Editing Semilattice (section)

{{Short description|Partial order with joins}}
{{stack|{{Binary relations}}}}
In [[mathematics]], a '''join-semilattice''' (or '''upper semilattice''') is a [[partially ordered set]] that has a [[join (mathematics)|join]] (a [[least upper bound]]) for any [[nonempty set|nonempty]] [[finite set|finite]] [[subset]]. [[Duality (order theory)|Dually]], a '''meet-semilattice''' (or '''lower semilattice''') is a partially ordered set which has a [[meet (mathematics)|meet]] (or [[greatest lower bound]]) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the [[inverse order]] and vice versa.

Semilattices can also be defined [[algebra|algebraically]]: join and meet are [[associativity|associative]], [[commutativity|commutative]], [[idempotency|idempotent]] [[binary operation]]s, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order. 

A [[lattice (order)|lattice]] is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by corresponding [[absorption law]]s.

{{Algebraic structures |Lattice}}