# Characterization of state spaces of Boolean algebras

A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces of Boolean algebras?

• Since it's not obviously documented, would you say what the state space is? I guess it's the set of functions $f:A\to [0,1]$ such that $f(1)=1$ and $f(xy)=f(x)+f(y)$ for all $x,y$ such that $xy=0$, with the compact topology induced by inclusion into $[0,1]^A$? – YCor Oct 4 '18 at 21:33
• @YCor You probably mean $f(x\lor y)$ in place of $f(xy)$? – მამუკა ჯიბლაძე Oct 5 '18 at 17:00
• Yes (I wanted to write $x+y$, which amounts to the same as $x\vee y$ when $xy=0$). – YCor Oct 5 '18 at 17:11
• You are right, this is what is meant by the state space. – Miroslav Korbelar Oct 5 '18 at 19:49
• @GerryMyerson sorry, I thought a block of 9 was better than 3 blocks of 3... (PS we should eventually erase these comments to unspam this post's comments) – YCor Oct 17 '18 at 12:15