Entropic groupoids or medial groupoids

Definition

A groupoid+(ab)(cd)=(ac)(bd).

Examples

• Any commutative semigroup(including any commutative monoid, Abelian group)
• Any medial quasigroup, medial semigroup.
• Define a binary operation on the reals by:

  x*y = ax + by,

  where a and b are fixed real numbers. The groupoid (R,*) is a medial groupoid.

Structure

Representation

Decision problems

Identity problem:

Word problem:

Spectra and growth

Finite spectrum:

Free spectrum:

Growth series:

History/Importance

References

Jaroslav Je\v{z}ek, Thom\acute{a}\v{s} Kepka, Medial Groupoids, 1983, Academia Praha.

Subsystems

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A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu