Matriksia

Matriksia is a theoretical framework described as a generalization of matrix algebra for handling multi-relational data and processes. It envisions a set of elements called matrikses, together with two binary operations: addition, making the set an abelian group, and a product that generalizes matrix multiplication. The product may be indexed by relation type and can be non-commutative or non-associative if the axioms are chosen accordingly. Many formulations include scalar multiplication by elements of a base field and an identity matriksia, giving a structure that resembles an algebra over that field.

A matriksia of finite order is often represented as an array-like object with indices drawn from a finite set. Composition rules are defined so that the shape of the output aligns with the input shapes, allowing consecutive products to be well defined. In common variants, matrikses can carry extra structure, such as a row-and-column partition, tensor blocks, or relation labels to indicate how entries should interact during multiplication.

Key properties include closure under addition and product, distributivity of the product over addition, and a trace-like mapping that yields a scalar. Depending on the chosen axioms, matriksia may conserve dimensions, support dimension-changing operations, or admit inverses for certain elements.

Applications are primarily theoretical and methodological, including modeling multi-relational networks, tensor-based data fusion, and abstractions of computation that blend linear and non-linear steps. As a concept, matriksia is used in thought experiments and in fictional or speculative contexts to explore generalized linear-algebra ideas beyond standard matrices.

See also: Matrix, Tensor, Hypermatrix, Algebra, Category theory.