On semi-vector spaces and semi-algebras

TL;DR

This paper explores the mathematical structure of semi-vector spaces and semi-algebras over nonnegative real numbers, introducing new concepts like eigenvalues and examining their topological properties, with applications in fuzzy set theory.

Contribution

It provides new theoretical results on semi-vector spaces and semi-algebras, including eigenvalues, topological properties, and constructions from semi-metrics and semi-inner products.

Findings

Eigenvalues and eigenvectors of semi-linear operators are characterized.

Topological properties such as completeness and separability are analyzed.

New families of semi-vector spaces are constructed from semi-metric and semi-inner product structures.

Abstract

It is well-known that the theories of semi-vector spaces and semi-algebras -- which were not much studied over time -- are utilized/applied in Fuzzy Set Theory in order to obtain extensions of the concept of fuzzy numbers as well as to provide new mathematical tools to investigate properties and new results on fuzzy systems. In this paper we investigate the theory of semi-vector spaces over the semi-field of nonnegative real numbers ${\mathbb R}_{0}^{+}$. We prove several results concerning semi-vector spaces and semi-linear transformations. Moreover, we introduce in the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing in some cases how to compute them. Topological properties of semi-vector spaces such as completeness and separability are also investigated. New families of semi-vector spaces derived from semi-metric, semi-norm, semi-inner product, among others are exhibited. Additionally, some results on semi-algebras are presented.