Quasi-algebra, a special sample of quasilinear spaces

TL;DR

This paper generalizes quasilinear spaces into quasi-algebras, introduces normed quasi-algebras and quasi-homomorphisms, and explores their properties and spectra, expanding the algebraic framework for such spaces.

Contribution

It introduces the concepts of quasi-algebras, normed quasi-algebras, and quasi-homomorphisms, extending the theory of quasilinear spaces with algebraic and spectral structures.

Findings

Defined quasi-algebras and normed quasi-algebras.

Introduced quasi-homomorphisms as quasilinear operators.

Established the concept of quasi-spectrum for quasi-algebras.

Abstract

Similar to linear spaces, many examples of quasilinear spaces have a notion of multiplication of the elements. To characterising these examples, in the present paper we generalize the notion of quasilinear spaces and introduce quasi-algebras and normed quasi-algebras with the help of what was done in constructing algebras. After examining some properties of these new spaces, we introduce the concept of quasi-homomorphisms that is an instance of a quasilinear operator. Moreover since the study of spaces with algebraic structures cannot be completed without the study of spectrums, we define the concept of quasi-spectrum on quasi-algebras as we expect it to be.