In mathematics, specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle S,} denoted ⋁ S , {\textstyle \bigvee S,} and similarly, the meet of S {\displaystyle S} is the infimum (greatest lower bound), denoted ⋀ S . {\textstyle \bigwedge S.} In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms. The join/meet of a subset of a totally ordered set is simply the maximal/minimal element of that subset, if such an element exists. If a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is also an (upward) directed set, then its join (if it exists) is called a directed join or directed supremum. Dually, if S {\displaystyle S} is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum.
Definitions
Partial order approach Let A {\displaystyle A} be a set with a partial order ≤ , {\displaystyle \,\leq ,\,} and let x , y ∈ A . {\displaystyle x,y\in A.} An element m {\displaystyle m} of A {\displaystyle A} is called the meet (or greatest lower bound or infimum) of x and y {\displaystyle x{\text{ and }}y} and is denoted by x ∧ y , {\displaystyle x\wedge y,} if the following two conditions are satisfied:
m ≤ x and m ≤ y {\displaystyle m\leq x{\text{ and }}m\leq y} (that is, m {\displaystyle m} is a lower bound of x and y {\displaystyle x{\text{ and }}y} ). For any w ∈ A , {\displaystyle w\in A,} if w ≤ x and w ≤ y , {\displaystyle w\leq x{\text{ and }}w\leq y,} then w ≤ m {\displaystyle w\leq m} (that is, m {\displaystyle m} is greater than or equal to any other lower bound of x and y {\displaystyle x{\text{ and }}y} ). The meet need not exist, either since the pair has no lower bound at all, or since none of the lower bounds is greater than all the others. However, if there is a meet of x and y , {\displaystyle x{\text{ and }}y,} then it is unique, since if both m and m ′ {\displaystyle m{\text{ and }}m^{\prime }} are greatest lower bounds of x and y , {\displaystyle x{\text{ and }}y,} then m ≤ m ′ and m ′ ≤ m , {\displaystyle m\leq m^{\prime }{\text{ and }}m^{\prime }\leq m,} and thus m = m ′ . {\displaystyle m=m^{\prime }.} If not all pairs of elements from A {\displaystyle A} have a meet, then the meet can still be seen as a partial binary operation on A . {\displaystyle A.}
If the meet does exist then it is denoted x ∧ y . {\displaystyle x\wedge y.} If all pairs of elements from A {\displaystyle A} have a meet, then the meet is a binary operation on A , {\displaystyle A,} and it is easy to see that this operation fulfills the following three conditions: For any elements x , y , z ∈ A , {\displaystyle x,y,z\in A,}
x ∧ y = y ∧ x {\displaystyle x\wedge y=y\wedge x} (commutativity),
x ∧ ( y ∧ z ) = ( x ∧ y ) ∧ z {\displaystyle x\wedge (y\wedge z)=(x\wedge y)\wedge z} (associativity), and
x ∧ x = x {\displaystyle x\wedge x=x} (idempotency).
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