is that it can attract the interest of other researchers in the field, for the benefits are potentially great: push on the use of categorical methods in analysis and analytical methods in category theory.

To the best of our knowledge, the only previous contribution in this line dates back to the 2008 Ph.D. thesis of G.S.H. Cruttwell [START_REF] Cruttwell | Normed Spaces and the Change of Base for Enriched Categories[END_REF]Ch. 3]. There, partially based on work by M. Grandis [START_REF] Grandis | Categories, norms and weights[END_REF], the author gives a categorical abstraction of normed groups by regarding a group norm as a (lax) monoidal functor from a compact closed category D to a monoidal category M. Then, he considers the category AbNorm with objects given by Abelian normed groups and arrows by group homomorphisms which are also weakly contractive maps. Lastly, he defines a normed (unital) ring R as a one-object category enriched over AbNorm and a normed module over R as an AbNorm-functor R → AbNorm (having once recognized that AbNorm is a category enriched over itself). Our approach is substantially different (despite of a few points in common). We do not focus our attention on a restricted class of familiar normed structures to describe them from the general perspectives of categories. Rather, we combine the language of categories with that of model theory to invent a general notion of "norm", which applies especially to any arbitrary model of any algebraic theory, to the degree that normed groups, normed rings and normed modules result as an instance of a general concept of "normed" structure.

Many ideas in this paper have been influenced by the prominent work of R. Lowen on approach spaces [START_REF] Lowen | Approach spaces. The missing link in the topology-uniformity-metric triad[END_REF] and F.W. Lawvere on algebraic theories [START_REF] Lawvere | Functorial Semantics of algebraic theories[END_REF] and generalized metric spaces [START_REF] Lawvere | Metric spaces, generalised logic, and closed categories[END_REF]. From a categorical point of view, extended pseudometric spaces and extended pseudoquasimetric spaces, along with their corresponding Lipschitz maps, have the best properties that one can actually expect from a category of metric spaces: It is possible, within them, to form quotients and take arbitrary products and coproducts. Dropping the attribute "extended" implies that, in general, only finite products and coproducts will exist, while curtailing the prefix "pseudo" affects the existence of quotients. Moving from these considerations, it seems quite reasonable, in search of a "good" answer to questions concerning the "real nature" of norms, to focus first on the weaker notions of seminorm and pseudoseminorm. This leads to one more basic insight, which has been central in this research and can be roughly outlined as follows.

Loosely speaking, a homomorphism of two algebraic structures of the same type, as described in the language of model theory, is a function between the underlying sets with the property of "preserving the operations". Then, one observes that, with a little effort of imagination, a seminorm, say, on a real vector space exhibits almost the same behaviour: (i) Its codomain is a special "reference structure". In the toy case that we are considering, this is the set of non-negative real numbers, herein denoted by R + 0 , together with its standard structure of totally ordered semiring, i.e., a ring without additive inverses (we do not intentionally regard R + 0 , in this paper, as an ordered semifield). (ii) It preserves the additive identity (a nullary operation). This has always been something subtle (and, hence, interesting) to our eyes: In the final analysis, one is basically requiring a seminorm to map a distinguished element a in the domain to a distinguished element b in the codomain, in a context where a and b play the same (algebraic) role, but still in a match lining up essentially different teams (both of them are identities, but in structures marked by significant differences). (iii) It relates a sum (of vectors) to a sum (of scalars) by means of an inequality. (iv) It equates the product of a scalar by a vector to a product of two scalars, which is informally the same as saying that it preserves the products, except that the one product and the other have very little in common, at least at a first glance.

That said, the next step is to give emphasis to something absolutely obvious, i.e., that equalities and inequalities, appearing in such a fundamental way in the (classical) definition of seminorms, have in common the property of being orders. Some of them are partial, as for the equality relation, while others are total, like in the case of the standard order on the set of real numbers, but they all are orders, i.e., reflexive, antisymmetric and transitive binary relations. And it is just by using orders and relaxing equalities to inequalities that we can manage to relate structures of different types and "let them play a good game." This intuition is strengthened by the inspection of other similar constructions encountered in various fields of the mathematical landscape. E.g., a group seminorm can be abstractly defined, based on common terminology and notation from model theory (cf. Remarks 2 and 3), as a function • from a group (G; +, -, 0 G ) to an ordered monoid (M ; +, 0 M ; ≤ M ) such that a+b ≤ M a + b for all a, b ∈ G and 0 G = 0 M , and this is, indeed, called a group norm if it is symmetric (with respect to the unary operation of negation) and a = 0 M for some a ∈ G if and only if a = 0 G . Likewise, an absolute value is defined, in the context of ring and field theory, as a function

| • | from a domain (D; +, • , -, 0 D ) to an ordered ring (R; +, • , -, 0 R ; ≤ R ) such that | • | is a group seminorm from the [Abelian] group (D; +, -, 0 D ) to the [Abelian] ordered monoid (R + 0 ; +, 0 R ; ≤ R ) such that |a • b| = |a| • |b| for all a, b ∈ D, where R + 0 := {a ∈ R : 0 R ≤ R a}.
Thus, it is naively apparent the existence of a common pattern among these definitions, and the primary goal of the paper is, indeed, to give an explicit formal description of such a pattern.

Basic notation and terminology. We set our foundations in the Neumann-Bernays-Gödel axiomatic class theory (NBG), as presented in [13, Ch. IV]. We use N for the non-negative integers and Z, Q and R according to their standard meaning. Unless differently stated, each of these sets will be endowed with its ordinary order and operations.

If X, Y are classes, D ⊆ X and f ⊆ D × Y is such that, for every x ∈ D, there exists only one y ∈ Y such that (x, y) ∈ f , we say that f is a (total) function (or map, mapping, or similia) D → Y , but also that f is a partial function from X to Y . In this case, D, X and Y are called, each in turn, the domain, the source and the target of f . In particular, we write dom(f ) for D and use the notation f : X → Y (an arrow with a vertical stroke) for a partial function f from X to Y . Note that, formally, a partial map from X to Y is an ordered triple (X, Y, f ) for which f is a function D → Y for some D ⊆ X. Yet, we will often identify (X, Y, f ) with f when it is convenient to do that and it is clear from the context which classes must be used as source and target. Lastly, if S ⊆ X and g is a function X → Y , then we denote by g |S , as is customary, the mapping S → Y : x → f (x) and refer to g |S as the restriction of g to S.

Furthermore, in dealing with a set I, we write |I| for its cardinality and use {X i } i∈I if we want to describe an indexed family of members of a certain class X. All indexed families considered in this paper are implicitly indexed by sets.

Organization. In Section 2, after having recalled some rudiments of model theory and given definitions useful to adapt them to our specific needs, we introduce prealgebraic [resp. subalgebraic] theories and prenorms [resp. subnorms] and prove the main result of the paper (i.e. Proposition 2.1), subsequently presenting the category of prealgebraic [resp. subalgebraic] Kmodels relative to a certain family of prealgebraic [resp. subalgebraic] theories (for K a category with finite products). Section 3 discusses prenormed [resp. subnormed] models and Section 4 shows how these are ultimately an abstraction of familiar normed structures, such as normed groups and valuated rings, by a number of examples. Lastly, in Appendix A, we provide a short introductory overview to category theory. The intent is twofold. On the one hand, this article is motivated by research in the field of numerical analysis, and therefore it aims to attract the interest not only of categorists but also of non-specialists in the are. On the other, we feel necessary to fix, once and for all, basic notation and terminology that we are going to use, both here and in future work, to deal with categories.

First-order prealgebraic theories and prenorms

In the traditional language of model theory and first-order logic [START_REF] Chang | Model Theory[END_REF], a (finitary single-sorted) signature, or type, is a triple σ = (Σ f , Σ r , ar), where Σ f and Σ r are disjoint sets not including logical symbols of the underlying formal language and ar is a map Σ f ∪ Σ r → N. The members of Σ f are called function symbols, those of Σ r relation symbols. For each symbol ζ ∈ Σ f ∪ Σ r , ar(ζ) is referred to as the ariety of ζ. A subsignature of σ is any signature σ 0 = (Σ f,0 , Σ r,0 , ar 0 ) such that Σ f,0 ⊆ Σ f , Σ r,0 ⊆ Σ r and ar 0 is the restriction of ar to Σ f,0 ∪ Σ r,0 . In addition to this, we say that σ is algebraic if Σ r = ∅ and balanced if there exists a bijection φ : Σ f → Σ r . Remark 1. A balanced signature σ = (Σ f , Σ r , ar) can be, and will be, systematically represented as ({(ς r , ̺ r )} r∈R ; ar), where {ς r : r ∈ R} = Σ f and {̺ r : r ∈ R} = Σ r .

Provided that σ i = (Σ f,i , Σ r,i , ar i ) is a signature (i = 1, 2), we define a signature homomorphism from σ 1 to σ 2 to be a map α

: Σ f,1 ∪ Σ r,1 → Σ f,2 ∪ Σ r,2 such that α(Σ f,1 ) ⊆ Σ f,2 , α(Σ r,1 ) ⊆ Σ r,2 and ar 2 (α(ζ)) = ar 1 (ζ) for every ζ ∈ Σ f,1 ∪ Σ r,1 .
If so, we write that α : σ 1 → σ 2 is a signature homomorphism. In addition to this, for σ 0 = (Σ f,0 , Σ r,0 , ar 0 ) a subsignature of σ 1 , we say that a signature homomorphism α 0 : σ 0 → σ 2 is the restriction of α to σ 0 if, regarded as a function, it is the restriction of α to Σ f,0 ∪ Σ r,0 . Lastly, we refer to the signature homomorphism j :

σ 0 → σ 1 sending each ζ ∈ Σ f,0 ∪ Σ r,0 to itself as the canonical injection σ 0 → σ 1 .
Remark 2. Pick σ = (Σ f , Σ r , ar) to be a signature. For k, ℓ ∈ N, suppose that Σ f and Σ r can be respectively partitioned into k families of function symbols {ς 1,r } r∈R1 , . . . , {ς k,r } r∈R k and ℓ families of relation symbols {̺ 1,s } s∈S1 , . . . , {̺ ℓ,s } s∈S ℓ . Then, σ is possibly denoted by ({ς 1,r } r∈R1 , . . . , {ς k,r } r∈R k ; {̺ 1,s } s∈S1 , . . . , {̺ ℓ,s } s∈S ℓ ; ar).

(

) 1
On another hand, assume that σ is balanced and let σ = ({(ς r , ̺ r )} r∈Σ f ; ar) (see Remark 1). Admit that there exists k ∈ N such that {(ς r , ̺ r )} r∈Σ f can be partitioned into k collections of the form {(ς 1,r , ̺ 1,r )} r∈R1 , . . . , {(ς k,r , ̺ k,r )} r∈R k . Then, we possibly write σ as

({(ς 1,r , ̺ 1,r )} r∈R1 ; . . . ; {(ς k,r , ̺ k,r )} r∈R k ; ar). (2)
These notations are further simplified, in the most obvious way, whenever a family of symbols consists of one element (i.e. is a singleton), to the extent of writing, for instance, (+, ⋆, 1; ≤, ∼; ar) in place of σ = ({+, ⋆, 1}, {≤, ∼}, ar) or (-, ; ⋆, ≃; ar) instead of ({-, ⋆}, { , ≃}, ar).

One-sorted finitary signatures form a category, denoted by Sgn. This has the class of (onesorted finitary) signatures as objects and all triples of the form (σ, τ, α) as morphisms, with σ and τ being signatures and α : σ → τ a signature homomorphism. As is customary, when there is no likelihood of ambiguity, we will use α as a shorthand of (σ, τ, α). The composition of two morphisms (σ, τ, α) and (τ, ω, β) is defined by the triple (σ, ω, β • Set α). The local identities and the maps of source and target are the obvious ones. With this in hand, let us assume henceforth that K is a category with (all) finite products (see Appendix A for notation and terminology). In our understanding, a (finitary single-sorted) structure over K, or K-structure, is then any 4-uple A = (A, χ, σ, I) consisting of (i) an object A of K, referred to as the carrier of the structure and denoted by |A|.

(ii) a "sorting" function χ :

N → obj(K) such that χ(1) = A and χ(n) ∈ K {A} n i=1 . (iii) a (finitary single-sorted) signature σ = (Σ f , Σ r , ar). (iv) an interpretation function I : Σ f ∪ Σ r → hom(K) sending an n-ary function symbol ς ∈ Σ f to an arrow f ς ∈ hom K (χ(n), A) and an n-ary relation symbol ̺ ∈ Σ r to a monomorphism m ̺ ∈ hom K (R, χ(n))
, whence an n-ary relation on A (see Definition 5). As is customary, a nullary function symbol ς is called a constant symbol, because its interpretation I(ς) can be identified with a distinguished "point" of A.

Remark 3. In the sequel, dealing with a (finitary single-sorted) K-structure A = (A, χ, σ, I), we systematically forget about χ and write simply A n in place of χ(n), with the implicit arrangement that this is only a shorthand for a distinguished representative in the isomorphism class of the product objects of K {A} n i=1 . Thus, to describe a K-structure, we use a 3-uple instead of a 4-uple and omit any further reference to any sorting function. In addition, given n points a 1 , a 2 , . . . , a n ∈ A, we write (a 1 , a 2 , . . . , a n ) in place of (a 1 , a 2 , . . . , a n ; A 0 ) (see Remark 21). Finally, as far as there is no danger of confusion, we do not make any notational distinction between a symbol ζ of σ and its interpretation under I, to the extent of using ζ for I(ζ): This will be especially the case when ζ is the function symbol of a binary operation, such as +, • or ⋆, or the relation symbol of a preorder, such as ≤ or . In these circumstances, for a, b ∈ A, we will write, e.g., a + b in place of I(+)(a, b) and a ≤ b for (a, b) ∈ I(≤) without additional explanation (see Remark 21).

Remark 4. Still for the sake of simplifying our notation, suppose that

A i = (A i , σ i , I i ) is a K-structure (i = 1, 2), let ϕ be a morphism A 1 → A 2 of K, and pick n ∈ N. Then, we agree to denote by ϕ n the arrow K (π 1,j • K ϕ, π 2,j ) n j=1 (indeed a morphism A n 1 → A n 2 )
, where {π i,j } n j=1 is the set of the canonical projections A n i → A i . In point of fact, ϕ n is unambiguously determined in the light of Remarks 3 and 18.

Remark 5. In the cases considered below to work out the basics of the abstract theory of normed structures, we will restrict ourselves to K-structures of type (A, σ, I), where σ is balanced and each relation symbol ̺ of σ, if any is present, is binary and its interpretation is a preorder [resp. a partial order] on A, in such a way that (A, I(̺)) is an object of Pre(K) [resp. Pos(K)], i.e., a preordered object [resp. a pod] of C (see Definition 6). When this occurs, A will be referred to as a prealgebraic [resp. subalgebraic] K-structure, and indeed as an algebraic K-structure if I(̺) is the equality relation on A for every ̺ ∈ Σ r . Algebraic structures over Set are precisely the (finitary single-sorted) structures traditionally studied by universal algebra. Remark 6. Say that A = (A, σ, I) is a K-structure, with σ = (Σ f , Σ r , ar), and assume that, for some k, ℓ ∈ N, it is possible to partition Σ f and Σ r , each in turn, into k families of function symbols {ς 1,r } r∈R1 , . . . , {ς k,r } r∈R k and ℓ families of relation symbols {̺ 1,s } s∈S1 , . . . , {̺ ℓ,s } s∈S ℓ (see Remark 2). In this case, A is possibly represented by

(A; {ς 1,r } r∈R1 , . . . , {ς k,r } r∈R k ; {̺ 1,s } s∈S1 , . . . , {̺ ℓ,s } s∈S ℓ ). ( 3
)
On another hand, admit that σ is balanced and let σ = ({(ς r , ̺ r )} r∈Σ f ; ar) (see Remark 1). Suppose that there exists

k ∈ N such that {(ς r , ̺ r )} r∈Σ f can be partitioned into k collections of the form {(ς 1,r , ̺ 1,r )} r∈R1 , . . . , {(ς k,r , ̺ k,r )} r∈Rm .
Then, we possibly denote A by

(A; {(ς 1,r , ̺ 1,r )} r∈R1 ; . . . ; {(ς k,r , ̺ k,r )} r∈R k ). (4)
Moreover, these notations are further simplified, in the most obvious way, if a family of symbols is a singleton, to the degree of writing, e.g., (A; +, ⋆, 1; ≤, ∼) in place of (A, σ, I) provided σ = ({+, ⋆, 1}, {≤, ∼}, ar) or (A; -, ; ⋆, ≃) instead of (A, σ, I) for σ = ({-, ⋆}, { , ≃}, ar).

Upon these premises, fix an infinite set V of (propositional) variables. One denotes by V ; σ the collection of all (well-formed) formulas in the variables V generated by σ according to the formation rules of first-order logic and says that a K-structure A = (A, σ, I) satisfies a formula φ ∈ V ; σ of n arguments x 1 , x 2 , . . . , x n ∈ V if the interpretation φ(A) of φ over A is a true statement, which is expressed by writing that A |= φ. Here, φ(A) is obtained from φ by (i) by replacing each variable x i with a point A 0 → A and each ζ ∈ Σ f ∪ Σ r occurring in the expression of φ with its interpretation under I. (ii) by interpreting expressions of the form I(ς)(a 1 , a 2 , . . . , a n ) and (a 1 , a 2 , . . . , a n ) ∈ I(̺),

where ς ∈ Σ f and ̺ ∈ Σ r are n-ary symbols and a 1 , a 2 , . . . , a n are points A 0 → A of K, according to Remarks 3 and 21. A σ-theory, or a theory of type σ, in the variables V is then any triple T = (V, σ, Ξ) such that Ξ is a (possibly empty) subset of V ; σ , while a K-model A of T , or equivalently a model of T over K, is a K-structure (A, σ, I) that satisfies every axiom φ ∈ Ξ. Such a condition is equivalently stated by writing A |= T and saying that A satisfies T : in this respect, σ will be also referred to as the signature of (T, A). If T = (V, σ, Ξ) is a theory, a subtheory of T is any theory T s = (V, σ s , Ξ s ) such that σ s is a subsignature of σ and Ξ s = Ξ ∩ V ; σ s , while a K-submodel of T is a K-model of a subtheory of T . If T s is a subtheory of T , we write T s ≤ T and possibly say that T is an extension, or a supertheory, of T s .

Remark 7. Let σ include, among its functional symbols, two binary symbols ∨ and ∧, a unary symbol u and a nullary symbol e. Then, consider the following formulas from V ; σ :

(a.1) ∀x, y, z ∈ V : (x ∨ y) ∨ z = x ∨ (y ∨ z). (a.2) ∀x ∈ V : x ∨ e = e ∨ x = x. (a.3) ∀x ∈ V : x ∨ u(x) = u(x) ∨ x = e. (a.4) ∀x, y, z ∈ V : x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). (a.5) ∀x, y, z ∈ V : (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z).
We refer, as is usual, to (a.1) as the axiom of associativity for the symbol ∨; to (a.2) as the axiom of neutrality for the pair (∨, e); to (a.3) as the axiom of inverses for the triple (∨, u, e); to (a.4) and (a.5), respectively, as the axioms of left and right distributiveness of ∧ over ∨. We list them here for future reference, they will be used later to deal with examples in Section 4.

Definition 1. We say that a theory T = (V, σ, Ξ) is prealgebraic if σ is a balanced signature of type (Σ f , Σ r , ar), where all relation symbols are binary and, for each ̺ ∈ Σ r , the axioms of T include at least the axiom of reflexivity: ∀x ∈ V : (x, x) ∈ ̺, and the axiom of transitivity: ∀x, y, z ∈ V : (((x, y) ∈ ̺) AND ((y, z) ∈ ̺))) =⇒ ((x, z) ∈ ̺). In addition to this, T will be called subalgebraic if it is prealgebraic and, for every ̺ ∈ Σ r , Ξ contains also the axiom of antisymmetry:

∀x, y ∈ V : (((x, y) ∈ ̺) AND ((y, x) ∈ ̺))) =⇒ (x = y). On another hand, T is called algebraic if Σ r is empty (no relational symbols are permitted) and algebraic over K if, for each K-model A = (A, σ, I) of T , there is one more K-model A a = (A, σ, I a ) of T such that I a (ς) = I(ς) for each ς ∈ Σ f and I a (̺) is the equality relation on A for all ̺ ∈ Σ r , so that A a is an algebraic K-structure (see Remark 5), referred to as an algebraization of A, on the one hand, and an algebraic K-model of T , on the other. Note that, if T is prealgebraic [resp. subalgebraic], any relation symbol of its signature will be interpreted, in any possible K-model A of T , as a preorder [resp. a partial order] on |A|.

Definition 2. A K-model A = (A, σ, I) of a theory T = (V, σ, Ξ), with σ = (Σ f , Σ r ,
ar), is said pivotal if the symbols of Σ r have all the same ariety and either Σ r = ∅ or there exists ̺ 0 ∈ Σ r such that I(̺) ⊆ I(̺ 0 ) for every ̺ ∈ Σ r . When this happens, I(̺ 0 ) is called the pivot of A.

Pivotal models will be crucial, later in this section, for the definition of the categories of prenormed and subnormed structures over a fixed "target" (see Section 3). Remark 8. A subalgebraic theory is always a prealgebraic theory. More interestingly, there is a canonical way to identify a K-model A = (A, σ, I) of an algebraic theory T = (V, σ, Ξ) with a K-model of a subalgebraic theory (in the same variables), so that any algebraic theory can be definitely identified with an subalgebraic theory. To see how, let σ = (Σ f , ∅, ar). For each ς ∈ Σ f , consider a binary relation symbol ̺ ς / ∈ Σ f , not already comprised among the basic symbols of the underlying logic, and set Σ r := {̺ ς } ς∈Σ f . Extend ar to the function ar e : Σ f ∪ Σ r → N by taking ar e (̺) := 2 for every ̺ ∈ Σ r and define σ e := (Σ f , Σ r , ar e ). Lastly, expand Ξ to a larger set of axioms, namely Ξ e , in such a way as to include all and only the axioms of reflexivity, symmetry and transitivity relative to every relation symbol ̺ ∈ Σ r . Then, T e = (V, σ e , Ξ e ) is a subalgebraic (and hence prealgebraic) theory and A can be identified with the model A e = (A, σ e , I e ) of T e defined by assuming that I e = I on Σ f and I e (̺) is the equality relation on A for each ̺ ∈ Σ r . On another hand, say that T is an arbitrary theory. Then, a slight modification of the above arguments shows that it is always possible to find a smallest prealgebraic [resp. subalgebraic] theory in the same variables and with the same signature as T , where "smallest" must be intended as "with the fewest possible axioms". This will be denoted by ♯ p T [resp. ♯ s T ] and called the prealgebraic [resp. subalgebraic] embodiment of T . Remark 9. When the set of variables V is well understood from the context, we will use simply (σ, Ξ) in place of (V, σ, Ξ) to indicate a theory in the variables V . Definition 6), we convey to denote by ℘(P, Q) the preorder induced on hom K (P, Q) by ≤ Q as follows: If f, g ∈ hom K (P, Q), we let (f, g) ∈ ℘(P, Q) if and only if f (x) ≤ Q g(x) for all x ∈ P . Observe that, in the special case where K = Set, ℘(P, Q) represents, up to an isomorphism, the preorder corresponding to the exponential [1, Chapter 6] of the pair (P, Q) in Pre(Set) for P being the trivial poset on P (see Example c.3 in Appendix A). Also, note that ℘(P, Q) is, in fact, a partial order on hom K (P, Q) when Q is a pod of K and K is such that two arrows f, g : A → B are equal if f (a) = g(a) for all a ∈ A: this is, e.g., the case for Set and Pos(Set) (cf. [1, p. 36]).

If P ∈ obj(K) and Q = (Q, ≤ Q ) is a preordered object of K (see
Lemma 2.1. Given that P ∈ obj(K) and

Q = (Q, ≤ Q ) and R = (R, ≤ R ) are preordered objects of K, let f, f 1 , f 2 ∈ hom K (P, Q) and g, g 1 , g 2 ∈ hom K (Q, R). The following hold: (i) If (g 1 , g 2 ) ∈ ℘(Q, R), then (g 1 • K f, g 2 • K f ) ∈ ℘(P, R). (ii) If (f 1 , f 2 ) ∈ ℘(P, Q) and g ∈ hom Pre(K) (Q, R), then (g • K f 1 , g • K f 2 ) ∈ ℘(P, R).
Proof. The first claim is obvious. As for the second, pick

x ∈ P . Since (f 1 , f 2 ) ∈ ℘(P, Q), it is f 1 (x) ≤ Q f 2 (x), from which g(f 1 (x)) ≤ R g(f 2 (x)
), for g is a monotonic function Q → R. By the arbitrariness of x ∈ P , this completes the proof.

Now, in what follows, let T i = (V, σ i , Ξ i ) be a prealgebraic [resp. subalgebraic] theory and

A i = (A i , σ i , I i ) a K-model of T i (i = 1, 2), with σ i = ({ς r,i , ̺ r,i } r∈Ri ; ar i ).
It is then possible to regard a signature homomorphism α : σ 1 → σ 2 as a pair (α 1 , α 2 ) of maps α 1 , α 2 : R 1 → R 2 by imposing that ς α1(r),2 = α(ς r,1 ) and ̺ α2(r),2 = α(̺ r,1 ) for each r ∈ R 1 . Hence, we systematically abuse notation and identify α with its "components" α 1 and α 2 , to the extent of writing ς α(r),2 in place of ς α1(r),2 and ̺ α(s),2 for ̺ α2(s),2 . In addition to this, set A r,i equal to (A i , ̺ r,i ) for every r ∈ R i (while having in mind Remark 3). The following definition is fundamental:

Definition 3. We say that A 1 is prehomomorphic [resp. subhomomorphic] to A 2 if there exist α ∈ hom Sgn (σ 1 , σ 2 ) and ϕ ∈ hom K (A 1 , A 2 ) such that, for each r ∈ R 1 , (i) (ϕ • K ς r,1 , ς α(r),2 • K ϕ n ) ∈ ℘(A n 1 , A α(r),2 ) (see Remark 4); (ii) ϕ is a monotonic arrow A r,1 → A α(r),2 in Pre(K) (see Example c.3 in Appendix A),
where n := ar 1 (ς r,1 ) (see Remark ). Then, we call Φ := (α, ϕ) a K-prenorm [resp. K-subnorm] or write that Φ : A 1 → A 2 is a prenorm [resp. subnorm] of K-models. In particular, a K-prenorm

A 1 → A 2 is said a K-homomorphism if I 2 (̺ α(r),2
) is the equality on A 2 for each r ∈ R 1 .

Remark 10. Definition 3 returns exactly the standard notion of a homomorphism of algebraic structures, as given in the framework of universal algebra, in the special case where T i is an algebraic theory over Set and T 1 = T 2 .

Remark 11. Let T ∅ denote the "empty theory" (V, σ ∅ , ∅), where σ ∅ = (∅, ∅, ∅). Clearly, T ∅ is a subalgebraic (indeed algebraic) theory and its K-models are all and only the triples (A, σ ∅ , ∅) such that A ∈ obj(K). Therefore, for T another prealgebraic [resp. subalgebraic] theory in the variables V , A ∅ a K-model of T ∅ and A a K-model of T , the K-prenorms [resp. K-subnorms] A ∅ → A are all and only the pairs (∅, ϕ) for which ϕ is a morphism

|A ∅ | → |A| in K.
Remark 12. Except for those required to turn T 1 and T 2 into prealgebraic [resp. subalgebraic] theories, the definition of a K-prenorm [resp. K-subnorm] A 1 → A 2 does not depend at all on the axioms that A 1 and A 2 have to satisfy as models of T 1 and T 2 , respectively. This ultimately means that other axioms, if any is present, do not play an active role in the foundations of the abstract theory so far developed. Rather, they (can) contribute to determining "extrinsic" properties of prenorms (and, later, prenormed structures), i.e., properties complementary to the inherent ones stemming directly from their very definition.

Remark 13. Clearly, for a K-prenorm A 1 → A 2 to exist, it is necessary that any n-ary function symbol of σ 1 has a corresponding n-ary function symbol in σ 2 , though it is not necessary that σ 1 is smaller than σ 2 (in the sense that the former contains less symbols than the latter).

Remark 14. If R 1 = ∅, the pair (α, ϕ) is a K-prenorm A 1 → A 2 if and only if α = (σ 1 , σ 2 , ∅) and ϕ ∈ hom K (A 1 , A 2 )
is monotonic in the sense of condition (ii) of Definition 3. Lastly, for A 1 an algebraic K-model of T 1 , (α, ϕ) is a K-prenorm A 1 → A 2 if and only if it satisfies condition (i) in the aforementioned definition (since the other, in this case, is automatically fulfilled).

Remark 15. Suppose that (α, ϕ) is a K-prenorm A 1 → A 2 and pick an index r ∈ R 1 , if any exists, such that ς r,1 is (interpreted as) a nullary operation of A 1 . It then follows from Definition 3 that (ϕ(c r,1 ), c α(r),2 ) ∈ ̺ α(r),2 , where c r,1 := ς r,1

• K id K (A 0 1 ) is a distinguished point of A 1 and c r,2 := ς α(r),2 • K id K (A 0 2 ) a distinguished point of A 2 .
In particular, if ̺ α(r),2 is antisymmetric, thus a partial order on A 2 , and c α(r),2 is the least element of (A 2 , ̺ α(r),2 ), in the sense that (c α(r),2 , a) ∈ ̺ α(r),2 for every a ∈ A 2 , this implies ϕ(c r,1 ) = c α(r),2 . On account of the worked examples examined in Section 4, such a result represents a minor but attractive byproduct of the framework set up in this work. In the ultimate analysis, it shows that there is no need to assume, say, that a group norm or a ring valuation, as defined in the traditional setting by taking them to be valued in R + 0 (cf. Examples e.3 and e.5 in Section 4), preserve the additive identities, for this is nothing but a consequence of the inherent properties of subnorms. Definition 4. Assume that A 2 is pivotal and denote its pivot by ≤. Take ς 2 to be a nullary function symbol in σ 2 (if any exists) and Φ = (α, ϕ) a K-prenorm

A 1 → A 2 . We say that Φ is (i) upward [resp. downward] semidefinite with respect to ς 2 if (ς 2 • K ϕ 0 )(a) ≤ ϕ(a) [resp. ϕ(a) ≤ (ς 2 • K ϕ 0 )(a)] for every a ∈ A 1 ; (ii) upward [resp. downward] definite (with respect to ς 2 ) if it is upward [resp. downward] semidefinite and (ς 2 • K ϕ 0 )(a) = ϕ(a) for all a ∈ A 1 \ ς1∈α -1 (ς2) (ς 1 • K id K (A 0 1
)), which is equivalently expressed by writing that (ς

2 • K ϕ 0 )(a) ϕ(a) [resp. ϕ(a) (ς 2 • K ϕ 0 )(a)] for every a ∈ A 1 \ ς1∈α -1 (ς2) (ς 1 • K id K (A 0 1 )
). (iii) indefinite (with respect to ς 2 ) if it is neither upward nor downward semidefinite. On another hand, we say that Φ is trivial if ϕ(a) = ϕ(b) for all a, b ∈ A 1 .

Upward (semi)definiteness abstracts and generalizes one of the most basic properties of standard norms, to wit, positive (semi)definiteness. More than this, Definition 4 suggests that, at least in principle, (semi)definitiness of norms has nothing really special to do with the additive identities in group-like, ring-like or module-like structures, as one might naively conclude from the classical perspective. Rather, it is an issue related to constants, all constants: Which one of them is more significant than the others strongly depends on the case at hand.

Proposition 2.1. Let T i = (V, σ i , Ξ i ) be a

prealgebraic [resp. subalgebraic] theory and

A i = (A i , σ i , I i ) a K-model of T i (i = 1, 2, 3). Suppose Φ = (α, ϕ) is a K-prenorm [resp. K-subnorm] A 1 → A 2 and Ψ = (β, ψ) a K-prenorm [resp. K-subnorm] A 2 → A 3 . Finally, take γ := β • Sgn α and ϑ := ψ • K ϕ. Then, Θ = (γ, ϑ) is a K-prenorm [resp. -subnorm] A 1 → A 3 .
Proof. Assume σ i = ({(ς r,i , ̺ r,i )} r∈Ri ; ar i ) and, for every r ∈ R i , take A r,i to be the preordered [resp. partially ordered] object (A i , ̺ r,i ) of K. Since Sgn and Pre(K) [resp. Pos(K)] are categories, γ is obviously a signature homomorphism σ 1 → σ 2 and ϑ a monotonic arrow A r,1 → A γ(r),3 for every r ∈ R 1 . Thus, it is left to prove that (ϑ

• K ς r,1 , ς γ(r),3 • K ϑ n ) ∈ ℘(A n 1 , A γ(r),3
) for each n-ary function symbol ς r,1 ∈ σ 1 . For this purpose, pick r ∈ R 1 and set n := ar 1 (ς r,1 ).

In particular, C Ts returns a "forgetful" functor to K in the extreme case where T s is the empty theory in the variables V . One question is, then, to establish under which conditions C Ts admits a left or right adjoint. Nevertheless, this and other properties of Pnr K (T ) and Snr K (T ) appear to to be strongly dependent on the actual characteristics of T , T s and K and will be considered non here but in a separate paper.

Prenormed models over a fixed target

Continuing with the notation of the previous section (unless explicitly overridden), assume henceforth that M 0 = (T 0 , A 0 ) is a fixed "target" in Pnr K (T) and suppose that A 0 is pivotal: Let us denote its pivot by ≤ and set A 0 := (|A 0 |, ≤). A prenormed K-model of T over M 0 is, then, any pair M = (M, Φ) such that M is another object of Pnr K (T) and Φ a K-prenorm M → M 0 : We refer to Φ as an M 0 -valued K-prenorm on M (or a K-prenorm on M with values in M 0 ), and indeed as an M 0 -valued K-subnorm on M (or a K-subnorm on M with values in M 0 ) if M and M 0 are both prenormed K-models of T. Observe that M can be well identified with the morphism Φ : M → M 0 of Pnr K (T).

Given a prenormed K-model

M i = (M i , Φ i ) of T over M 0 (i = 1, 2), with M i = (T i , A i ) and Φ i = (α i , ϕ i ), we define a K-short morphism M 1 → M 2 to be any morphism (β, ψ) : M 1 → M 2 in Pnr K (T) such that β • Sgn α 2 = α 1 and (ϕ 2 • K ψ, ϕ 1 ) ∈ ℘(|A 1 |, A 0 ).
The terminology is prompted by the fact that the latter condition is ultimately equivalent to saying, in a much more familiar notation, that ψ(a) 2 ≤ a 1 for all a ∈ |A 1 |, with

• i := ϕ i . In particular, we write that Ψ is a K-isometry M 1 → M 2 if ψ(a) 2 = a 1 for all a ∈ |A 1 |.
Remark 16. As shown shortly, prenormed K-models of T over M 0 and K-short morphisms thereof give rise to a further category, besides Pnr K (T). In the many-sorted case (discussed in the second part of the present work), this provides a full abstraction of the usual category of left modules over a fixed valuated ring (normed spaces over a fixed valuated field can be viewed as a special case of these), with morphisms given by weakly contractive linear transformations between the underlying vector spaces. This sounds intriguing, for it seems to suggest that, from the perspective of the framework set up in this paper, the "right choice" about the kind of morphisms to be considered in relation to normed structures, based only on abstract nonsense reasoning (and especially regardless of any further considerations relevant to applications), should "naturally" fall on short maps. Simply for the fact that common alternatives available in the "localized" context of normed spaces, such as bounded transformations or continuous functions between the standard topologies induced by the norms on the underlying sets, are ruled out, for they are not even possible, in the setting where we are planting the foundations of the general theory of normed structures.

Lemma 3.1. Let M i = (M i , Φ i ) be a prenormed K-model of T over M 0 (i = 1, 2, 3) and suppose that Ψ and Θ are, respectively, K-short maps M 1 → M 2 and M 2 → M 3 . The composition of Θ with Ψ in Pnr K (T) is then a K-short map M 1 → M 3 . Proof. Let Ψ = (β, ψ) and Θ = (γ, ϑ) and set M i = (T i , A i ), Φ i = (α i , ϕ i ) and • i := ϕ i . It is enough to show that (ϑ • K ψ)(a) 3 ≤ a 1 for all a ∈ |A 1 |.
But this is straightforward since, by hypothesis, ϑ(ψ(a)) 3 ≤ ψ(a) 2 on the one hand, and ψ(a) 2 ≤ a 1 on the other, so that the conclusion follows from the transitivity of ≤.

Define C o as the class of prenormed K-models over M 0 and C h as that of triples (M 1 , M 2 , Ψ) such that M i is a prenormed K-model of T over M 0 and Ψ a K-short morphism M 1 → M 2 . Then, take s and t to be the maps

C h → C o : (M 1 , M 2 , Φ) → M 1 and C h → C o : (M 1 , M 2 , Φ) → M 2
, respectively, and denote by i the function C o → C h sending a prenormed K-model M = (M, Φ) to the triple (M, M, ε), where ε is the identity M → M in Prn K (T). Lastly, let c be the partial function

C h × C h → C h specified as follows: Pick m = (M 1 , M 2 , Ψ) and n = (N 1 , N 2 , Θ) in C h . If M 2 = N 1 , then c(m, n) is not defined.
Otherwise, based on Lemma 3.1, assume Ψ = (β, ψ) and Θ = (γ, ϑ) and set c(m, n) equal to the triple (M 1 , N 2 , Π), where Π := (γ

• Sgn β, ϑ • K ψ).
It is easy to verify that the 6-uple (C o , C h , s, t, i, c) gives a category. We call it the category of prenormed K-models of T over M 0 and refer to its objects as prenormed K-models of T over M 0 . This category will be denoted, in general, by Pnr K (T; M 0 ), and especially by Pnr K (T ; M 0 ) in the case where T = {T, T 0 } for some prealgebraic theory T (possibly equal to T 0 ). When M 0 is a subnormed K-model of T, then the objects in Pnr K (T; M 0 ) that are, indeed, subnormed Kmodels of T over M 0 , form a full subcategory of Pnr K (T; M 0 ). This is written as Snr K (T; M 0 ), or as Snr K (T ; M 0 ) for T = {T, T 0 }, and called the category of subnormed K-models over M 0 .

A detailed study of the properties of these categories is beyond the scope of the present paper and we just restrict ourselves to a couple of considerations. The first is that Snr K (T; M 0 ) is a full subcategory of Pnr K (T; M 0 ) whenever M 0 is a subnormed K-model of T. As for the second, assume T = {T, T 0 } and T s = {T s , T 0 }, where T and T s are prealgebraic [resp. subalgebraic] theories and T s ≤ T . Then, the "forgetful" functor C Ts : Pnr K (T) → Pnr K (T s ) [resp. C Ts : Snr K (T) → Snr K (T s )] defined by the end of Section 2 gives rise to another "forgetful" functor

E Ts : Pnr K (T ; M 0 ) → Pnr K (T s ; M 0 ) [resp. E Ts : Snr K (T ; M 0 ) → Snr K (T s ; M 0 )] by sending (i) a prenormed [resp. subnormed] K-model (M, Φ) of T over M 0 to (C Ts (M), C Ts (Φ)), the last being regarded as a prenormed [resp. subnormed] K-model of T s over M 0 ; (ii) a K-short morphism Ψ : (M 1 , Φ 1 ) → (M 2 , Φ 2 ) of Pnr K (T ; M 0 ) [resp. Snr K (T ; M 0 )] to the K-short morphism (C Ts (M 1 ), C Ts (Φ 1 )) → (C Ts (M 2 ), C Ts (Φ 2 )) of Pnr K (T s ; M 0 ) [resp. Snr K (T s ; M 0 )].
In particular, E Ts returns a "forgetful" functor to K in the extreme case where T s is the empty theory in the variables V Remark 11). It is then interesting to ask when E Ts admits adjoints. However, the question, along with other properties of Pnr K (T ; M 0 ) and Snr K (T ; M 0 ), critically depends on the specificity of K, T and T s and will be investigated in future work.

Some worked examples

Unless explicitly overridden, the notation throughout is based on that of the previous section.

Here, we show how the framework developed so far succeeds to capture all the essential features of the notion itself of norm as this is intended in the classical approach to the theory of normed groups, valuated rings and similar one-sorted structures (modules and vector spaces will be discussed in Part II). In each of the examples examined, it is K = Set. Thus, we omit any further reference to K and use, e.g., "model" in place of "K-model", "prenorm" [resp. "subnorm"] instead of "K-prenorm" [resp. "K-subnorm"], Pnr(•) for Pnr K (•), and so on. We focus on a family T consisting of two theories T = (σ, Ξ) and T 0 = (σ 0 , Ξ 0 ), possibly equal to each other. That said, we take A = (A, σ, I) to be an algebraic model of T and A 0 = (A 0 , σ 0 , I 0 ) a pivotal prealgebraic [resp. subalgebraic] model of T 0 (keep Remark 14 in mind). We set M := (T, A) and M 0 := (T 0 , A 0 ) and denote the pivot of A 0 by ≤. Furthermore, in the light of Remark 13, we assume that σ is a subsignature of σ 0 and concentrate only on M 0 -valued prenorms [resp. subnorms] on M of the form Φ = (α, • ) such that α is the canonical injection σ → σ 0 , hence identifying Φ with • by a convenient abuse of notation. Lastly, for T s a subtheory of T , we use C Ts for the "forgetful" functor Pnr(T ) → Pnr(T s ) defined by the end of Section 2. Now, we pick a distinguished set of (non-logical) function symbols, Σ f = {+, ⋆, u, 0, 1}, and a distinguished set of (non-logical) relation symbols, Σ r = {≤ + , ≤ ⋆ , ≤ u , ≤ 0 , ≤ 1 }, and introduce a "reference signature" σ ref = (Σ f , Σ r , ar ref ), where ar ref is defined in such a way that +, ⋆ and every member of Σ r are binary, u is unary, and 0 and 1 are nullary. Then, as is usual, we call (s.1) σ sgrp := (+, ≤ + ; ar sgrp ) the signature of semigroups; (s.2) σ mon := (+, ≤ + ; 0, ≤ 0 ; ar mon ) the signature of monoids; (s.3) σ grp := (+, ≤ + ; u, ≤ u ; 0, ≤ 0 ; ar grp ) the signature of groups; (s.4) σ rg := (+, ≤ + ; ⋆, ≤ ⋆ ; 0, ≤ 0 ; ar rg ) the signature of semirings; (s.5) σ rig := (+, ≤ + ; ⋆, ≤ ⋆ ; 0, ≤ 0 ; 1, ≤ 1 ; ar rig ) the of unital semirings; (s.6) σ rng := (+, ≤ + ; ⋆, ≤ ⋆ ; u, ≤ u ; 0, ≤ 0 ; ar rng ) the signature of rings; (s.7) σ ring := (+, ≤ + ; ⋆, ≤ ⋆ ; u, ≤ u ; 0, ≤ 0 ; 1, ≤ 1 ; ar rng ) the signature of unital rings. Here, ar sgrp , ar mon , etc are the appropriate restrictions of ar ref to {+, ≤ + }, {+, ≤ + , 0, ≤ 0 }, etc. We then say that a subnorm • : M → M 0 , if any exists, is an M 0 -valued semigroup [resp. group] subnorm (on M) if T is the smallest subalgebraic theory of type σ sgrp [resp. σ grp ], an M 0 -valued monoid subnorm if T is the smallest subalgebraic theory of type σ mon , an M 0 -valued semiring [resp. ring] subnorm if T is the smallest subalgebraic theory of type σ rg [resp. σ rng ], and an M 0 -valued subnorm of unital semirings [resp. unital rings] if T is the smallest subalgebraic theory of type σ rig [resp. σ ring ] (cf. Remark 8). Thus, a subnorm • : M → M 0 is (e.1) an M 0 -valued semigroup subnorm (on M) if and only if

a + b ≤ + a + b for all a, b ∈ A. (9)
It is common that A is a model of the subalgebraic theory T sgrp of semigroups, which is the smallest subalgebraic theory of signature σ sgrp including the axiom of associativity (see Remark 7) for the symbol +. (e.2) an M 0 -valued monoid subnorm if and only if C ♯sTsgrp ( • ) is an M 0 -valued semigroup subnorm on C ♯sTsgrp (M) and 0 ≤ 0 0 (see Remark 8). Typically, A models the subalgebraic theory T mon of monoids, to wit, the smallest subalgebraic extension of T sgrp comprising the axiom of neutrality (see Remark 7) for the symbol 0. When this happens, motivated by the "classical theory", we refer to an M 0 -valued monoid subnorm • (on M) which is upward semidefinite [resp. definite] with respect to 0 as an M 0 -valued monoid seminorm [resp. norm], and call M a seminormed [resp. normed] monoid over M 0 . One usually takes M 0 to be the additive monoid of the non-negative real numbers with its standard (order and algebraic) structure: then, Remark 15 implies 0 = 0. (e.3) an M 0 -valued group subnorm if and only if

u(a) ≤ u u( a ) for all a ∈ A (10)
and C ♯sTmon ( • ) is an M 0 -valued monoid subnorm on C ♯sTmon (M). Commonly, A is a model of the subalgebraic theory T grp of groups, the smallest subalgebraic extension of T mon containing the axiom of inverses (see Remark 7) for the triple (+, u, 0). In these cases, u is usually represented by the symbol -, so the above Equation ( 10) reads as:

a ≤ u u( a ) for all a ∈ A [START_REF] Lowen | Approach spaces. The missing link in the topology-uniformity-metric triad[END_REF] and indeed as:a ≤a for all a ∈ A if T grp ≤ T 0 . Therefore, one concludes that group subnorms are, in some sense, "naturally negative" as far as we look at them as morphisms between structures of the very same type, i.e., groups. While intriguing, this is not completely satisfactory, for the relevant case of standard (positive definite) group norms [2, p. 5] is not covered. However, similar structures can be brought within the scope of our framework in the light of one trivial consideration: That the target of a standard group norm is taken to be R + 0 , which is everything but a group. With this in mind, the most obvious workaround is to assume that T 0 is not a supertheory of T , but instead the smallest subalgebraic σ grp -theory. Then, u can be interpreted as the identity map on A 0 and Equation [START_REF] Lowen | Approach spaces. The missing link in the topology-uniformity-metric triad[END_REF] becomes:a ≤ u a for all a ∈ A. If ≤ is a partial order, it follows from here that • is necessarily symmetric. This is another byproduct of our approach. It suggests that "asymmetric group norms" (cf. [2, Remark 2]) do not really exist as such: They can, e.g., as monoid norms but not as group norms, which is absolutely reasonable if we think of the fact that an "asymmetric group norm" is ultimately defined without any specific requirement about inverses. Starting from these considerations, we then refer to an M 0 -valued group subnorm • (on M), which is upward semidefinite [resp. definite] with respect to 0, as an M 0 -valued group seminorm [resp. norm] (on M), and call M a seminormed [resp. normed] group over M 0 . (e.4) an M 0 -valued semiring subnorm if and only if

a ⋆ b ≤ ⋆ a ⋆ b for all a, b ∈ A (12)
and C ♯sTmon ( • ) is an M 0 -valued monoid subnorm on C ♯sTmon (M). Note how this suggests that norms on ring-like structures are "inherently submultiplicative": "Multiplicativeness" is covered by assuming that ≤ ⋆ is (interpreted as) the equality relation on A 0 , and the same applies to different operations, to the extent that, from an abstract point of view, there is no apparent reason to focus on the one rather than the others. Motivated by the terminology of the theory of valuated rings, we then refer to an M 0valued semiring subnorm • (on M), which is upward semidefinite [resp. definite] with respect to 0 and "multiplicative" with respect to ⋆, as an M 0 -valued semiring semivaluation [resp. valuation] (on M), and call M a semivaluated [resp. valuated] semiring over M 0 . In most applications, A and A 0 will be models of the subalgebraic theory T rg of semirings, i.e., the smallest subalgebraic extension of T mon containing the axioms of left and right distributiveness of ⋆ over + and the axiom of associativity for ⋆. E.g., this is the case with the semiring of non-negative real numbers (with the usual structure inherited from the real field). (e.5) an M 0 -valued ring subnorm if and only if C ♯sTgrp ( • ) is an M 0 -valued group subnorm on C ♯sTgrp (M) and C ♯sTrg ( • ) is an M 0 -valued group semiring subnorm on C ♯sTrg (M). Thus, all the considerations previously made on group and semiring subnorms also apply to ring subnorms. In particular, we refer to an M 0 -valued ring subnorm • (on M), which is upward semidefinite [resp. definite] with respect to 0 and "multiplicative" with respect to ⋆, as an M 0 -valued ring semivaluation [resp. valuation] (on M), and then call M a semivaluated [resp. valuated] ring over M 0 . In common cases, A is a model of the subalgebraic theory T rng of rings, i.e., the smallest subalgebraic theory containing both the axioms of T grp and those of T rg . (e.6) an M 0 -valued subnorm of unital semirings if and only if C ♯sTrg ( • ) is an M 0 -valued semiring subnorm on C ♯sTrg (M) and 1 ≤ 1 1. The same considerations previously made on the symbol 0 in the case of group subnorms apply to 1. Furthermore, mimicking the case of semiring subnorms, we refer to an M 0 -valued subnorm • of unital semirings (on M), which is upward semidefinite [resp. definite] with respect to 0 and "multiplicative" with respect to ⋆, as an M 0 -valued semivaluation [resp. valuation] of unital semirings (on M), and then call M a semivaluated [resp. valuated] unital semiring over M 0 . If

• is a M 0 -valued semivaluation of unital semirings (on M) for which 1 is a unit in M 0 , i.e., an invertible element with respect to ⋆, and ≤ ⋆ is compatible with ⋆, in the sense that

a 1 ⋆ a 2 ≤ ⋆ b 1 ⋆ b 2 for a i , b i ∈ A and a i ≤ ⋆ b i , then 1 = 1 ⋆ 1 ≤ ⋆ 1 ⋆ 1 implies 1 = 1
, and hence 1 = 1 (one more unexpected outcome of our approach). In relevant applications, A will model the subalgebraic theory T rig of unital semirings, i.e., the smallest subalgebraic extension of T rg including the axiom of neutrality for 1, as for the non-negative real numbers with their usual order and algebraic structure. (e.7) an M 0 -valued subnorm of unital rings if and only if C ♯sTrig ( • ) is an M 0 -valued subnorm of unital semirings on C ♯sTrig (M) and C ♯sTgrp ( • ) is an M 0 -valued group subnorm on C ♯sTgrp (M). The very same considerations previously made in the case of group subnorms and subnorms of unital semirings apply to ring subnorms. We refer to an M 0 -valued subnorm • of unital rings (on M), which is upward semidefinite [resp. definite] with respect to 0 and "multiplicative" with respect to ⋆, as an M 0 -valued semivaluation [resp. valuation] of unital rings (on M), and then call M a semivaluated [resp. valuated] unital ring over M 0 . Typically, A models the subalgebraic theory T ring of unital rings, the smallest subalgebraic extension of T rig including the axiom of inverses for the triple (+, u, 0). M 0 -valued semigroup subnorms, monoid subnorms, etc are defined and characterized in the very same way, by replacing "subalgebraic" with "prealgebraic" and ♯ s with ♯ p in all of their occurrences in the above discussion. Furthermore, most of the considerations made in the subalgebraic case still hold in the prealgebraic one, except for those based on Remark 15.

Field valuations and norms of vector spaces over a fixed valuated field, together with variants thereof, will be discussed in Part II as special instances of many-sorted subnormed structures.

Appendix A. A resumé of the very basics of category theory

Following [START_REF]nLab[END_REF], we define a category as a 6-uple (C o , C h , s, t, i, c), where C o and C h are classes, the former referred to as the collection of objects, the latter as the collection of morphisms or arrows; s and t are functions C h → C o which assign, to every arrow, its source and target; i is a further function C o → C h sending each object A to a distinguished morphism, called the (local) identity on A; c is a partial operation C h × C h → C h (called composition) whose domain is the class of all pairs ((f, g 1 ), (g 2 , h)) ∈ C h × C h with g 1 = g 2 ; and all is accompanied by the usual axioms that s, t, id and • are required to satisfy, to wit,

(i) s(c(f, g)) = s(f ) and t(c(f, g)) = t(g) for every (f, g) ∈ dom(c). (ii) s(i(A)) = t(i(A)) = A. (iii) c(f, c(g, h)) = c(c(f, g), h) for all f, g, h ∈ C h such that (f, g), (g, h) ∈ dom(c). (iv) c(f, i(t(f ))) = c(i(s(f )), f ) = f for each f ∈ C h .
Property (iii) is spelled by saying that c is associative. If C is a category, one denotes obj(C) the class of its objects and hom(C) the one of its arrows. We write src C and trg C for the functions mapping a morphism to its source and target, respectively, and id C for that sending an object A to the identity on A. Let C be a category and {A i } i∈I an indexed family of objects of C. We write C {A i } i∈I , whenever it exists, for the C-product of the A i 's, i.e., for the class of all pairs (P, {π i } i∈I ) such that P ∈ obj(C), π i ∈ hom C (P, A i ) for each i ∈ I and the following universal property is satisfied: if (Q, {ω i } i∈I ) is any other pair with Q ∈ obj(C) and ω i ∈ hom C (Q, A i ), there exists a unique morphism u :

Q → P such that ω i = π i • C u for all i ∈ I. The universal arrow u is denoted by C (ω i , π i ) i∈I , or equivalently by (ω 1 , π 1 ) × C (ω 2 , π 2 ) × C • • • × C (ω n , π n ) if I
is finite and 0 = n := |I|, and one refers to P as a product object of the A i 's and to the π i 's as the canonical projections from P (cf. [1, Section 2.4]).

Remark 18. One same product object cannot have two different classes of projections associated with it in the product of the A i 's, when this exists. Therefore, if P is a product object of an indexed collection {A i } i∈I of objects of C and {π i } i∈I the family of the corresponding canonical projections, one can write P ∈ C {A i } i∈I to mean that (P, {π i } i∈I ) ∈ C {A i } i∈I if there is no need to make an explicit reference to the π i 's.

Remark 19. If there is no likelihood of confusion, any explicit reference to C is dropped in these and other similar notations and one writes, e.g., • in place of • C , ∼ = for ∼ =C, and so on. In particular, if {A i } i∈I is an indexed subfamily of obj(C) with a product and C is clear from the context, one writes i∈I A i instead of C {A i } i∈I and A I if the A i 's are all equal to one same object A. Moreover, i∈I A i is written as A 1 × A 2 × • • • × A n if I is finite and 0 = n := |I|, and indeed as A n if the A i 's are all equal to one same object A.

Remark 20. If C has a terminal object ⊤ [1, Section 2.2] and I is a set, then ⊤ I exists and ⊤ ∈ ⊤ I , i.e., ⊤ is isomorphic to all of its own powers. Definition 5. Let {A i } i∈I be an indexed subfamily of obj(C). Provided that the A i 's have a product, a relation on {A i } i∈I (in C) is then any monomorphism [1, Section 2.1] ̺ ∈ hom(R, P ) for which P ∈ C {A i } i∈I . This is called an n-ary relation if I is finite and n := |I|, and indeed an n-ary relation on A if the A i 's are all equal to one same object A.

Remark 21. If C has a terminal object ⊤ and A ∈ obj(C), one refers to an arrow a ∈ hom(C) with src(a) ∼ = ⊤ as a point of A [1, Section 2.3] and defines a point of C as any morphism a ∈ hom(C) such that src(a) ∼ = ⊤: a point is, a fortiori, a monomorphism. Subsequently, one writes a ∈ A to mean that a is a point of A and f (a) in place of f • a for f ∈ hom(C) with src(f ) = A. Taking this in mind, pick a family {a i } i∈I of points of C with a i ∈ A i and let ϕ ∈ hom(P ), where (P, {π i } i∈I ) ∈ i∈I A i . Then, based on Remark 20, we use (i) (a 1 , a 2 , . . . , a n ; ⊤) for the morphism n i=1 (a i • u i , π i ); (ii) ϕ(a 1 , a 2 , . . . , a n ; ⊤) in place of ϕ • (a 1 , a 2 , . . . , a n ; ⊤); (iii) (a 1 , a 2 , . . . , a n ; ⊤) ∈ ̺ if there is a monic µ : ⊤ → R such that (a 1 , a 2 , . . . , a n ; ⊤) = ̺ • µ, where u i is the unique arrow ⊤ → src(a i ) of C (indeed an isomorphism). The notation is suggestive of the fact that a point can be regarded as a generalization of the set-theoretic notion of element and is unambiguous since, as observed before, the π i 's are uniquely determined by the datum of P , and P is clearly understood as the target of ϕ and ̺. Moreover, since the condition underlying the latter of these notations does not really depend on the actual choice of ⊤ as a representative of the isomorphism class of the terminal objects of C, we can further simplify our notational system by writing (a 1 , a 2 , . . . , a n ) ∈ ̺ for (a 1 , a 2 , . . . , a n ; ⊤) ∈ ̺. Definition 6. Assume C is a category with binary products and a terminal object ⊤, and hence with all finite products [1, Section 2.6]. Pick A ∈ obj(C) and let ̺ : R → P be a binary relation on A, where P ∈ A 2 . We say that ̺ is reflexive if (a, a) ∈ ̺ for every a ∈ A; antisymmetric if, for all parallel a, b ∈ A, it holds that (a, b), (b, a) ∈ ̺ only if a = b; transitive if, for all a, b, c ∈ A, one has that (a, c) ∈ ̺ whenever (a, b), (b, c) ∈ ̺. Then, ̺ is called a preorder (on the object A) if it is reflexive and transitive, and a partial order if it is an antisymmetric preorder: In the former case, we refer to (A, ̺) as a preordered object of C; in the latter, as a partially ordered object, or pod. These definitions are different from those analogously given for internal relations, especially in reference to the notion of congruence [START_REF] Nlab | Congruence[END_REF]. Yet, they are more than suitable for our purposes.

Thus, we have fixed the bulk of the terminology and notation used here and in future work to deal with categories. However, before concluding with this appendix, we add the definition of four basic categories that will be considered at several points (cf. The local identities and the maps of source and target are specified in the most obvious way. This completes our brief introduction to the categorical language. For anything else not explicitly mentioned here, the interested reader can refer to [START_REF] Awodey | Category Theory[END_REF].

  For A, B ∈ obj(C), we adopt the notation (f : A → B) ∈ C to mean that f ∈ hom(C), src C (f ) = A and trg C (f ) = B. This simplifies to the usual f : A → B, or one says that f is an arrow A → B, when C is clear from the context. We use • C for the composition law of C and hom C (A, B) for the collection of arrows (f : A → B) ∈ C, or simply hom C (A) when A = B. Lastly, for (f, g) ∈ dom(• C ), we write g • C f in place of • C (f, g) and refer to g • C f , as is customary, as the composition of g with f (cf. [1, Section 1.3]). Remark 17. We denote ∼ =C the equivalence on obj(C) defined by: A ∼ =C B for A, B ∈ obj(C) if and only if there is an isomorphism u : A → B. Then, for A ∈ obj(C), we indicate by iso C (A) the equivalence class of A in the quotient of obj(C) by ∼ =C and refer to it as the isomorphism class of A in C. If B, C ∈ iso C (A), one says that B and C are isomorphic (cf. [1, Section 1.5]).

[ 1 ,

 1 Section 1.4]):(c.1) Rel, the category having sets as objects and all triples R of type (X, Y, R) as morphisms, where X and Y are sets and R is a subset of the Cartesian product X × Y . When this does not lead to confusion, we identify R with R. The composition of two relations R : X → Y and S : Y → Z is defined by the triple (X, Z, T ), where T ⊆ X × Z and (x, z) ∈ T for x ∈ X and z ∈ Z if and only if (x, y) ∈ R and (y, z) ∈ S for some y ∈ Y . (c.2) Set, the subcategory of Rel whose morphisms are functions. (c.3) Pre(C), the category having as objects the preordered objects of a given category C with finite products and as morphisms all triples f of type (P, Q, f ), where P = (P, ̺) and Q = (Q, η) are preordered objects of C and f ∈ hom C (P, Q) is such that (f (x), f (y)) ∈ η for all x, y ∈ hom C (⊤, P ) with (x, y) ∈ ̺, where ⊤ is any terminal object of C. We call f a monotonic arrow P → Q and use f as a shorthand of f when there is no danger of confusion. The composition of two morphisms (P, Q, f ) and (Q, R, g) is (P, R, g • C f ). (c.4) Pos(C), the full subcategory of Pre(C) whose objects are pods of C.

Acknowledgements

I am thankful to Prof. Gilles Godefroy (IMJ, UPMC) for the afternoons spent with me on the board, and to Prof. Yvon Maday (LJLL, UPMC) and the Research European Agency for the opportunity of being in Paris and drawing inspiration and continuous stimuli for my research from the environment. Moreover, I am profoundly grateful to my former PhD advisor, Giovanni Angiulli (DIMET, Università di Reggio Calabria), and Andrea Gagna (Università di Torino) for many fruitful discussions and their precious friendship, and also to Fosco Loregian (Università di Padova), Giorgio Mossa (Università di Pisa), Mauro Porta (Università di Milano), and Giovanni Viglietta (Università di Pisa) for suggestions and useful hints.

The author is funded from the European Community's 7th Framework Programme (FP7/2007-2013) under Grant Agreement No. 276487 (project ApProCEM).

Since, by hypothesis, (ϕ

) and ψ is a monotonic morphism A α(r),2 → A γ(r),3 , it follows from the second point of Lemma 2.1 that

By the associativity of • K , this in turn is equivalent to

On the other hand, again by hypothesis, (ψ

). Hence, the first point of Lemma 2.1 implies that

Using once more the associativity of • K , along with the fact that

this equation can be rearranged in the form:

In the light of Equation ( 6), it then follows that (ϑ

) is a preorder on hom K (A n 1 , A 3 ). And this ultimately proves, by the arbitrariness

With this in hand, suppose T is a given collection of prealgebraic [resp. subalgebraic] theories in the variables V . We take C o to be the class of all pairs (T, A) for which T ∈ T and A |= T , and

We let s and t be, each in turn, the maps It is then routine to check that (C o , C h , s, t, i, c) is a category. We call it the category of prealgebraic [resp. subalgebraic] K-models of T. It will be denoted, in general, by Pnr K (T) [resp. Snr K (T)], and especially written as Pnr K (T ) [resp. Snr K (T )] in the case where T consists of a unique prealgebraic [resp. subalgebraic] theory T (in the variables V ). In the latter occurrence, whenever T is implied by the context, we use A in place of (T, A) to mean an object of Pnr K (T ) [resp. Snr K (T )]. A thorough investigation of the properties of these categories is behind the scope of the present paper: it will, in fact, be the subject of a subsequent article. For the moment, we restrict ourselves to a few trivial remarks and observations. First, it is clear that Snr K (T) is contained in Pnr K (T) as a full subcategory, so that we can partially reduce the study of the former to the study of the latter. Second, suppose T = (V, σ, Ξ) is a prealgebraic [resp. subalgebraic] theory and T s = (V, σ s , Ξ s ) a prealgebraic [resp. subalgebraic] subtheory of T . Then, there exists an obvious "forgetful" functor C Ts : Pnr K (T ) → Pnr K (T s ) [resp. C Ts : Snr K (T ) → Snr K (T s )] defined by mapping (i) a prealgebraic [resp. subalgebraic] K-model M = (T, A) of T , with A = (A, σ, I), to the pair (T s , A s ), where A s := (A, σ s , I |σs ); (ii) a morphism (α, ϕ) : M 1 → M 2 of Pnr K (T ) [resp. Snr K (T )] to the K-prenorm [resp.

K-subnorm] (α s , ϕ) : C Ts (M 1 ) → C Ts (M 2 ), where α s is the restriction of α to σ s .