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hilbert_algebras [2012/07/17 10:07]
jipsen
hilbert_algebras [2016/09/02 09:28] (current)
jipsen
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$x\to(y\to x)=1$ $x\to(y\to x)=1$
-$(x\to(y\to z))\to((x\to y)\to(x\to y))=1$+$(x\to(y\to z))\to((x\to y)\to(x\to z))=1$
$x\to y=1\mbox{ and }y\to x=1 \Longrightarrow x=y$ $x\to y=1\mbox{ and }y\to x=1 \Longrightarrow x=y$
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====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&\mbox{ if $a\le b}\\ b&\mbox{ otherwise}\end{array}\right.$. 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====