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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>
+)]