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How can I recognize, that some lattice is boolean algebra ? I know the lattice has to be distributive and complemented to be boolean algebra. But I do not understand, how to apply this into examples.

What about those example - I have to figure out, if is lattice distributive, complemented or boolean algebra:

a) $X = {1,2,3,12,18,30. 180}$
b) $X = {1,2,4,6,7,10,60,420}$
c) $X = {1,2,3,7,6,14,21,42}$

and I really do not know how to figure it out :( Can you explain me this, please?

PS Sorry for my English, I have done the best...

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    B&C has 8 entries, A has 7. Is this an error? – 2017-01-06

1 Answers 1

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The first thing to do is figure out what the ordering on your lattice is - you've just given three lists of numbers, and a list of numbers is not a lattice. To me, it looks like the order relation here is divisibility - so $x \wedge y$ is the greatest number in the list that divides both $x$ and $y$, while $x \vee y$ is the least number on the list that is a multiple of both $x$ and $y$. For example, in (b), $6 \vee 7 = 420$, and $6 \wedge 7 = 1$. If that interpretation is not the interpretation used in your class or textbook, the rest of this answer won't be particularly helpful.

Determining distributivity or complementation is just a question of checking a rule. For example, to determine whether it's complemented, just check whether for every $x$ there is a $y$ so that $x \wedge y$ is the minimum element and $x \vee y$ is the maximum. For (a), that doesn't work - $2 \vee 3 = 12$, $2 \vee 1 = 2$, and $2 \vee x = x$ for all the other numbers in the list. So (a) is not complemented. On the other hand, (c) is - for example, the complement of $3$ is $14$.

This is not complete - I've left a lot for you to do. But if my assumption about the lattices is correct, then this should get you started.

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    Thank you a lot. I helped me to understand what complementation is:) and what about distribution ? Is there any simple rule to check distributivity? – 2017-01-08