Derivations on algebras of one-point compactification of affine semigroups

TL;DR

This paper investigates derivations on the semigroup algebra of the one-point compactification of affine semigroups, linking their structure to derivations on the original semigroup algebra within a metrizable topology.

Contribution

It provides a detailed analysis of derivations on the algebra of the one-point compactification of affine semigroups, extending understanding of their algebraic and topological properties.

Findings

Characterization of derivations on the compactified semigroup algebra

Relation between derivations on the compactified and original semigroup algebras

Insights into the topology-induced structure of derivations

Abstract

For any affine semigroup $S$ the set $S\cup\{\infty\}$ has a natural structure of semigroup, additionally if $S$ is endowed with the discrete topology, the semigroup $S\cup\{\infty\}$ can be studied as the one-point compactification of $S$. In this article we study the derivations on the semigroup algebra $\mathbb{C}[S\cup\{\infty\}]$ in relation to the derivations on the semigroup algebra $\mathbb{C}[S]$ considering the metrizable topology on $\mathbb{C}[S\cup\{\infty\}]$ induced by the one-point compactification topology of $S\cup\{\infty\}$.