We call the pair $(P,\le)$ a *partially ordered set* or a *poset*. When the relation is clear from context, we shall usually write $P=(P,\le)$. A poset is *locally finite* if for every pair $x,y \in P$, the set $\{z \in P : x \le z \le y\}$ is finite.

Given a partially ordered set $P=(P,\le)$, there is also the *dual partial order* $\le^{op}$ on $P$, for which $x \le^{op} y$ if and only if $y \le x$. The partially ordered set $P^{op}=(P,\le^{op})$ is called the *dual poset to $P$*.

![A poset $P$, its Alexandroff topology (Definition \[df:alexandroff\]), the dual poset $P^{op}$, and sheaves over each (Definition \[df:sheaf\_poset\]). The gluing axiom appears in Definition \[df:sheaf\].[]{data-label="fig:lattice_sheaves"}](lattice_sheaves){width="4in"}

Given a subset $A \subseteq P$ of a poset, the *infimum of $A$* is the unique greatest element in $P$ less than or equal to each element of $A$ if such an element exists. We will write $\bigwedge A$ for the infimum of $A$ if it exists. Similarly, the *supremum of $A$*, written $\bigvee A$ is the unique least element in $P$ greater than or equal to each element of $A$, if such an element exists. If $A=\{A_1,A_2\}$, we usually write $\bigwedge A = A_1 \wedge 