Constructions of color schemes. (English) Zbl 0532.20042

The collection of colors in a general type of color scheme can be made into a multi-valued ”color algebra” which satisfies natural group-like axioms. (The models of these axioms are called polygroups.) Configurations present in the scheme are reflected in the arithmetic of the color algebra. It is not easy to decide whether a given polygroup is isomorphic to the color algebra of some scheme. This article is a survey of recent methods that have been used to construct such isomorphisms. In section 1 a general iterative procedure is used to build color schemes with a prescribed set of forbidden configurations. In section 2 a strong form of this inductive process leads to a random-type scheme whose properties are easy to control and for which there is an easy sufficient condition for existence. The third section shows that various important classes of color algebras (for example, double coset algebras) are closed under the direct product operation and a ”wreath product” operation. An application to ultragroups is given.


20N99 Other generalizations of groups
08A02 Relational systems, laws of composition
05B30 Other designs, configurations
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