Abstract :
The subject of this paper is the sets of prefixes and the sets of overlaps of traces (elements of free partially commutative monoids), when the prefixes or overlaps are ordered in one natural way. Lattice-theoretic characterizations, and related properties, are developed. Specifically, it is shown that the collection of prefix-sets of traces constitutes precisely the class of finite distributive lattices and that the overlap-set of a trace is a sublattice of its prefix-set, so the overlap-sets of traces also form finite distributive lattices. Several characterizations of the class of overlap-lattices of traces are given; for example, incomparable join-irreducible elements of such lattices must meet at the minimum element of the lattice.
