 The intersection of an arbitrary collection of ideals is again an ideal, which allows for the definition of a smallest ideal containing some non-empty subset of , and is called the ideal "generated" by .An Ideal generated by a singleton is called a principal ideal.A band in a Riesz space is defined to be an ideal with the extra property, that for any element in for which its absolute value is the supremum of an arbitrary subset of positive elements in , that is actually in .