Shuffle products and other operations in the class of primitive spaces

Abstract

A Boolean space is a compact Hausdorff zero-dimensional space. Boolean spaces arise as Stone duals of Boolean algebras and the duals of the countable Boolean algebras are the countably baaed (or equivalently metrizable) Boolean spaces. Among the latter spaces, primitive spaces and more particularly finitary spaces form two very peculiar subclasses. A countably baaed Boolean space is primitive if it admits a basis of pi clopen sets, where X is pi if for each clopen U C X, either X z U or X Z X \ U. It is finitary if the set of homeomorphism types of its pi clopen subsets is finite. These spaces have been introduced through utterly different means by Hanf [2], Pierce [21] and also by Paljutin 1201. They were also studied by Dobbertin [I], Heindorf [12], Myers [18], the author [6] and others ([17], [23] . . . ). Primitive spaces can be handled by easy-to-use combinatorial tools : their diagrams ([2] or [7]) and the structure of their homeomorphism types is at present fairly well understood ([22], [12], [7]). Their pl ace in Ketonen’s hierarchy ([13]) is well-known, at least in the finitary case ([12]). In the class of primitive spaces, Tarski’s cube problem (X+X+X E X implies X+X E X, see [2l] or [S]) has a positive answer, but otherwise the diagram technique enables to produce easy examples of pathological behaviour of product of spaces ([22], [S]). It should be noted that the primitive spaces bear some surprising relationships with the variable-free fragment of the modal system K4 ([9]) and that Paljutin has used finitary spaces to obtain some decidability results ([20]). Finally, it turns out that many Lindenbaum-Tarski algebras of important first-order theories (such as abelian groups, well-orders and linear orders, Boolean algebras, . . ) are primitive ([lS]). It is well-known that the class of primitive spaces is closed under the formation of finite topological sums and finite Cartesian products ([22], [24]). In this paper, we introduce other operations on Boolean spaces under which the class of primitive