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If S is a finite semigroup, and if
K is a field, under what conditions is there a group G such that the semigroup
algebra KS is isomorphic to the group algebra KG?
The following theorems are proved:
1. Let S have odd order n, and let K be either a real number field or GF(q),
where q is a prime less than any prime divisor of n. If KlS≅KG for a group G, then
S is a group.
2. Let K be a cyclotomic field over the rationals, and let G be an abelian group.
Then KG≅KS for a semigroup S that is not a group if and only if for some prime p
and some positive integer k,K contains all pk-th roots of unity and the cyclic group
of order pk is a direct factor of G.
3. Let S be a commutative semigroup of order n, and let K = GF(p), where p is a
prime not exceeding the smallest prime dividing n. If K1S≅KG for a group G, then
S is a group.
The semigroup ring of a semilattice is also considered.
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