Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Are there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that $$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in I}a_{if(i)}$$ ?

where $\bigvee$ stands for $\sup$ and $\bigwedge_{f:I\to J}\bigvee_{i\in I}a_{if(i)}$ is $\inf\{\bigvee_{i\in I}a_{if(i)}\mid f:I\to J\text{ is any function} \}$.

share|cite|improve this question

1 Answer 1

up vote 3 down vote accepted

Yes as soon as $X$ is not atomic. See Handbook of Boolean algebras, Theorem 14.5 (page 215 in Volume 1)

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.