Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If I take a lattice and I add the following axioms:

  • $a\vee b = a$ or $a\vee b = b$
  • $a\wedge b = a$ or $a\wedge b = b$

do I get a total order?

I suppose, in this case we would define $a \leq b$ to hold whenever $a\wedge b = a$.

share|cite|improve this question
Yes. And we always take $a\leq b$ to be equivaent to $a\wedge b=a$. – Michael Greinecker Oct 5 '12 at 16:01
up vote 1 down vote accepted


Given $a$ and $b$, by the assumptions you made, either $a\le b$ or $b\le a$ or both. In the case that both are true, obviously $a=b$.

To check transitivity, assume $a\le b$ and $b\le c$. This means, according to the definition, that $a\land b=a$ and $b\land c=b$. Then $a\land c = (a \land b)\land c = a\land (b\land c) = a\land b = a$, that is $a\le c$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.