mátrxions

Mátrxions are a hypothetical mathematical construct used in theoretical discussions of linear transformations and tensor-network representations. They are conceived as elemental units that pair a linear operator with an additional scalar attribute, often described as a charge. Formally, for a fixed dimension d, a mátrxion is a pair M = (A, q) where A is a d×d matrix and q is a scalar from the underlying field (real or complex). The collection of all mátrxions of dimension d is denoted Md.

A central operation on mátrxions is a binary composition, often written as ∘, defined by M ∘ N = (AB, q + p) when M = (A, q) and N = (B, p). This composition is associative and has an identity element I = (Id, 0), where Id is the d×d identity matrix. Under this operation, Md forms a semigroup, and properties of the matrices A and B influence the combined transformation AB.

In networks, a finite arrangement of mátrxions can be connected along edges to model the sequential application of linear transformations. Contracting such a network yields a global transformation obtained by multiplying the local matrices along paths, with the total charge given by the sum of the individual charges. In many discussions, networks are analyzed under a fixed total charge, echoing a conservation-like principle.

Etymology and usage: The term blends “matrix” with “ion,” signaling a unit that carries both a linear transformation and a scalar attribute. In fictional or speculative frameworks, mátrxions provide a compact way to represent composite linear effects and their aggregate properties in distributed systems or tensor-network models.

See also: matrix, tensor network, matrix product state, semigroup, tensor contraction.