In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly.

For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic filter $G$ it follows that

$\Sigma (A \cap M) \in G$

and by genericity

$\Pi_{A \in M} \Sigma(A \cap M) \in G$.

What is the meaning of $\Sigma$ and $\Pi$ of an intersection? Does $\Sigma$ of sets mean their sum, and by that their union?

  • 1
    $\begingroup$ $A$ is an antichain in a Boolean algebra, so $\Sigma$ denotes a supremum and $\Pi$ an infimum in that algebra. $\endgroup$ – Miha Habič Aug 29 '15 at 13:00
  • $\begingroup$ Some people may write $\bigvee A\cap M$ or $\bigvee_{x\in A\cap M} x$ instead of $\sum(A\cap M)$. $\endgroup$ – Goldstern Aug 31 '15 at 17:09

I'm a few thousand miles away from my copy of Jech's book, but I think the $A$ in the passage you're quoting must be a subset of a complete Boolean algebra (perhaps an antichain to suggest the notation $A$). If so, then $\Sigma$ means the Boolean sum, also known as the join and as the least upper bound. So $\Sigma(A\cap M)$ would be the join, in the Boolean algebra, of all the elements of $A\cap M$. Similarly, $\Pi$ means the Boolean product, also known as the meet and as the greatest lower bound.

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