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right_hoops [2018/08/04 15:53] (current)
jipsen created
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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>
 +)]