How to reconcile these three definitions of an order?

Question

I have of late come across three definitions of a (total or partial) order and am not sure how to establish an equivalence among them.

Here's Definition 4.1-1 on page 210 in Introductory Functional Analysis With Applications by Erwine Kreyszig.

A partially ordered set is a set $M$ on which there is defined a partial ordering, that is, a binary relation which is written $\leqq$ and satisfies the conditions

(PO1) $a \leqq a$ for every $a \in M$. (Reflexivity)

(PO2) If $a \leqq b$ and $b \leqq a$, then $a = b$. (Antisymmetry)

(PO3) If $a \leqq b$ and $b \leqq c$, then $a \leqq c$. (Transitivity)

"Partially" emphasizes the fact that $M$ may contain elements for which neither $a \leqq b$ nor $b \leqq a$ holds. Then $a$ and $b$ are called incomparable elements. In contrast, two elements $a$ and $b$ are called comparable elements if they satisfy $a \leqq b$ or $b \leqq a$ (or both).

A totally ordered set or chain is a partially ordered set such that every two elements of the set are comparable. In other words, a chain is a partially ordered set that has no incomparable elements.

Here's the definition of an order relation on page 24 in Topology by James R. Munkres, 2nd edition.

A relation $C$ on a set $A$ is called an order relation (or a simple order, or a linear order) if it has the following properties:

(1) (Comparability) For every $x$ and $y$ in $A$ for which $x \neq y$, either $x C y$ or $y C x$.

(2) (Non-reflexivity) For no $x$ in $A$ does the relation $x C x$ hold.

(3) (Transitivity) If $x C y$ and $y C z$, then $x C z$.

Finally, here's Definition 1.5 on page 3 of Principles of Mathematical Analysis by Walter Rudin, 3rd edition.

Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties:

(i) If $x \in S$ and $y \in S$ then one and only one of the statements $$ x < y, \ \ \ x = y, \ \ \ y < x$$ is true.

(ii) If $x, y, z \in S$, if $x < y$ and $y < z$, then $x < z$.

Now are these three definitions mutually in agreement? If so, how? Or, are there different schools of thought amongst mathematicians on this matter?

Answer 1

The last two are exactly the same.

In fact, you can rewrite $(1)$ as $$\forall x,y,\ (x=y\vee xCy\vee yCx)$$

And $(3)$ applied to $x=z$ becomes $$\forall x,y,\ ((xCy\wedge yCx)\rightarrow xCx)$$

So, since $xCx$ never holds by $(2)$, the statements $[x=y]$, $[xCy]$ and $[yCx]$ are mutually exclusive (i.e., $(i)$).

Of course, both $(1)$ and $(2)$ are special cases of $(i)$.

Those two are not equivalent to the first one: for one thing, $((PO1)\wedge (2))\iff M=\emptyset$.

In fact, in the notation of the first definition (the one I've been taught, actually), a relation such as in the last two is called a strict total order. If it does not satisfy $(1)$, it's called "strict (partial) order" or, to be precise, you define a stirct partial order to be a relation $<$ satisfying:

• $[\forall x,y,z,\ ((x<y\wedge y<z)\longrightarrow x<z)]$

• $[\forall x,\neg x<x]$ (you can exchange this one with $[\forall x,y,\ \neg (x<y\wedge y<x)]$ and get an equivalent definition)

And a strict total order is a strict partial order satisfying $(1)$.

You can check that the following are equivalent:

a) $R$ is a strict partial (total) order on $M$.

b) $R$ satisfies $(2)$ and the relation $S$ defined by $$xSy\stackrel{\text{def}}\iff (x=y\vee xRy)$$ is a partial (total) order on $M$.

c) there exists a partial (total) order $Q$ on $M$ such that $$\forall x,y,\ (xRy\longleftrightarrow (x\ne y\wedge xQy))$$