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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>
 +)]