Basis and Dimension of Exponential Vector Space

TL;DR

This paper extends the algebraic theory of vector spaces to exponential vector spaces (evs), defining basis and dimension within this framework, and explores their properties and conditions for existence.

Contribution

It introduces the concepts of basis and dimension for evs, providing necessary and sufficient conditions for their existence and analyzing their properties.

Findings

Not all evs contain a basis.

Equality of dimension is an evs property, but the converse is not.

Every evs has a subevs with all lower possible dimensions.

Abstract

Exponential vector space [shortly \emph{evs}] is an algebraic order extension of vector space in the sense that every evs contains a vector space and conversely every vector space can be embedded into such a structure. This evs structure consists of a semigroup structure, a scalar multiplication and a partial order. In this paper we have developed the concepts of basis and dimension of an evs by introducing the ideas of orderly independent set and generating set with the help of partial order and algebraic operations. We have found that like vector space, an evs does not contain basis always. We have established a necessary and sufficient condition for an evs to have a basis. It was shown that equality of dimension is an evs property but the converse is not true. We have studied the dimension of subevs and found that every evs contains a subevs with all possible lower dimensions. Lastly we have computed basis and dimension of some evs which help us to explore the theory of basis by creating counter examples in different aspects.