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We have a poset $(X, \sqsubseteq)$, and we define operations $+$ and $\cdot$ by $x+y=inf(x, y)$ and $x\cdot y=sup(x, y)$ ($+$ can be seen as union in sets and $\cdot$ as intersection in sets).

The question is: show that $+$ and $\cdot$ satisfy the absorption property $x+(x\cdot y)=x$. There is a suggestion to first settle $\textit 0$ and $\textit 1$ and also make use of symmetry.

It has the following Hasse diagram:

HasseDiagram

My idea is that I should translate $x+(x\cdot y)=x$ to $inf$ and $sup$, like this:

$sup(1, inf(1, 0))=\\sup(1, 0)=\\1$

But to me, that feels like not proving that much. Also, I didn't make use of symmetry as was suggested.

How can I tackle this?

egreg
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user95514
  • 13

2 Answers2

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I prefer to denote the operations by $x\land y=\inf(x,y)$ and $x\lor y=\sup(x,y)$, because $+$ and $\cdot$ might suggest wrong ideas.

Those operations are well defined because your poset is a lattice. Now, by definition, $x\land y\sqsubseteq x$, so $x$ is an upper bound of the subset $\{x,x\land y\}$. Can the least upper bound be different from $x$?

When you have realized an answer to the question, you'll have proved that $x\lor(x\land y)=x$.

You can also do this by cases; prove the property for $x=0$, for $x=1$ and for the case when neither $x$ or $y$ is $0$ nor $1$. And for this it's sufficient to check $x=p$ and $y=q$, as permuting the names of $p$, $q$ and $r$ doesn't change the Hasse diagram (in a substantial way).

egreg
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You can use $a \leq b$ if and only if $a = a \wedge b$ if and only if $a \vee b = b$.

You know $a \leq a \vee b$; see if you can use the above to prove one absorption law. And you know $a \wedge b \leq a$; use the above to prove the other absorption law.

user43208
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  • Thank you for your comment. I'm still unclear with what you mean and what I should do with it though. Are $a$ and $b$ analogous to $x$ and $y$ respectively? – user95514 Sep 18 '13 at 11:53
  • Yes, they are analogous (it makes no difference what names you give variables). But write $x \leq x \vee y$ (instead of $a \leq a \vee b$) in the second paragraph. Now compare this with the first paragraph: what here are playing the role of $a$ and $b$ in the first paragraph? – user43208 Sep 18 '13 at 12:37