Intersecting Multisets and Applications to Multiset Languages

We show that the family Sem = mREG = mCF of regular multiset languages is closed under applications of finite and iterated multiset-union, finite and iterated multiset-intersection, and multiset-subtraction. For the class Sem(šk) of semilinear subsets of šk, or that of Žk, Sem := ⋃k≥0 Sem(Žk), this amounts to verify, that the component-wise maximum, minimum or non-negative subtraction of pairs of elements from two semi-linear sets is again semi-linear, and that the iterated application can be replaced by a fixed finite application of multisetintersection, respectively multiset-subtraction. We solve the three questions about closure properties that remained open in [KuMi 01,KuMi 02], verify that the family mMON is not closed with respect to multiset-intersection, and correct a small mistake in a proof in [EiSc 69,Bers 79].