order_algebras - MathStructures

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=====Order algebras=====

Abbreviation: **OrdA**
====Definition====
An \emph{order algebra} is a structure $\mathbf{A}=\langle A,\cdot
\rangle $, where $\cdot $ is an infix binary operation such that


$\cdot $ is idempotent:  $x\cdot x=x$


$(x\cdot y)\cdot x=y\cdot x$


$(x\cdot y)\cdot y=x\cdot y$


$x\cdot ((x\cdot y)\cdot z)=x\cdot(y\cdot z)$


$((x\cdot y)\cdot z)\cdot y=(x\cdot z)\cdot y$

Remark: 


==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be order algebras. 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====

====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 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====
[[Bands]] 

====Superclasses====
[[Groupoids]] 


====References====

[(Ln19xx>
)]

Trace: • cancellative_residuated_lattices • conjugative_binars • involutive_lattices • stably_compact_spaces • left_cancellative_semigroups • de_morgan_monoids • residuated_idempotent_semirings • cancellative_commutative_semigroups • complete_distributive_lattices • pseudo_basic_logic_algebras