an:03989361
Zbl 0612.05017
Deza, M.-M.; Rosenberg, I. G.
General convolutions motivated by designs
EN
Acta Univ. Carol., Math. Phys. 27, No. 2, 49-66 (1986).
00165921
1986
j
05B30
t-designs; convolution
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.
J.E.Graver