multisequences

A multisequence is a mathematical object that generalizes the concept of a sequence. Instead of each term being a single element, each term in a multisequence is itself a multiset. A multiset is like a set, but it allows for multiple occurrences of the same element. For instance, while the set {1, 2, 3} contains each element once, the multiset {1, 1, 2, 3, 3, 3} contains the element 1 twice and the element 3 three times.

Therefore, a multisequence can be thought of as a list of multisets. For example, a multisequence might look like {{a, a}, {b}, {c, c, c}, {a, b, c}}. Here, the first term is the multiset {a, a}, the second term is the multiset {b}, and so on.

The study of multisequences can arise in various contexts, particularly in combinatorics and discrete mathematics, where generalizing sequence properties to account for multiplicity is useful. For example, one might consider operations on multisequences, such as concatenation or element-wise operations, analogous to operations on standard sequences. The properties of convergence or limits, if defined for multisequences, would need to account for the varying multiplicities within the terms. While not as commonly encountered as standard sequences, multisequences offer a richer framework for describing collections of objects where both order and repetition are significant.