Characters of finite semigroups

Abstract Let S be a finite semigroup and let J1,…, Jr be the regular j -classes of S. Then the main theorem of this paper shows that ch S ≈ ch H1 X ··· X ch Hr where, for example, ch S denotes the character ring of S and H1,…, Hr are maximal subgroups of J1,…, Jr, respectively. As a consequence of this result, two representations of S are equivalent if and only if they are equivalent on the subgroups of S. Further, we show that each character of S can be uniquely expressed as an integral linear combination of what we term standard irreducible characters. As a consequence of this, the analog of an important theorem of Brauer [1] holds for finite semigroups; namely, every character of a finite semigroup can be expressed as an integral linear combination of characters induced from linear characters of its elementary subgroups.