Stuart D. Scott Department of Mathematics, University of Auckland, Auckland, New Zealand

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Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).

  • [1]

    E. Aichinger. Congruence lattices forcing nilpotency. J. Algebra Appl., 17(2):1850033, 2018.

  • [2]

    P. Higgins. Groups with multiple operations. Proc. Lon. Math. Soc., 6(3):366416, 1956.

  • [3]

    H. Neumann. Varieties of groups and their associated nearrings. Math. Z., 65:3669, 1956.

  • [4]

    G. Pitz. Nearrings. Revised Edition. North-Holland. Amsterdam (1983)

  • [5]

    R. Schafer. An introduction to non-associative algebras. Academic Press, New York (1966)

  • [6]

    S. Scott. Binary groups, 1137, 2019. Manuscript.

  • [7]

    S. Scott. N-solubility and N-nilpotency in tame N-groups. Alg. Coll., 5(4):425448, 1998.

  • [8]

    S. Scott. Tame nearrings and N-groups. Proc. Edin. Math. Soc., 23:275296, 1980.

  • [9]

    S. Scott. The structure of Ω-groups. Nearrings, nearfields and K -loops, Kluever Acad. Pub., Netherlands, 1997, 47137.

  • [10]

    S. Scott. The Z -constrained conjecture. Nearrings and Nearfields. Hamburg, 2003, 69168. Springer, Dordrecht, 2005.

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