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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},
 +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>
 +)]