A generalization of metric spaces

Question

Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i.e. commutativity, associativity, $a+0 = a$, and $a< b$ iff there is $c>0$ such that $b =a+c$.

Say a pair $(X,d)$ is an “$L$-metric space” if $d : X^2 \to L$ is a function satisfying the metric space axioms (only the codomain is not necessarily $\mathbb R$).

Question 1: Has this or something similar been studied?

Question 2: Say a topological space is “weakly metrizable” when there exists $L$ and $d$ as above inducing the topology. When is a space weakly metrizable?

Answer 1

It follows from your assumptions that for $a<b\in L$ there is a unique $c$ such that $a+c=b$ and that $L$ is a cancellative monoid: $a+c=b+c$ implies $a=b$. Also addition preserves the order. A cancellative commutative monoid embeds in an abelian group. If I'm not mistaken $L$ embeds in an ordered abelian group $\Lambda$ such that the positive elements correspond to the non-zero elements of $L$.

Metric spaces where the codomain of the distance function is an ordered abelian group - that is, $\Lambda$-metric spaces - have been defined as a first step to defining $\Lambda$-trees. Ian Chiswell has written an introduction to this subject.

As for the topology that such a metric gives rise to, note that if $\Lambda$ has a minimum non-zero convex subgroup, then this subgroup is archimedean, and thus the topology is $\mathbb{R}$-metrisable. If there is no such convex subgroup, then the balls of radius $\Lambda_0$ - that is, the sets $B(x,\Lambda_0)=\{y\in X:d(x,y)\in\Lambda_0\}$ - where $\Lambda_0$ ranges through the non-zero convex subgroups, gives a base for the topology which is therefore $0$-dimensional. And in fact replacing $\Lambda$ by the (linearly ordered) set of its convex subgroups and putting $d'(x,y)$ equal to the convex subgroup spanned by $d(x,y)$ gives a generalised ultrametric equivalent to $d$.

Generalised ultrametric spaces have been studied for example by Priess-Crampe and Ribenboim. (Of course if $L\subseteq\mathbb{R}_{\geq 0}$, then these are ultrametric spaces.)

Answer 2

My PhD thesis was on this topic, focusing mostly on a combinatorial approach (rather than topological). So what I write below is directed toward Question 1.

I call a structure $\mathcal{R}=(R,+,\leq,0)$ a distance magma if $(R,\leq,0)$ is a total order with least element $0$, and $+$ is a commutative binary operation on $R$ which preserves the order ($a\leq b\Rightarrow a+c\leq b+c$). I call $\mathcal{R}$ a distance monoid if the operation is associative. The last axiom you mention about "solving inequalities" didn't play a special role. (Also, I use $\mathcal{R}$ instead of $L$ since eventually there will be a first order language.)

My own interest started with the special case of fixing a countable set $R\subseteq\mathbb{R}_{\geq 0}$ such that $0\in R$ and $R$ is closed under $x+_R y:=\sup\{z\in R:x\leq y+z\}$, which determines a distance magma $\mathcal{R}=(R,+_R,\leq,0)$. The first main question was:

Question. When is there a countable, universal and ultrahomogeneous $\mathcal{R}$-metric space (so, in other words, a Urysohn-like space relative to the distance set $R$)?

For example if $R=\mathbb{Q}_{\geq0}$ then then the answer is yes, witnessed by the classical rational Urysohn space. Another example is $R=\{0,1,2\}$ which yields the random graph (where $1$ designates an edge and $2$ designates no edge).

It turned out that this question was answered by Delhomme, LaFlamme, Pouzet, and Sauer. There is a more technical general characterization, but, in the special case above, there is a really nice fact:

Fact. There is an $\mathcal{R}$-Urysohn space if and only if the operation $+_R$ is associative.

The idea is that associativity of $+_R$ is combinatorially capturing the ability to amalgamate three-point $\mathcal{R}$-metric spaces, which leads to amalgamation for the class of all finite $\mathcal{R}$-metric spaces.

The "smallest" example of a set $R$ as above for which there is no $\mathcal{R}$-metric spaces is $\{0,1,2,4\}$.

In any case, it became more natural to work in a more general axiomatic framework. Given a countable distance magma $\mathcal{R}$, there is a natural notion of an $\mathcal{R}$-metric space, and a countable $\mathcal{R}$-Urysohn space exists if and only if $\mathcal{R}$ is a distance monoid. So assume $\mathcal{R}$ is a distance monoid and let $\mathcal{U}_{\mathcal{R}}$ denote the $\mathcal{R}$-Ursyosn space. In other words, $\mathcal{U}_{\mathcal{R}}$ is the unique (up to isometry) countable $\mathcal{R}$-metric space such that every finite $\mathcal{R}$-metric space embeds as a subspace of $\mathcal{U}_{\mathcal{R}}$, and any partial isometry between two finite subspaces of $\mathcal{U}_{\mathcal{R}}$ extends to a (total) isometry $\mathcal{U}_{\mathcal{R}}$.

I consider $\mathcal{U}_{\mathcal{R}}$ as a first-order structure in a relational language $L_{\mathcal{R}}$ containing binary relations $d_r(x,y)$ for $r\in R$, which are interpreted as "$d(x,y)\leq r$". Let $T_{\mathcal{R}}$ be the complete theory of $\mathcal{U}_{\mathcal{R}}$ in this language.

If $\mathcal{R}$ is finite then $T_{\mathcal{R}}$ is $\aleph_0$-categorical, but for infinite distance monoids this fails. For example, if $\mathcal{R}=(\mathbb{Q}_{\geq 0},+,\leq,0)$ (so $\mathcal{U}_{\mathcal{R}}$ is the rational Urysohn space), then there are plenty of non-isolated $2$-types over $\emptyset$, which describe "new distances" from irrationals and infinitesimal cuts. For example, $\{\neg d_0(x,y)\}\cup\{d_r(x,y)\leq r:r>0\}$ is a finitely satisfiable $2$-type, which describes two elements at nonzero infinitesimal distance.

In general, any distance $r\in R$ can be viewed as a quantifier-free $2$-type: $\{\neg d_s(x,y):s<r\}\cup\{d_s(x,y):r\leq s\}$. This produces a canonical embedding of $R$ in the set of quantifier-free $2$-types over $\emptyset$, and the ordering on $R$ extends naturally to this space (in a way very similar to a Dedekind-MacNeille completion). A more interesting fact is that the monoid operation also extends to the (quantifier-free) $2$-types. Specifically, given $2$-types $p,q$, define $p+q$ to be the supremum (which exists) of the set of $2$-types $r$ such that the $3$-type $p(x,y)\cup p(y,z)\cup r(x,z)$ is finitely satisfiable. (So $p+q$ is the largest distance that you can consistently put on a triangle with distances $p$ and $q$.)

So altogether, we have a new distance monoid $\mathcal{R}^*$, built from the quantifier-free $2$-types, which extends $\mathcal{R}$. $\mathcal{R}^*$ behaves a little like a "nonstandard" or saturated extension of $\mathcal{R}$ (but it is not exactly this).

In my thesis I proved a characterization of quantifier elimination for $T_{\mathcal{R}}$ in terms of the behavior of $\mathcal{R}^*$.

Theorem. $T_{\mathcal{R}}$ has QE if and only if for any standard element $r\in R$, the operation $x\mapsto x+r$ is continuous on $\mathcal{R}^*$ (where here continuity is with respect to the order topology on $\mathcal{R}^*$).

So QE does happen in most "nice" situations, for example if $\mathcal{R}$ is finite (in which case $\mathcal{R}^*=\mathcal{R}$), or if $\mathcal{R}=(\mathbb{Q}_{\geq 0},+,\leq,0)$. But it can fail by "poking holes" in nice monoid. An example is $\mathcal{R}=(R,+_R,\leq,0)$ where $R=\mathbb{Q}\cap (\{0\}\cup[2,3)\cup (3,\infty))$.

The QE result was really only the starting point, since I was mainly interested in model-theoretic neostability properties of $T_{\mathcal{R}}$ under the assumption of QE. The rest of the thesis was about characterizing such properties (stability, simplicity, SOP$_n$, forking, elimination of imaginaries, etc.) via algebraic/combinatorial properties of $\mathcal{R}$. It turned out to be quite successful: most model-theoretic properties of $T_{\mathcal{R}}$ are controlled by $\mathcal{R}$.

The model-theoretic results became two papers: arXiv 1502.05002 and arXiv 1504.02427. I wrote another paper (arXiv 1509.04950) about extending partial isometries (i.e., the "Hrushovksi property"), which built on results of Solecki about rational-valued metric spaces. What I did was generalized further by Hubička, Konečný, and Nešetřil arXiv 1902.03855. They also proved the Ramsey property for these classes (arXiv 1710.04690).

Question #2 is definitely interesting, but nothing I did was in that direction since I effectively viewed my spaces as discrete. But there might be something in these two older articles:

1. Alsina & Trillas, On natural metrics, Stochastica 2 (1977).
2. Narens, Field embeddings of generalized metric spaces, Victoria Symposium on Nonstandard Analysis, Springer, Berlin (1974).

There is also some more recent work by Etedadialiabadi, Gao, Le Maître, and Melleray, which I haven't read closely, but looks related.

Answer 3

There has been a lot of work around a related idea:

F. W. Lawvere, Metric spaces, generalized logic, and closed categories,

Rendiconti del seminario matématico e fisico di Milano, 1973 - Springer

The analogy between $dist (a, b)+ dist (b, c)≥ dist (a, c)$ and $hom (A, B)⊗ hom (B, C)→ hom(A, C)$ is rigorously developed to display many general results about metric spaces as consequences of a «generalized pure logic» whose «truth-values» are taken in an arbitrary closed category.

It might be useful to look at this to get an idea of where such a theory can go. You would need to follow up citations. The paper was reprinted in TAC reprints see http://www.tac.mta.ca/tac/reprints/index.html.

Answer 4

For completion's sake and regarding Question 1, let me add a slightly more general concept: scaled spaces.

Let $M$ be a set, let $X$ be a totally ordered set and let $0$ be a symbol such that $0<x$ for all $x\in X$. An $X$-valued scale on $M$ is a map $d:M\times M\to X\cup\{0\}$ such that for all $x,y,z\in X$:

1. $d(x,y)=0\Leftrightarrow x=y$;
2. $d(x,y)=d(y,x)$; and
3. $d(x,z)\leq\max\{d(x,y),d(y,z)\}$.

The space $(M,X,d)$ is called a scaled space (ultrametric space if $X\subset(0,\infty)$).

This concept was studied by H. Ochsenius and W. H. Schikhof and then applied to the study of Banach spaces over fields with an infinite rank valuation. To initiate in this topic I recommend three articles:

1. H. Ochsenius and W. H. Schikhof, “Banach spaces over fields with an infinite rank valuation,” p-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math. 207, 233–293 (Marcel Dekker, 1999).
2. H. Ochsenius and W. H. Schikhof, “Norm Hilbert spaces over Krull valued fields,” Indagat. Math. 17 (1), 65–84 (2006).
3. A. Barria Comicheo, Generalized Open Mapping Theorem for $X$-normed spaces, p-Adic Numbers, Ultrametric Analysis and Applications, vol. 11, (2), 2019, pp. 135--150.

Regarding Question 2, in the context of scaled spaces, it was proved in [1] that a scaled space $(M,X,d)$ is ultrametrizable if and only if $M$ is discrete or there exist $s_1>s_2>\dots$ in $X$ such that $\lim_n s_n=0$.

Answer 5

There are several results in topology about metrics taking values in (the positive cone of) a partially ordered Abelian group $\mathbb{G}=(G, <, +, 0)$. The first occurrence of this spaces I'm aware of is dated 1950 by Sikorski (MR0040643 - R. Sikorski, Remarks on some topological spaces of high power). He literally started a new branch of topology that investigated this kind of objects, called $\omega_\mu$-metrizability.

There are also several results about the topological nature of spaces with metrics taking values in other kind of structures. The wider framework I'm aware of has been studied by Reichel in 1977 (MR0458373 - H. C. Reichel, Some results on distance functions). I'll resume here some of its results:

He consider totally ordered semigroups with minimum $(S,<, 0, +)$, i.e. structures such that $(S,<, 0)$ is a total order with minimum $0$, $(S, +)$ is an Abelian semigroup and $<$ is translation invariant, i.e. $a<b$ implies $a+c<b+c$ for every $c$ (I believe you also wanted to assume this kind of invariance?).

We say that a $S$ is continuous if whenever a sequence $\langle x_i\mid i<\gamma\rangle$ is coinitial in $(S\setminus\{0\},<)$, then $\langle x_i+x_i\mid i<\gamma\rangle$ is coinitial as well in $(S\setminus\{0\},<)$. Notice that every structure of the form you described is a continuous totally ordered semigroup with minimum in this sense.

Let us call $S$-metrizable a space that has a metric in the structure $S$. It turns out that the behavior of such class of spaces is influenced mostly by the coinitiality of $(S, <)$, i.e. the size of the smallest sequence of elements of $S\setminus \{0\}$ converging to $0$. We call such cardinal number the degree of $S$.

Reichel proved the following:

Theorem 2, countable (Reichel): For a given topological space $(X,\tau)$, the following are equivalent:

1. $X$ is $S$-metrizable for some continuous totally ordered semigroup $S$ of degree $\omega$.
2. $X$ is metrizable (with standard metric over real numbers).

(He also characterized spaces that are metrizable over non-continuous totally ordered semigroup $S$ (Theorem 1 of the paper cited above)).

When the degree of $S$ is uncountable, the situation is easier.

Theorem 2, uncountable (Reichel): For a given topological space $(X,\tau)$, the following are equivalent:

1. $X$ is $S$-metrizable for some continuous totally ordered semigroup $S$ of degree $\kappa>\omega$.
2. $X$ is $S$-metrizable for $S$ the set of positive elements of some totally ordered Abelian group of degree $\kappa>\omega$.

Combining this with other results in literature, in second case one get the following:

Theorem: For a given topological space $(X,\tau)$, the following are equivalent:

1. $X$ is $S$-metrizable for some continuous totally ordered semigroup $S$ of degree $\kappa>\omega$.
2. $X$ is $S$-metrizable for every continuous totally ordered semigroup $S$ of degree $\kappa>\omega$.
3. $X$ is $S$-metrizable for $S$ the set of positive elements of some totally ordered Abelian group of degree $\kappa>\omega$.
4. $X$ is ultrametrizable over some total order of coinitiality $\kappa>\omega$ (i.e. the $+$ operation of $S$ coincide with the maximum between elements).
5. $X$ is a subset of $\lambda^\kappa$ with bounded topology for some $\lambda$.
6. ...(there are several other topological characterizations).