right_hoops - MathStructures

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=====Right hoops=====

====Definition====
A \emph{right hoop} is a structure $\mathbf{A}=\langle A,\cdot,/,1\rangle $ of type $\langle 2,2,0\rangle$ such that

$\langle A,\cdot ,1\rangle $ is a [[monoid]]

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

$x/x=1$

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


Remark: 
This definition shows that right hoops form a variety.

Right hoops are partially ordered by the relation $x\leq y \iff
y/x=1$.

The operation $x\wedge y = (x/y)\cdot y$ is a meet with respect to this order.


====Definition====
A \emph{right hoop} is a structure $\mathbf{A}=\langle A,\cdot,/,1\rangle $ of type $\langle 2,2,0\rangle$ such that

$x\cdot y = y\cdot x$

$x\cdot 1 = x$

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

$x/x=1$

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


====Definition====
A \emph{right hoop} is a structure $\mathbf{A}=\langle A,\cdot,/,1\rangle $ of type $\langle 2,2,0\rangle$ such that

$\langle A,\cdot ,1\rangle $ is a [[commutative monoid]]

and if $x\le y$ is defined by $y/x = 1$ then

$\le$ is a partial order,

$/$ is the right residual of $\cdot$, i.e., $\ x\cdot y\le z \iff x\le z/y$, and

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


==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be hoops. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 

$h(x\cdot y)=h(x)\cdot h(y)$, $h(x/y)=h(x)/h(y) $, $h(1)=1$

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  | |
^[[Congruence modular]]  | |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &2\\
f(4)= &8\\
f(5)= &24\\
f(6)= &91\\
f(7)= &\\
\end{array}$

====Subclasses====
[[hoops]] 

====Superclasses====
[[Porrims]] 

====References====

[(Ln19xx>
)]

Trace: • m-zeroid • bck-lattices • complemented_modular_lattices • pseudo_basic_logic_algebras • commutative_rings • shells • cancellative_partial_monoids • finite_semilattices_with_identity • generalized_boolean_algebras • semilattices_with_zero