0. Definitions

A relation $\leq$ on a set $P$ is a partial order relation if

(a)	$x \leq x$	(reflexivity)
(b)	$x \leq y$ and $y \leq x$ imply $x = y$	(antisymmetry)
(c)	$x \leq y$ and $y \leq z$ imply $x \leq z$	(transitivity)

$x \geq y$ means $y \leq x$; $x < y$ means $x \leq y$ and $x \neq y$; $x > y$ means $y < x$.

$\langle P, \leq \rangle$ is a partially ordered set (or poset or partly ordered set or ordered set) if $\leq$ is a partial order relation on $P$. (Generally we just say, ``the partially ordered set $P$''.) In the following, $P$ and $Q$ refer to partially ordered sets.

The relation $\leq$ is a total order relation on $P$ if also

(d) for all $x,y$, either $x \leq y$ or $y \leq x$.

In this case, $\langle P, \leq \rangle$ is a chain or totally ordered set or linearly ordered set. (In contrast, if instead no two distinct elements are related, then $P$ is an antichain.)

In $P$, $a$ covers $b$ if $a > b$ and there is no $c$ with $a > c > b$.

The Hasse diagram of a finite partially ordered set $P$ is a diagram indicating the elements of $P$ by circles or dots, connected by lines that indicate the coverings in $P$. (No lines are drawn horizontal; a non-horizontal line from $b$ up to $a$ indicates that $a$ covers $b$.)

A map $f:P\rightarrow Q$ is said to be isotone if $f$ preserves order: $x \leq y \Rightarrow f(x) \leq f(y)$. It is possible, however, for an isotone map to take two unrelated elements to two related elements, or even to the same element.

A map $f:P\rightarrow Q$ is said to be an isomorphism if $f$ is one-to-one and onto and both $f$ and its inverse are isotone. In this case, $P$ and $Q$ are isomorphic.

Note. The best way to show that two partially ordered sets $P,Q$ are isomorphic is to define maps $f:P\rightarrow Q$ and $g:Q\rightarrow P$, show that $f$ and $g$ are isotone, and show that $f$ and $g$ are inverse to each other, in the sense that $g(f(p))=p$ and $f(g(q))=q$ for all $p \in P, q \in Q$. (It is not enough to define $f$ and show that $f$ is isotone, one-to-one, and onto.)

For partially ordered sets $P$, $Q$, the direct product partial order on the set $P \times Q$ is the coordinatewise ordering: $\langle p, q \rangle \leq \langle p', q' \rangle$ $\Leftrightarrow$ $p \leq p'$ and $q \leq q'$.

The Hasse diagram of $P \times Q$ can be drawn as a copy of $Q$ for each element of $P$, with $P$ used as a guide for the placement of the copies and for the coverings between them.

A direct product $P _ 1 \times \dots \times P _ n$ or $\prod _ {i \in I} P _ i$ is defined similarly.