On Finite Bases for Some PBD-Closed Sets

Let \(H_a^1 = \{v: v \geq a, v \equiv 0,1 \pmod{a}\}\). 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 \(H^a = \{v: v \geq a+1, v \equiv 1 \pmod{a}\}\) for \(a = 5\) and \(6\).