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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>
 +)]