Two families of Newman lattices

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 l b 1 , . . . , and a move is the insertion or removal of a consecutive pair of letters xx-~ or x i x . . . . 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 . . . .