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if for all a, b, and x:

: If x ≤ a ∨ b then there exist a and b such that a <= a, b <= b and x = a ? b' .

In either case, a' and b' need not be unique.These definitions are justified by the fact that given any lattice L, the following statements are all equivalent:

* L is distributive as a meet-semilattice
* L is distributive as a join-semilattice
* L is a distributive lattice.

Thus any distributive meet-semilattice in which binary joins exist is a distributive lattice. A join-semilattice

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