An analysis of Pierce's work on compact zero-dimensional spaces of finite type and of Hanfs work on primitive Boolean algebras shows that it is possible to obtain a description of the semiring generated by all primitive Boolean algebras in term of simple quasi-ordered systems.
All Boolean algebras considered here are assumed to be denumerable. If B is (such) a Boolean algebra and a e B, let B(a) denote the Boolean algebra {x ~ B Ix ~a}. The algebra B is pseudo-indecomposable (abbreviated p.i.) if for all a~B, either B-~B(a) or B~B(a¢), where a c denotes the complement of a. The element aeB is p.i. if B(a) is p.i. A Boolean algebra B is primitive if it is p.i. and each dement is the sup of a finite family of disjoint p.i. elements, it is quasi-primitive if it is a finite direct product of primitive Boolean algebras. By a result of Williams, the free product of quasi-primitive Boolean algebras is again quasi-primitive and we denote by ~g the semiring of all (isomorphism classes of) quasi-primitive Boolean algebras with product and free product.
To have a more concrete version of s~, we need to recall Pierce's concept of quasi-ordered system (Q.O. system). A Q.O. system is merely a set equipped with a binary transitive relation--which we shall always denote by R. A morphism between Q.O. systems Q and Q' is a map h:Q ~ Q' such that hR(q)= Rh(q), where R(q)={p[pRq}. A Q.O. system is simple if any morphism from Q is iniective. Q.O. systems make two complementary appearances in the theory of quasi-primitive Boolean algebras.
(1) Let B be a primitive Boolean algebra. Define S(B)= {Ix]ix p.i. in B} where [x] is the isomorphism type of B(x); and let [x]R[y] whenever B(x)xB(y)=B(y). Then Hard and Williams proved that S(B) is a simple Q.O. system that characterizes B (up to isomorphism). Note that it is possible in some cases (following Pierce) to have an utterly different description
of S(B).
(2) A Q.O. semigroup is a structure D = (D;., R) where (D, .) is a semigroup and (D, R) is a compatible O.O. system (i.e. xRy implies xzRyz and zxRzy). Let Pi(~) denote the Q.O. semigroup of all (isomorphism classes of) primitive Boolean algebras, with free product and relation R defined by BRB' if B x B' ~ B'. Then it is a consequence of a theorem of Pierce that Pi(~) determines ~, up to isomorphism. Moreover, there exists a universal simple Q.O. system V, which contains a unique representative of each isomorphism class of countable simple Q.O. systems (take the direct limit of all countable simple O.O. systems). This system V can be uniquely endowed with a multiplication which makes it into a O.O. semigroup isomorphic with Pi(~).
Thus the remaining problems to elucidate the structure of s~ are of the following kind: describe V (that is, give necessary and sufficient conditions for a countable Q.O. system to be simple), and characterize the Q.O. semigroup V. These problems are solved in the case of well-founded Q.O. systems.