Definition 7

A preordered set E is said to be right directed (resp. left directed) if every subset of two elements of E is bounded above (resp. bounded below).

In place of "right directed" we shall often use the expression "directed with respect to the relation $\leqslant$", and analogous expressions when the preorder relation is denoted by some other sign. For example, if $\mathfrak{S}$ is a set of subsets of a set A, we say that $\mathfrak{S}$ is directed with respect to the relation $\subset$ (resp. $\supset$) if, for each subset $\{\mathrm{X}, \mathrm{Y}\}$ consisting of two elements of $\mathfrak{S}$, there exists $\mathrm{Z} \in \mathfrak{S}$ such that $\mathrm{X} \subset \mathrm{Z}$ and $\mathrm{Y} \subset \mathrm{Z}$ (resp. $\mathrm{X} \supset \mathrm{Z}$ and $\mathrm{Y} \supset \mathrm{Z}$).

Examples (1) An ordered set which has a greatest element is right directed. \* (2) In a topological space, a fundamental system of neighbourhoods of a point is directed with respect to the relation $\supset$. (3) The set of submodules of finite type of an arbitrary module is directed with respect to the relation $\subset$. \*