# Differences

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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}$\end{array}$