BINOMIAL SEMIGROUPS

The set S of ordered monomials in the variables x(1),...,x(n) is calle d a binomial semigroup if, as a semigroup, it can be defined via a set of generators {x(1),...,x(n)} and a set of n(n -1)/2 quadratic relati ons of the type x(j)x(i) = x(i')x(j') where j > i and i' < j', i' < j, such that each pair with i' < j' appears precisely once in the right- hand side. These semigroups were studied by Gateva-Ivanova and Van den Bergh in their investigations of binomial skew polynomial rings. They are also an example of semigroups of I-type, a condition which appear ed naturally in the work of Tate and Van den Bergh. In this paper see study the structure of binomial semigroups and we investigate the heig ht one prime ideals of their binomial skew polynomial rings. In partic ular, we give a representation theorem of such semigroups as a product of binomial semigroups on fewer generators and we prove that binomial semigroups have (torsion-free) solvable groups of quotients. It is sh own that binomial semigroups are Noetherian maximal orders in their qu otient group and have trivial normalizing class group. Quotient rings and localizations with respect to height one primes of the binomial sk ew polynomial ring are described. It follows that binomial skew semigr oup rings are Noetherian maximal orders with principal homogeneous hei ght one prime ideals. (C) 1998 Academic Press.