Example 2

Let E be an ordered set. For each $x \in E$ let $U_x$ be the set of all $y \in E$ such that $y \geqslant x$. The mapping $x \to U_x$ is a strictly decreasing mapping of E into $\mathfrak{P}(E)$ (ordered by inclusion); indeed, the relation $x \leqslant y$ is equivalent to $U_x \supset U_y$.

An injective monotone mapping of an ordered set E into an ordered set F is strictly monotone; the converse is usually not true, because it may happen that $f(x) = f(y)$ when neither of the relations $x \leqslant y$, $x \geqslant y$ is true (cf. no. 12, Proposition 11).

¶ A bijective mapping $f$ of an ordered set E onto an ordered set E' is an isomorphism of E onto E' (no. 3) if and only if both $f$ and its inverse mapping are increasing.

¶ If I is an ordered index set, a family of subsets $(X_\iota)_{\iota \in I}$ of a set E is said to be increasing if $\iota \to X_\iota$ is an increasing mapping of I into $\mathfrak{P}(E)$, ordered by inclusion (in other words, if $\iota \leqslant \varkappa$ implies $X_\iota \subset X_\varkappa$). Similarly we define a decreasing, strictly increasing, or strictly decreasing family of subsets $(X_\iota)_{\iota \in I}$.