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A lower semilattice is a partially ordered set S in which each pair of elements has a greatest lower bound.
A upper semilattice is a partially ordered set S in which each pair of elements has a least upper bound.
Note that it is not normally necessary to distinguish lower from upper semilattices, because one may be converted to the other by reversing the partial order. It is normal practise to refer to either structure as a semilattice and it should be clear from the context whether greatest lower bounds or least upper bounds exist.
Alternatively, a semilattice can be considered to be a commutative band, that is a semigroup which is commutative, and in which every element is idempotent. In this context, semilattices are important elements of semigroup theory and play a key role in the structure theory of commutative semigroups.
A partially ordered set which is both a lower semilattice and an upper semilattice is a lattice.
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"semilattice" is owned by mclase.
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Cross-references: lattice, commutative semigroups, theory, idempotent, semigroup, band, commutative, clear, structure, normal, partial order, necessary, least upper bound, greatest lower bound, partially ordered set
There are 9 references to this entry.
This is version 3 of semilattice, born on 2002-08-19, modified 2002-08-19.
Object id is 3317, canonical name is Semilattice.
Accessed 6587 times total.
Classification:
| AMS MSC: | 20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous) | | | 06A12 (Order, lattices, ordered algebraic structures :: Ordered sets :: Semilattices) |
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Pending Errata and Addenda
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