Exponential Algebra: As an Extension of Topological Algebra

In this book, we have introduced the concept of `\textit{exponential algebra}' (in short \textit{ealg}) by defining an internal multiplication on an evs over some field $ K $. We have explained that the concept of exponential algebra can be thought of as a generalisation of `algebra' in the sense that every exponential algebra contains an algebra; conversely, any algebra can be embedded into an exponential algebra. We develop a quotient structure on an ealg $X$ over some field $K$ by using the concept of congruence and topologise it. We introduce the concept of \emph{ideal}, \emph{semiideal} and \emph{maximal ideal} of an ealg. We have shown that the hyperspace $\com{\X}{}$ (the set of all nonempty compact subsets of a Hausdorff topological algebra $\X$) is a topological exponential algebra over the field $\K$ of real or complex. We explore the function spaces in light of exponential algebra. It has been shown that the space of positive measures $\mathscr M(G)$ on a locally compact Housdorff topological group $G$, which are finite on each compact subset of $G$ is a topological ealg. Finally, we found a topological ealg with the help of Hausdorff metric.