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ABSTRACT

In this note we present some algorithms to deal with nearrings, the appropriate algebraic structure to study non-linear functions. This is similar the role of rings in the theory of linear functions or that of groups for permutations. In particular, we give efficient algorithms that deal with big nearrings that are given by a small set of generators. In this context, generating involves composition as well as point-wise addition. In the extreme case, one transformation of a group of order n can generate a set of up to nn transformations.

REFERENCES

• 1: E. Aichinger. Interpolation with near-rings of polynomial functions. Master's thesis, University of Linz, 1994.
• 2: E. Aichinger. Local interpolation near-rings as a frame-work for the density theorems. In Contributions to General Algebra, volume 9, pages 27- 36. Verlag HSlder-Pichler-Tempsky, Wien- Verlag B.G. Teubner, Stuttgart, 1995.
• 3: E. Aichinger and C. NSbauer. The cardinalities of the endomorphism near-rings I(G), A(G), and E(G) for all groups G with IGI _< 31. In G. Saad and M. J. Thomsen, editors, Near-rings, near-fields and K-loops, pages 175-178. Kluwer Acad. Publisher, 1997.
• 4: G. Betsch. Primitive near-rings. Math. Z., 130:351-361, 1973.
• 5: J. Clay. Nearrings: Geneses and Applications. Oxford University Press - Oxford, New York, Tokyo, 1992.
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• 8: S. D. Scott. Involution near-rings. Prac. Edinburgh Math. Sac. (2), 22(3):241-245, 1979.
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INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

Additional Classification:
  G. Mathematics of Computing

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

General Terms:
Algorithms

Keywords:
groups, near-rings, non-linear transformations