directoids - MathStructures

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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>
)]

Trace: • equivalence_relations • semirings • process_algebras • boolean_semigroups • cancellative_partial_monoids • amalgamation_property • complete_lattices • suggestions_or_comments • boolean_monoids • action_lattices