Binomial Semigroups

The setSof ordered monomials in the variablesx1,…,xnis called abinomial semigroupif, as a semigroup, it can be defined via a set of generators {x1,…,xn} and a set ofn(n − 1)/2 quadratic relations of the typexjxi = xi′xj′, wherej > iandi′ < j′,i′ < j, such that each pair withi′ < 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 ofI-type, a condition which appeared naturally in the work of Tate and Van den Bergh. In this paper we study the structure of binomial semigroups and we investigate the height one prime ideals of their binomial skew polynomial rings. In particular, 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 shown that binomial semigroups are Noetherian maximal orders in their quotient group and have trivial normalizing class group. Quotient rings and localizations with respect to height one primes of the binomial skew polynomial ring are described. It follows that binomial skew semigroup rings are Noetherian maximal orders with principal homogeneous height one prime ideals.