Linear representations of regular rings and complemented modular lattices with involution

TL;DR

This paper investigates faithful representations of regular *-rings and complemented modular lattices with involution in orthosymmetric sesquilinear spaces, analyzing their class correspondences and structural properties within Universal Algebra.

Contribution

It establishes a detailed correspondence between classes of spaces and classes of representable algebraic structures, including closure properties and conditions for a one-to-one correspondence.

Findings

Classes of spaces closed under ultraproducts and finite dimensional subspaces have their associated classes closed under subalgebras, homomorphic images, and ultraproducts.

Finite dimensional spaces generate the classes of representable structures.

Under certain restrictions, a one-to-one correspondence between classes of spaces and algebraic structures is achieved.

Abstract

Faithful representations of regular $\ast$-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between classes of spaces and classes of representables is analyzed; for a class of spaces which is closed under ultraproducts and non-degenerate finite dimensional subspaces, the latter are shown to be closed under complemented [regular] subalgebras, homomorphic images, and ultraproducts and being generated by those members which are associated with finite dimensional spaces. Under natural restrictions, this is refined to a $1$-$1$-correspondence between the two types of classes.