Probabilistic Multiset

Multisets allow repeated elements, but in many applications their multiplicities are uncertain. We study probabilistic multisets in which each element is assigned a probability mass function on [Formula: see text]. We develop independent additive superposition, max-based union, min-based intersection, truncated difference, capacity relative complement, and scalar multiplication, and examine their algebraic properties. Pointwise probability mass envelopes are distinguished from genuine probabilistic operations and are placed in an ambient profile lattice. For bounded multiplicities, first order stochastic order yields a complete bounded distributive lattice. We also investigate expected multiplicity, expected cardinality, total variation and Wasserstein geometries, operation stability properties, and overlap similarity. Applications to word frequency modeling, genomic [Formula: see text]-mer counts, and uncertain graph neighborhood aggregation illustrate the framework.