# Differences

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hoops [2010/07/29 15:46] (current)
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+=====Hoops=====
+====Definition====
+A \emph{hoop} is a structure $\mathbf{A}=\langle A,\cdot,\rightarrow,1\rangle$ of type $\langle 2,2,0\rangle$ such that
+
+
+$\langle A,\cdot ,1\rangle$ is a [[commutative monoids]]
+
+
+$x\rightarrow ( y\rightarrow z) = (x\cdot y)\rightarrow z$
+
+
+$x\rightarrow x=1$
+
+
+$(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$
+
+
+Remark:
+This definition shows that hoops form a variety.
+
+Hoops are partially ordered by the relation $x\leq y \iff +x\rightarrow y=1$.
+
+The operation $x\wedge y = (x\rightarrow y)\cdot x$ is a meet with
+respect to this order.
+
+
+==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:
+
+$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$, $h(1)=1$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  |decidable |
+^[[Quasiequational theory]]  |decidable |
+^[[First-order theory]]  | |
+^[[Locally finite]]  |no |
+^[[Residual size]]  |unbounded |
+^[[Congruence distributive]]  |yes |
+^[[Congruence modular]]  |yes |
+^[[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====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +f(7)= &\\ +\end{array}$
+
+====Subclasses====
+[[Wajsberg hoops]]
+
+[[Idempotent hoops]]
+
+[[Commutative generalized BL-algebras]]
+
+====Superclasses====
+[[Pocrims]]
+
+[[Generalized hoops]]
+
+
+====References====
+
+[(Ln19xx>
+)]