Notice that we do not need all of the information contained in configurations, just those labellings that can be changed by future catenations. By Proposition 7, instead of (x, y) we can consider a reduced configuration defined as a pair [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
Another direction for further study is the behaviour of infinite catenations, like the usual infinite word asymptotics (cf.
(2005) and admits a natural definition of catenation. We use the attribute "directed" to emphasize the way figures are catenated; this should not be confused with the meaning of "directed" in e.g.
Definition 2 (Catenation) Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be directed figures.
Abusing this notation, we will also write [X.sup.[??].sub.m] to denote the set of all figures that can be composed by [[??].sub.m] catenation from figures in X [subset or equal to] [[SIGMA].sup.[??].sub.m].
1 (London, 1796), 225-26, presents stuttering as an instance of broken "catenations
," trains of thought: "Thus in the common impediment of speech, when the association of the motions of the muscles of enunciation with the idea of the word to be spoken is disordered, the great voluntary efforts, which distort the countenance, prevent the rejoining of the broken associations."