On Finite Bases for Some PBD-Closed Sets

Abstract

Let $H_a^1=\{v:v\ge a,v\equiv 0,1(\mathrm{mod}a)\}$. It is well known that such sets are PBD-closed. Finite bases are found for these sets for $a=6,7,$ and $8$. At the same time, we improve the result of Mullin in [6] about finite bases of $H^a=\{v:v\ge a+1,v\equiv 1(\mathrm{mod}a)\}$ for $a=5$ and $6$.