# Differences

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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> | ||

+ | )] |

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