Some properties of partly-associative operations

Abstract

Three properties of a group-operation are (i) it is associative: (xy)z=x(yz); (ii) it is regular: a=b if ax=bx or if ya=yb; and (iii) it is reversible: ax= ya =b is solvable for x and y. These definitions may readily be generalized. For example, associative property may be stated as the two continued products which can be formed from same three elements in same order are equal (for all values of elements concerned). Under a (v+1)-ary operation, v+1 continued products can be formed from 2v+1 elements in order. For any given operation, some, none, or all of these may be equal. If some are equal, operation is partlyassociative. If in addition operation is regular and reversible, then there are numbers j and k, v being a multiple of k and k of j, such that pth continued product is equal to (p+q)th if p is a multiple of j and q of k. (Partly associative operations, J. London Math. Soc. vol. 24 (1949) pp. 260-271.) Such an operation is (j, k)-associative. If j = k = 1 (that is, if all continued products are equal) operation is, if reversible, that of a polyadic group. (E. L. Post, Polyadic groups, Trans. Amer. Math. Soc. vol. 48 (1940) pp. 208350.) A fundamental theorem about polyadic groups is that a polyadic operation can be regarded as continued product of a group operation. (Op. cit. pp. 218-219.) The proof of this involves setting up an equivalence such that an ordered set can replace any equivalent ordered set in a polyadic product without changing value of product. (Op. cit. p. 217.) The continued-product theorem can be generalized to apply to (1, k)-associative operations (Theorem H of present paper) and replacement theorem to (j, k)-associative operations (Theorem E). Other replacement theorems are proved in part 2. They do not require full reversibility and I have stated them with only properties actually required for proofs. They can be summed up (in somewhat less general forms than in text) as follows: Let (a, 3, y) be either (1, -1, 1), (0, 0, 1), (0, 1, 0), or (1, 0, 0). Then if * is a 0and v-reversible (j, k)-associative operation, if