# On the implication $T_{0} \Rightarrow T_{3 \frac{1}{2}}$ for some topological protomodular algebras

## TL;DR

This paper introduces right-cancellable topological protomodular algebras and proves that such algebras satisfying the T0 separation axiom are necessarily completely regular, linking algebraic properties with topological regularity.

## Contribution

It establishes a new connection between algebraic cancellability and topological regularity in protomodular algebras, expanding understanding of their structure.

## Key findings

- Right-cancellable topological protomodular algebras are completely regular under T0 separation.
- The notion of right-cancellable protomodular algebra is introduced.
- A key implication T0 ⇒ T3 1/2 is demonstrated for these algebras.

## Abstract

The notion of a right-cancellable protomodular algebra is introduced. It is proved that a right-cancellable topological protomodular algebra that satisfies the separation axiom $T_{0}$ is completely regular.