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hoops [2010/07/29 15:46]
127.0.0.1 external edit
hoops [2018/08/04 15:39] (current)
jipsen
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A \emph{hoop} is a structure $\mathbf{A}=\langle A,\cdot,\rightarrow,1\rangle $ of type $\langle 2,2,0\rangle$ such that 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 monoid]]
-$\langle A,\cdot ,1\rangle $ is a [[commutative monoids]] +
$x\rightarrow ( y\rightarrow z) = (x\cdot y)\rightarrow z$ $x\rightarrow ( y\rightarrow z) = (x\cdot y)\rightarrow z$
- 
$x\rightarrow x=1$ $x\rightarrow x=1$
- 
$(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$ $(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$
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The operation $x\wedge y = (x\rightarrow y)\cdot x$ is a meet with The operation $x\wedge y = (x\rightarrow y)\cdot x$ is a meet with
respect to this order. respect to this order.
 +
 +
 +====Definition====
 +A \emph{hoop} is a structure $\mathbf{A}=\langle A,\cdot,\rightarrow,1\rangle $ of type $\langle 2,2,0\rangle$ such that
 +
 +$x\cdot y = y\cdot x$
 +
 +$x\cdot 1 = x$
 +
 +$x\rightarrow ( y\rightarrow z) = (x\cdot y)\rightarrow z$
 +
 +$x\rightarrow x=1$
 +
 +$(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$
 +
 +
 +====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 monoid]]
 +
 +and if $x\le y$ is defined by $x\rightarrow y = 1$ then
 +
 +$\le$ is a partial order,
 +
 +$\rightarrow$ is the residual of $\cdot$, i.e., $\ x\cdot y\le z \iff y\le x\rightarrow z$, and
 +
 +$(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$.
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====Basic results==== ====Basic results====
 +
 +Finite hoops are the same as [[generalized BL-algebras]] (= divisible residuated lattices) since the join always exists in a finite meet-semilattice with top, and since all finite GBL-algebras are commutative and integral.
====Properties==== ====Properties====
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f(1)= &1\\ f(1)= &1\\
f(2)= &1\\ f(2)= &1\\
-f(3)= &\\ +f(3)= &2\\ 
-f(4)= &\\ +f(4)= &5\\ 
-f(5)= &\\ +f(5)= &10\\ 
-f(6)= &\\ +f(6)= &23\\ 
-f(7)= &\\+f(7)= &49\\
\end{array}$ \end{array}$