semilattice
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.
| 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 |