Postulates for abelian groups and fields in terms of non-associative operations
B. A. Bernstein · Transactions of the American Mathematical Society · 1938
1. Object. The object of this paper is to present sets of postulates for abelian groups and fields in terms of the non-associative (and non-commutative) operations - and /, the inverses of + and X> in a field. The postulates will thus treat directly the properties of the inverse operations in a field, properties important from the standpoint of operations in general, but perhaps not sufficiently emphasized in the usual treatment of groups and fields. Three sets of postulates will be given for fields. In each set, the postulates free from `/, taken by themselves, will form a set of postulates for abelian groups. Unlike other sets of postulates for fields known to me, the sets offered contain no (unconditioned) existence proposition other than one demanding that the class contain at least two elements. The consistency, necessariness, and sufficiency of the postulates are established by the' usual methods. The postulates will be found to be simple and natural. 2. Postulates (F) for fields. A class K of elements will be a (non-trivial) field with respect to a pair of binary operations if K, -, / satisfy the postulates N, S1-S3, Dr-D6 following: