Lexicographic semigroupoids

The natural lexicographic semigroupoids associated with Cantor product spaces indexed by countable linear orders are classified. Applications are given to the classification of triangular operator algebras which are direct limits of upper-triangular matrix algebras. 0. Introduction Consider a Cantor space which is presented explicitly as an infinite product of finite topological spaces. The product presentation provides an equivalence relation R consisting of the pairs (x,y) of points x and y which disagree in at most finitely many coordinates. This equivalence relation supports a natural locally compact totally disconnected topology which makes R a principal groupoid. It is well-known that in the case of countable products such topological equivalence relations are classified by the generalised integer obtained from the formal product of the cardinalities of the component spaces. Furthermore, this classification is closely related to the classification of C*-algebras that are infinite tensor products of matrix algebras, the so-called UHF C*-algebras. See, for example, Renault [10] and Power [7]. In the present paper we consider antisymmetric topological binary relations which are the lexicographic topological subrelations arising from infinite products indexed by general countable linear orderings. These natural semigroupoids are classified and their automorphism groups determined. This and related results enable us to give applications to the classification of triangular operator algebras which are themselves lexicographic products in an algebraic sense. The binary relations may also be viewed as the (semigroupoid) lexicographic products of total orderings on finite sets, and in fact our methods are applicable to lexicographic products of connected antisymmetric finite partial orders. Although applications to approximately finite operator algebras provide our primary motivation, it seems clear that lexicographic subrelations are interesting in their own right. In § 1 we recall how the generalised integer associated with the presentation of the Cantor space gives a complete invariant for the associated approximately finite groupoid. In §2 we classify the lexicographic products in the case of indexing by a countable dense