# Differences

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directoids [2010/07/29 15:46] (current)
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+=====Directoids=====
+Abbreviation: **Dtoid**
+====Definition====
+A \emph{directoid} is a structure $\mathbf{A}=\langle A,\cdot +\rangle$, where $\cdot$ is an infix binary operation such that
+
+
+$\cdot$ is idempotent:  $x\cdot x=x$
+
+
+$(x\cdot y)\cdot x=x\cdot y$
+
+
+$y\cdot(x\cdot y)=x\cdot y$
+
+
+$x\cdot ((x\cdot y)\cdot z)=(x\cdot y)\cdot z$
+
+Remark:
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be directoids. A morphism from $\mathbf{A}$
+to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+
+$h(xy)=h(x)h(y)$
+
+====Examples====
+Example 1:
+
+====Basic results====
+The relation $x\le y \iff x\cdot y=x$ is a partial order.
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  | |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  | |
+^[[Locally finite]]  | |
+^[[residual size]]  |unbounded |
+^[[Congruence distributive]]  |no |
+^[[Congruence modular]]  |no |
+^[[Congruence n-permutable]]  |no |
+^[[Congruence regular]]  |no |
+^[[Congruence uniform]]  |no |
+^[[Congruence types]]  |semilattice (5) |
+^[[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====
+[[Semilattices]]
+
+====Superclasses====
+[[Groupoids]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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