# The categorical equivalence between disjunctive sequent calculi and algebraic L-domains

## TL;DR

This paper demonstrates a categorical equivalence between disjunctive sequent calculi and algebraic L-domains, using logical states and consequence relations to connect syntactic and semantic structures.

## Contribution

It introduces a purely syntactic representation of algebraic L-domains and establishes a categorical equivalence with disjunctive sequent calculi via consequence relations.

## Key findings

- Categorical equivalence between disjunctive sequent calculi and algebraic L-domains.
- Logical states as a bridge between syntax and semantics.
- Representation of Scott-continuous functions through consequence relations.

## Abstract

This paper establishes a purely syntactic representation for the category of algebraic L-domains with Scott-continuous functions as morphisms. The central tool used here is the notion of logical states, which builds a bridge between disjunctive sequent calculi and algebraic L-domains. To capture Scott-continuous functions between algebraic L-domains, the notion of consequence relations between disjunctive sequent calculi is also introduced. It is shown that the category of disjunctive sequent calculi with consequence relations as morphisms is categorical equivalent to that of algebraic L-domains with Scott-continuous functions as morphisms.