# Differences

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medial_groupoids [2010/07/29 15:46] (current)
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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},
+\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>
+)]