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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
##### Keywords:
t-designs; convolution
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