When we talk of e.g. the natural numbers equipped with a non-standard order , what does "equipped" mean?

Question

A question for "real" mathematicians who have become better acculturated to math-speak than this philosopher!

If you read a phrase like

... the natural numbers equipped with the evens-before-odds order ...

just what do you understand by equipped?

[I have my suspicions, of course, but I won't prejudice the comments/answers by saying ...!]

Answer 1

I find equipped to be a word that is very evocative of the correct idea: like a worker equipped with a tool, or a phone equipped with a feature, I would refer to $A$ equipped with $B$ if I want to refer to them together as a single object, but with $A$ having a certain precedence.

If I wanted to formalize it, I would say that "$A$ equipped with $B$" means the ordered pair $(A,B)$, but with the caveat that the pair may be referred to as simply "$A$" if desired.

Other common uses of the word in mathematics are equipping sets with operations and topologies.

Answer 2

This means there are two types of structure - here a set and an ordered set - which are of the form $X$ and $(X, \Sigma)$, for some additional thing $\Sigma$ - here an order on $X$.

People simply mean that one adds/attaches $\Sigma$ to $X$, or constructs a new thing $(X,\Sigma)$ in which $X$ is naturally embedded (i.e. there is a surjective structure-of-$X$-preserving function $\pi:(X,\Sigma)\to X$ which naturally extracts $X$). This isn't at all mysterious.

Answer 3

A (partial) order is really a tuple $(X, \prec)$ where $X$ is some set and $\prec$ is a subset of $X \times X$ that satisfies the order axioms: reflexivity, transitivity, antisymmetry.

So you say a set $X$ is equip with an ordering if there is some subset of $X \times X$, denoted $\prec$, such that $(X, \prec)$ is a (partial) ordering.

So the natural numbers $\mathbb{N}$ can be equipped with the usual ordering. This means $\prec$ is $<$, where $<$ is the familiar order.

However, there are other ordering of the set $\mathbb{N}$. These are refered to as nonstandard orderings of $\mathbb{N}$.

You suggested something called the even before odd. Although, I have never heard of such an ordering, this how I would intrepret it: Let $<$ be the familar ordering on $\mathbb{N}$. Let $E$ denote the even natural numbers. Let $O$ denote the odds. Define $\prec$ on $\mathcal{N}$ as follows.

$a \prec b \text{ if and only if } (a \in E \wedge b \in O) \vee (a, b \in E \wedge a < b) \vee (a, b \in O \wedge a < b)$

Informally, you first check if $a$ is even and $b$ is odd. If so, then $a \prec b$. If not, check if $a$ and $b$ are both even (similarly odd) and $a < b$ (in the standard ordering), if so then $a \prec b$.

You can check this is a linear ordering on the set $\mathbb{N}$. Moreover it is a well-ordering. If you are familar with ordinals, you can check this is isomorphic to $\omega + \omega$.