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Computeralgebra and the Systematic Construction of Finite Unlabeled Structures

In: Computer Algebra in Scientific Computing

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  • Adalbert Kerber

    (University of Bayreuth, Dep. of Mathematics)

Abstract

This review is concerned with mathematical structures that can be defined as equivalence classes on finite sets. The method used is to replace the equivalence relation by a finite group action and then to apply all what is known about such actions, i.e. to apply a mixture of quite general methods, taken from combinatorics as well as from algebra. For this purpose group actions will be introduced, enumerative methods will be reported, but the main emphasize is put on the constructive aspects, the generation of orbits representatives, and several applications of these methods, in particular to graph theory, design theory, coding theory and to mathematical chemistry. These methods have been successfully implemented in various computeralgebra packages like MOLGEN (for the generation of molecular graphs and applications to molecular structure elucidation) as well as in DISCRETA (for the evaluation of combinatorial designs and linear codes as well as other finite discrete structures). Finally we shall discuss actions on posets, semigroups, lattices, where the action is compatible with the order of the lattice or the composition of the semigroup.

Suggested Citation

  • Adalbert Kerber, 2000. "Computeralgebra and the Systematic Construction of Finite Unlabeled Structures," Springer Books, in: Victor G. Ganzha & Ernst W. Mayr & Evgenii V. Vorozhtsov (ed.), Computer Algebra in Scientific Computing, pages 249-271, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-57201-2_20
    DOI: 10.1007/978-3-642-57201-2_20
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