Structure diagrams for primitive Boolean algebras

Abstract:

If $S$ and $T$ are structure diagrams for primitive Boolean algebras, call a homomorphism $f$ from $S$ onto $T$ right-strong iff whenever $xTf(t)$, there is an $s$ such that $f(s) = x$ and $sSt$; let RSE denote the category of diagrams and onto right-strong homomorphisms. The relation “$S$ structures $\mathfrak {B}$” between diagrams and Boolean algebras induces a 1-1 correspondence between the components of RSE and the isomorphism types of primitive Boolean algebras. Up to isomorphism, each component of RSE contains a unique minimal diagram and a unique maximal tree diagram. The minimal diagrams are like those given in a construction by William Hanf. The construction which is given for producing maximal tree diagrams is recursive; as a result, every diagram $S$ structures a Boolean algebra recursive in $S$.