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Medial groupoids
Definition
A \emph{medial groupoid} is a structure G=⟨G,⋅⟩, where ⋅ is an infix binary operation such that
⋅ mediates: (x⋅y)⋅(z⋅w)=(x⋅z)⋅(y⋅w)
Morphisms
Let G and H be medial groupoids. A morphism from G to H is a function h:G→H that is a homomorphism:
h(xy)=h(x)h(y)
Jaroslav Jezek, Tomas Kepka,\emph{Equational theories of medial groupoids}, Algebra Universalis, \textbf{17}1983,174–190MRreview
Jaroslav Jezek, Tomas Kepka,\emph{Medial groupoids}, Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved, \textbf{93}1983,93MRreview
Examples
Example 1: ⟨S,∗⟩, where ⟨S,+,⋅⟩ is any commutative semiring, a,b∈S, and x∗y=a⋅x+b⋅y.
Basic results
Properties
Finite members
f(1)=1f(2)=f(3)=f(4)=f(5)=f(6)=f(7)=