Pughmatrix

Pughmatrix is an algebraic concept that generalizes the classical matrix by embedding contextual information into each entry. In a pughmatrix, every matrix entry carries a weight and an auxiliary token called a pug, which records a short-lived state, provenance, or layer-specific context. The idea is to extend ordinary matrix algebra to handle data whose meaning can depend on transient context without leaving the algebraic framework.

A pughmatrix is defined over a chosen weight semiring (for example, a set with two operations resembling addition and multiplication, such as ⊕ and ⊗). Each entry is a pair (w, p) consisting of a weight w from the semiring and a pug p from a separate tag set. Matrix addition combines weights with the semiring operation ⊕ and merges pugs through a context-merge rule. Matrix multiplication proceeds as standard, but when combining paths, weights are multiplied via ⊗ and the associated pugs are merged through a context function that is associative. An identity pughmatrix has ones on the diagonal paired with an identity pug, and zeros off the diagonal with a neutral pug.

Conceptually, pughmatrices are used to model systems where data carry provenance or transient context. They appear in theoretical investigations of context-aware linear algebra and weighted automata, where the pug tokens help track how context evolves along computation or transmission. They also offer a way to study multi-layer or multi-criteria structures within a single algebraic object, connecting ideas from matrices, semirings, and category-theoretic formulations of computation.

See also matrices, semirings, weighted automata, and category theory.