Medial groupoids

A medial groupoid is a structure G = (G,·), where · is an infix binary operation such that
· mediates: (x·y)·(z·w) = (x·z)·(y·w).

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, Tomás Kepka, Equational theories of medial groupoids, Algebra Universalis 17 (1983) 174--190 MRreview]

[Jaroslav Jezek, Tomás Kepka, Medial groupoids, Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved 93 (1983) 93 MRreview]

(S,*), where (S, + ,·) is any commutative semiring, a,b ∈ S, and x*y = a·x + b·y.

Classtype	variety
Equational theory
Quasiequational theory
First-order theory
Locally finite	no
Residual size	unbounded
Congruence distributive	no
Congruence modular	no
Congruence n-permutable	no
Congruence regular	no
Congruence uniform	no
Congruence extension property
Definable principal congruences
Equationally definable principal congruences	no
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

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[Medial semigroups]?
[Commutative medial groupoids]?