# Tight and cover-to-join representations of semilattices and inverse semigroups

## TL;DR

This paper explores the relationship between tight and cover-to-join representations of semilattices and inverse semigroups, demonstrating their equivalence under certain conditions, which simplifies the construction of universal objects.

## Contribution

It shows that a slight extension of tight representations and proper co-domain selection make tight and cover-to-join representations equivalent, enabling interchangeable use in universal constructions.

## Key findings

- Tight and cover-to-join representations are equivalent under specific extensions.
- The equivalence allows substitution in the construction of universal objects.
- The study clarifies the relationship between different representation notions in algebraic structures.

## Abstract

We discuss the relationship between tight and cover-to-join representations of semilattices and inverse semigroups, showing that a slight extension of the former, together with an appropriate selection of co-domains, makes the two notions equivalent. As a consequence, when constructing universal objects based on them, one is allowed to substitute cover-to-join for tight and vice-versa.