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.