Multissetes

Multissetes are mathematical structures that generalize ordinary sets by allowing multiple copies of elements. In standard terminology they are called multisets or bags, but the form Multissetes appears in some literature as an alternative name.

A multiset M over a base set X is specified by a multiplicity function m: X → N0, where N0 denotes the nonnegative integers. The value m(x) gives how many copies of x are contained in M. If m(x) > 0 for finitely many x, M is finite; the support of M is supp(M) = { x ∈ X : m(x) > 0 }. The cardinality of M is |M| = ∑_{x∈X} m(x).

Operations include multiset addition (M ⊎ N with (M ⊎ N)(x) = m_M(x) + m_N(x)), intersection (min(m_M(x), m_N(x))), and union (max(m_M(x), m_N(x))). Subtraction M \ N is defined by (M \ N)(x) = max(m_M(x) − m_N(x), 0). The power multiset P(M) consists of all submultisets of M, and its elements correspond to all choices of multiplicities not exceeding those in M.

Examples: M = {a, a, b, c, c} has multiplicities m(a)=2, m(b)=1, m(c)=2 and |M|=5. In the special case where every m(x) is 0 or 1, M is a standard set.

Applications appear in combinatorics, probability with indistinguishable objects, database theory, and formal language theory, where generating functions and multiset coefficients count arrangements and selections with repetition. Multissetes thus provide a flexible framework for counting with multiplicities and for modeling collections where elements may appear multiple times.