Commutative Semigroups

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
                      

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

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PCP: Point and Click Proofs, Peter Jipsen, 2001

Adapted from J.Pedersen's Catalogue of Algebraic Systems