The subvarieties of a field

<p>Number systems which satisfy part but not all of the postulates for a field are called subvarieties of a field. The purpose of this paper is the determination of as great as possible a number of such varieties by suitable definitions of the class of elements and of the two operations involved.</p> <p>Two postulate systems are considered. The first gives rise to 284 varieties, instances of all of which are given for infinite classes of elements, and of all except three for finite classes.</p> <p>Of the 8192 combinations of postulates arising from the second system, not more than 1146 can be consistent. Instances are given of 1054 of these. As the postulates of this system are not independent, no conclusion has been reached regarding the remaining cases.</p>