Partial orders for Counter data structures

A counter data structure provides an abstract data type for a multiset . A multiset is much like a classic set, but augmented to track the multiplicity of each of its member elements. For example, $[1, 1, 1, 2, 3]$ is a multiset consisting of $1$ repeated thrice, and each of $2$ and $3$ once. If you define $A$ and $B$ to be multisets in a universe of objects $U$, and $m_A, m_B$ their respective…