A upper semilatticePlanetmathPlanetmath 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 orderMathworldPlanetmath. It is normal practise to refer to either structureMathworldPlanetmath 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 commutativePlanetmathPlanetmathPlanetmath band, that is a semigroupPlanetmathPlanetmath which is commutative, and in which every element is idempotentMathworldPlanetmathPlanetmath. 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 latticeMathworldPlanetmath.

Title semilattice
Canonical name Semilattice
Date of creation 2013-03-22 12:57:23
Last modified on 2013-03-22 12:57:23
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 20M99
Classification msc 06A12
Related topic Lattice
Related topic Poset
Related topic Idempotent2
Related topic Join
Related topic Meet
Related topic CompleteSemilattice
Defines lower semilattice
Defines upper semilattice