Property D Cyclic Neofields

Abstract

In 1948 L. J. Paige introduced the notion of a neofield (N,\oplus,\cdot) as a set N with two binary operations, generally referred to as addition (\oplus) and multiplication (\cdot) such that (N,\oplus) is a loop with identity 0 and (N-\{0\},\cdot) is a group, with both left and right distribution of multiplication over addition. The neofield was considered a generalization of a field and its application was for the coordinatizing of projective planes and related geometry problems. In 1967 A.D. Keedwell introduced the notion of property D cyclic neofields in relation to his study of latin squares and their application to projective geometry. In particular, the existence of a property D cyclic neofield guarantees the existence of a pair of orthogonal latin squares. Keedwell provides a theorem for the existence of property D cyclic neofields with a set of conditions on a sequence of integers.We provide an alternate condition for Keedwell's existence theorem that requires only one criteria for each condition in contrast to Keedwell's two criteria. We then establish a set of conditions for the existence of commutative property D cyclic neofields that require a sequence half as long as for Keedwell's existence theorem. We also examine subneofields of property D cyclic neofields and consider their application to extending known neofields to higher order property D cyclic neofields.