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Semilattice
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==Distributive semilattices== Surprisingly, there is a notion of "distributivity" applicable to semilattices, even though distributivity conventionally requires the interaction of two binary operations. This notion requires but a single operation, and generalizes the distributivity condition for lattices. A join-semilattice is '''distributive''' if for all {{math|1=''a'', ''b'',}} and {{math|1=''x''}} with {{math|1=''x'' ≤ ''a'' ∨ ''b''}} there exist {{math|1=''a' '' ≤ ''a''}} and {{math|1=''b' '' ≤ ''b''}} such that {{math|1=''x'' = ''a' '' ∨ ''b' ''.}} Distributive meet-semilattices are defined dually. These definitions are justified by the fact that any distributive join-semilattice in which binary meets exist is a distributive lattice. See the entry [[distributivity (order theory)]]. A join-semilattice is distributive if and only if the lattice of its [[ideal (order theory)|ideals]] (under inclusion) is distributive.
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