Bounded lattice

Remarks

The two forms agree: $x \vee 0 = x$ for all $x$ says $0 \le x$ for all $x$, and $x \wedge 1 = x$ says $x \le 1$. Then also $x \wedge 0 = 0$ and $x \vee 1 = 1$. Example: the power set of a set $X$, ordered by $\subseteq$, is a bounded lattice with $0 = \emptyset$ and $1 = X$. Every subset $A$ satisfies $\emptyset \subseteq A \subseteq X$. Non-example: $(\Z, \le)$ is a lattice with $x \vee y = \max(x, y)$ and $x \wedge y = \min(x, y)$, but it has no greatest element. Every finite nonempty lattice $L$ is bounded, with $1 = \bigvee L$ and $0 = \bigwedge L$, the join and the meet of all its elements.