## Definition

- A
*semilattice*is a commutative semigroup - A
*(meet) semilattice*is a partially ordered set ## Basic Results

- Every semilattice is a meetsemilattice (Interactive PCP), (complete PCProof)
- Every meetsemilattice is a semilattice (Interactive PCP), (complete PCProof)
- By duality, every semilattice is a joinsemilattice and vice versa.

## Examples

- Any lattice
- Some semilattices that aren't lattices:
- this minimal 3-element example:
* | 0 1 2 --------- 0 | 0 0 0 1 | 0 1 0 2 | 0 0 2

- this minimal 3-element example:

## Representation

- Any semilattice is isomorphic to a semilattice of subsets of some set, under the operation of union (or intersection). In fact a meetsemilattice is isomorphic to it's lattice of principal ideals, under intersection.
## Decision problems

**Equational Theory**: Decidable**Quasi-equational Theory**: Decidable**First order theory**: Undecidable## Spectra and growth

**Finite spectrum**:**Free spectrum**:## Subvarieties

- only the variety of one-element groupoids
## Subclasses

- reducts of lattices
## Expansions

- lattices
- semilattice ordered algebraic structures

Adapted from J.Pedersen's Catalogue of Algebraic Systems