lineaarien

Lineaarien is a theoretical framework in mathematics and computer science that generalizes linearity through the study of linear relations rather than maps. It treats objects together with a distinguished linear relation as basic units and uses relational composition and addition to form new relations. The central idea is that many problems defined by linear constraints can be expressed relationally, enabling algebraic manipulation without committing to a particular representation.

Definitions and structure: A Lineaarien object consists of a set V equipped with a subset L of V × V called the linear relation. The relation L is considered linear if it is closed under appropriate linear combinations: for all (u,v) and (u′,v′) in L and all scalars α,β, the pair (αu + βu′, αv + βv′) is in L. The identity relation id_V and the zero relation 0_V serve as basic examples. Morphisms in Lineaarien are relation-preserving maps whose graphs align with the target relations; more generally, a morphism f: V → W is compatible with the respective relations L ⊆ V × V and M ⊆ W × W if (v, f(v)) ∈ M for all (v, f(v)) in L.

Constructions and representations: A common representation uses matrices: a matrix A defines a lineaarinen relation L_A = { (x, Ax) | x in the domain }. Linear maps between vector spaces are special cases of Lineaarien morphisms, and subspaces induce restricted relations that reflect inclusion.

Applications: The framework is used to model linear constraint systems, dataflow analyses, and optimization problems where constraints are inherently linear but need a relational, representation-agnostic treatment.

Relation to existing theories: Lineaarien connects to standard linear algebra as a special case and relates to relation algebra and certain categorical approaches that emphasize relational composition over functional maps. See also linear algebra, relation algebra, and category theory.