Complexity of the word problem for commutative semigroups of fixed dimension

Summary In this paper we investigate the computational complexity of the word problem for commutative semigroups of fixed dimension. It is shown that for commutative semigroups of dimension k, k 6, the word problem is complete for symmetric linear space, providing another complete problem for this symmetric complexity class. We also show that in the case of one generator, the word problem is solvable in polynomial time.