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General convolutions motivated by designs. (English) Zbl 0612.05017
Let V be a finite set and let P(V) denote the power set of V; let \(X+Y\) denote the Boolean sum of the subsets X,Y\(\subseteq V\). We consider the collection of all functions from P(V) into the integers. Note that certain of these functions represent, in a natural way, the t-designs on V. A useful tool in studying t-designs and their generalizations is the convolution: \[ (f*g)(Z)=\sum_{X+Y=Z}f(x)g(Y). \] In this paper, the authors study the properties of the convolution in a very general setting: The set V is no longer required to be finite; the power set is replaced by an ideal in (P(V),\(\subseteq)\); Boolean \(''+''\) sum is replaced by a far more general operation ”\(\circ ''\) closed on the ideal; and the integers are replaced by an associative, commutative ring with identity or even a more general structure.
Reviewer: J.E.Graver
MSC:
05B30 Other designs, configurations
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