Ω-vector spaces

Abstract

In this work, we introduce the concept of Ω-vector spaces, extending the framework of Ω-algebras by incorporating a vector space structure over a field. These structures are defined over a complete lattice and equipped with an Ω-valued equality, which replaces the classical relation of being equal. We provide an equivalent characterization of Ω-vector spaces via cut-quotient structures and prove that each cut induces a classical vector space. Furthermore, we introduce the notion of Ω-vector subspaces and investigate the lattice-theoretic properties of their collection, including intersections and sums. Finally, we show an application for approximately solving systems of linear equations in this context. Several examples illustrate the theory, highlighting the algebraic richness and structural consistency of Ω-vector spaces.