## Definition

- A
*commutative semigroup*is a semigroup ## Basic Results

## Examples

- Any semilattice
- Any abelian group
- Any constant semigroup
- A commutative semigroup that isn't one of the above:
- this minimal 3-element example:
* | 0 1 2 --------- 0 | 0 1 2 1 | 1 2 2 2 | 2 2 2

- this minimal 3-element example:

## Representation

## Decision problems

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

**Finite spectrum**:**Free spectrum**: 1,## Subvarieties

- countably many, including semilattices, constant semigroups
## Expansions

- commutative monoids

Adapted from J.Pedersen's Catalogue of Algebraic Systems