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]] | + | |
| $x\rightarrow ( y\rightarrow z) = (x\cdot y)\rightarrow z$ | $x\rightarrow ( y\rightarrow z) = (x\cdot y)\rightarrow z$ | ||
| - | |||
| $x\rightarrow x=1$ | $x\rightarrow x=1$ | ||
| - | |||
| $(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$ | $(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$ | ||
| Line 25: | Line 21: | ||
| 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 | ||
| + | |||
| + | $(x\rightarrow y)\cdot x = (y\rightarrow x)\cdot y$. | ||
| Line 36: | Line 60: | ||
| ====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==== | ||
| Line 60: | Line 86: | ||
| 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}$ | ||
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