A partially-ordered set (poset) $(X,\succceq)$ is a set with a reflexive, transitive, and antisymmetric binary relation~$\succceq$.
$x\in X$ is an upper bound for $A\subset X$ if $x\succceq y$ for all $y\in A$. $x$ is a supremum of $A$ if it is an upper bound for $a$ and there is no upper bound $y$ for $A$ with $x\succ y$. Similarly for lower bounds.
A lattice is a poset $(X,\succceq)$ in which each pair of elements $x,y\in X$ has a supremum $x\vee y\in X$ and an infimum $x\wedge y\in X$.
A lattice is complete if every subset $A$ of $X$ has both a lub and a glb in $X$.
$A\subset X$ is as big as $B\subset X$ in the strong set ordering, $A\sqsupseteq B$ if for all $x\in A$ and $y\in B$, $x\vee y\in A$ and $x\wedge y\in B$.