AN ARITHMETIC OF COMPLETE PERMUTATIONS CONSl?RAMRi, I: AN EXPOSITION OF THE GENERAL THEORY

We develop an arithrretic of complete permutations of sy-ilmetric, integral bases; this arithmetic is comparable to that of perfect systems of difference sets with which there are several interrelations. Super-position of permutations provides the addition of this arithmetic. Addition if facilitated by complete permutations with a certain “rplitting” property, allowing them to be pulled apart and reassembled. The split permutations also provide a singular direct product for complete permutations in conjunction with the multiplicatiou (direct product) of the arithmetic which itself derives from that for perfect systems of digerence ccts. We pay special attention to complete permutations satistying constraints both fixed and variable; this is equivalent to embedding partial complete permutations iu complete permutations. In the sequel, using this arithmetic, we investigate the spectra + certain constraints with respect to central integral bases which are of interest for the purpose of giving further constructions either of complete permutations with constraints or of irregular, extreme1 perfect systems of difference sets.