Differences
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hilbert_algebras [2012/07/17 10:08] jipsen |
hilbert_algebras [2016/09/02 09:28] (current) jipsen |
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| $x\to(y\to x)=1$ | $x\to(y\to x)=1$ | ||
| - | $(x\to(y\to z))\to((x\to y)\to(x\to y))=1$ | + | $(x\to(y\to z))\to((x\to y)\to(x\to z))=1$ |
| $x\to y=1\mbox{ and }y\to x=1 \Longrightarrow x=y$ | $x\to y=1\mbox{ and }y\to x=1 \Longrightarrow x=y$ | ||
| Line 28: | Line 28: | ||
| ====Examples==== | ====Examples==== | ||
| - | Example 1: Given any poset with top element 1, $\langle A,\le, 1\rangle$, define $a\to b=\left\{\begin{array}{ll}1&\text{ if $a\le b$}\\ b&\text{ otherwise}\end{array}\right.$. Then $\langle A,\to,1\rangle$ is a Hilbert algebra. | + | Example 1: Given any poset with top element 1, $\langle A,\le, 1\rangle$, define $a\to b=\begin{cases}1&\text{ if $a\le b$}\\ b&\text{ otherwise.}\end{cases}$ Then $\langle A,\to,1\rangle$ is a Hilbert algebra. |
| ====Basic results==== | ====Basic results==== | ||
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