How is the order "less than" on Natural number a total order?

Question

First of all, I'm confused between the terms Order and Relation; I know what a relation on a set means, but don't know exactly a what order is. I may have used these terms as if they almost mean the same.

In wikipedia: order theory, it says -

Some orders, like "less-than" on the natural numbers and alphabetical order on words, have a special property: each element can be compared to any other element, i.e. it is smaller (earlier) than, larger (later) than, or identical to. However, many other orders do not. Consider for example the subset order on a collection of sets: though the set of birds and the set of dogs are both subsets of the set of animals, neither the birds nor the dogs constitutes a subset of the other. Those orders like the "subset-of" relation for which there exist incomparable elements are called partial orders; orders for which every pair of elements is comparable are total orders.

What I conclude from this text is that, if a relation can be applied between any two elements of set, then that relation is a total order.

In the first example, they said that in natural numbers, under the relation "less-than", every kind of pairs (not ordered, which means (1, 2) = (2, 1)) is comparable, does that mean that all these pairs fulfill the statement e1 < e2, where e1 and e2 are elements of the given pair? If yes, then how does identical element's pair fulfill this statement?

consider pairs: (1, 2), (4, 1), (5, 5)
1st and 2nd pairs fulfill the statement e1 < e2: 1 < 2 and 1 < 4, but 3rd pair doesn't as 5 < 5 is false.

So, How is the order "less than" on Natural number a total order?

Answer 1

There really are two notions: strict partial order, and non-strict partial order. People usually prefer one to the other, and use "partial order" to refer to that, and use the additional adjective to refer to the remaining one.

Let $X$ be a set. A non-strict partial order on $X$ is a (binary) relation $\preceq$ on $X$ which is:

1. Reflexive: for all $x\in X$, $x\preceq x$;
2. Anti-symmetric: for all $x,y\in X$, if $x\preceq y$ and $y\preceq x$, then $x=y$.
3. Transitive: for all $x,y,z\in X$, if $x\preceq y$ and $y\preceq z$, then $x\preceq z$.

We say that the non-strictly partially ordered set $(X,\preceq)$ is totally ordered (or is "linearly non-strictly ordered") if and only if for all $x,y\in X$, either $x\preceq y$ or $y\preceq x$ (or both, in the case where $x=y$). This is sometimes called the "totality" property.

Now let $X$ be a set. A strict partial order on $X$ is a (binary) relation $\lt$ on $X$ which is:

1. Irreflexive: for all $x\in X$, $x\not\lt x$;
2. Transitive: for all $x,y,z\in X$, if $x\lt y$ and $y\lt z$, then $x\lt z$.

(Sometimes people also add "asymmetry": if $x\lt y$ then $y\not\lt x$; but note that this follows from 1 and 2: if $x\lt y$ and $y\lt x$, then transitivity would yield $x\lt x$, which contradicts irreflexivity).

A strictly partially ordered set $(X,\lt)$ is totally ordered if and only if it satisfies trichotomy:

Trichotomy: for all $x,y\in X$, exactly one of the following three things happen:

1. $x=y$;
2. $x\lt y$;
3. $y\lt x$.

The two notions are equivalent in the following sense:

Theorem. Let $X$ be a set.

1. If $\preceq$ is a non-strict partial order on $X$, then define the binary relation $\lt$ on $X$ by $$x\lt y\iff x\preceq y\text{ and }x\neq y.$$ Then $(X,\lt)$ is a strictly partially ordered set. Moreover, $(X,\preceq)$ is a total order if and only if $(X,\lt)$ is a total order (that is, trichotomic).
2. If $\lt$ is a strict partial order on $X$, then define the binary relation $\preceq$ on $X$ by $$x\preceq y\iff x\lt y\text{ or }x=y.$$ Then $(X,\preceq)$ is a non-strict partially ordered set. Moreoever, $(X,\lt)$ is a total order (that is, trichotomic) if and only if $(X,\preceq)$ is a total order.

It is not hard to establish this theorem. It means, in spirit, that one can begin with either the concept of "non-strict partial order", and use it to make "strict partial order" a derived notion; or begin with the notion of "strict partial order" as your initial concept, and make non-strict partial orders a derived notion.

Historically, number theorists and those that start with the Natural numbers beginning at $1$, took strict partial orders to be the "initial" or "more primitive" notion of order, as the order on $\mathbb{N}$ was defined by $$a\lt b\iff \exists x\in\mathbb{N}\text{ such that }a+x=b.$$ Which is the "strictly less than" notion of the usual order of $\mathbb{N}$. By contrast, those who began with set theory (and defined $\mathbb{N}$ using von Neumann ordinals so that they began at zero) considered non-strict partial orders (the canonical example of which is the subset relation, $A\subseteq B$) to be the more "primitive" notion, with strict orders being defined in terms of the partial orders instead.

That means that when someone says "partially ordered set", you kind of need to know whether they mean "non-strict" or "strict" partial order. This is usually understood from context or from choice of symbols ($\leq$ vs. $\lt$; $\subseteq$ vs. $\subset$, etc). Then "total order" will mean either that for every $x$ and $y$ you have either $x\leq y$ or $y\leq x$, or both; or else that for all $x$ and $y$, you have exactly one of $x\lt y$, $y\lt x$, or $x=y$ (if you are working with strict orders).

Whichever notion you are using (strict order or non-strict order), the Natural numbers with their usual ordering are totally ordered, since they satisfy either totality (for non-strict order); or trichotomy (for strict order).