Let A be a Banach algebra. In [1, 2] R. Arens showed how to construct two multiplications on A** which make A** into a Banach algebra. The two multiplications are, in general, distinct; Arens said that A was regular if they coincided. In this paper we are concerned with the regularity of Banach algebras of the form l^S) and ^(S.eu), where S is a (discrete) semigroup and co is a weight function on S. Earlier results are to be found in [3, 8]. In [8, Theorem 2], Young obtains necessary and sufficient conditions for /x(5) to be regular; in [3] are given some conditions for the regularity of l^S, co). In this paper we obtain new conditions for each of these; we can then obtain our main result that l^S, co) is regular whenever US) is. In [5], H. A. M. Dzinotyiweyi introduces the notion of the spaces AP (S,co) and WAP (S, co) of weighted (weakly) almost periodic functions on a semigroup S. As is clear from [5], these two definitions have a number of shortcomings. In this paper we shall offer alternative definitions of these spaces. With these definitions we can show that WAP (S, co) = lw(S, co) if and only if l^S, co) is regular and obtain a condition for AP (S,co) to equal / OS, co).