On Theories with a Combinatorial Definition of "Equivalence"

The name "combinatorial theory" is often given to branches of mathematics in which the central concept is an equivalence relation defined by means of certain "allowed transformations" or "moves." A class of objects is given, and it is declared of certain pairs of them that one is obtained from the other by a "move"; and two objects are regarded as "equivalent" if, and only if, one is obtainable from the other by a series of moves. For example, in the theory of free groups the objects are words made from an alphabet a, b, , a-, b', and a move is the insertion or removal of a consecutive pair of letters xx-l or x-lx. In combinatorial topology the objects are complexes, and the allowed moves are "breaking an edge" by the insertion of a new vertex, or the reverse of this process.' In Church's "conversion calculus"2 the rules II and III are ''moves" of this kind. In many such theories the moves fall naturally into two classes, which may be called "positive" and "negative." Thus in the free group the cancelling of a pair of letters may be called a positive move, the insertion negative; in topology the breaking of an edge, in the conversion calculus the application of Rule II (elimination of a X), may be taken as the positive moves. In theories that have this dichotomy it is always important to discover whether there is what may be called a "theorem of confluence," namely, whether if A and B are "equivalent" it follows that there exists a third object, C, derivable both from A and from B by positive moves only. A closely connected problem is the search for "endforms," or "normal forms," i.e. objects which admit no positive move. It is obvious that in a theory in which the confluence theorem holds no equivalence class can contain more than one end-form, but there remains the question whether in such a class any random series of positive moves must terminate at the end-form, or whether infinite series of moves may also exist. The purpose of this paper is to make a start on a general theory of "sets of moves" by obtaining some conditions under which the answers to both the above questions are favorable. The results are essentially about "partially-ordered" systems, i.e. sets in which there is a transitive relation >, and sufficient conditions are given for every two elements to have a lower bound (i.e. for the set to be "directed") if it is known that every two "sufficiently near" elements have a lower bound. What further conditions are required for the existence of a greatest lower bound is not relevant to the present purpose, and is reserved for a later discussion.