medial_groupoids - MathStructures

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=====Medial groupoids=====

====Definition====
A \emph{medial groupoid} is a structure $\mathbf{G}=\langle G,\cdot\rangle$, where $\cdot $ is an infix binary operation such that


$\cdot$ mediates:  $(x\cdot y)\cdot(z\cdot w)=(x\cdot z)\cdot (y\cdot w)$
==Morphisms==
Let $\mathbf{G}$ and $\mathbf{H}$ be medial groupoids. A morphism from $\mathbf{G}$
to $\mathbf{H}$ is a function $h:G\rightarrow H$ 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--190[[MRreview]]

Jaroslav Jezek, Tomas Kepka,\emph{Medial groupoids},
Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved,
\textbf{93}1983,93[[MRreview]]


====Examples====
Example 1: $\langle S,*\rangle$, where $\langle S,+,\cdot\rangle$ is any commutative semiring, $a,b\in S$, and $x*y=a\cdot x+b\cdot y$.


====Basic results====

====Properties====
^[[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 def. pr. cong.]]  |no |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
\end{array}$

====Subclasses====
[[Medial semigroups]] 

[[Commutative medial groupoids]] 

====Superclasses====
[[Groupoids]] 


====References====

[(Ln19xx>
)]

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