Research Statement

My current research focuses broadly on Heyting algebras and closely related structures. In them I am particularly interested in the study of technical questions on the structure of the category of Heyting algebras, such as providing descriptions of ordered compactifications of ordered topological spaces, or describing free objects in these and associated categories; and properties of classes of Heyting algebras, such as the study of lattices of varieties and quasivarieties, the distribution of properties like structural completeness, interpolation or tabularity; as well as the study non-standard rules, such as Pi-2-rules, in their universal-algebraic aspects and their computational properties.

I am broadly interested in universal algebra, model theory and category theory, and especially their interaction.

In a previous academic life I was a researcher in sociology of culture. I worked on policy analysis and briefly on agent-based modelling. You can find most of my work related to that in this link.

Published Articles

Π2-rule systems and inductive classes of Gödel algebras

Authors: Rodrigo Nicolau Almeida

Annals of Pure and Applied Logic, Volume 176, Issue 4, April 2025 • 2025

In this paper we present a general theory of Π2-rules for systems of intuitionistic and modal logic. We introduce the notions of -rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools.

Polyatomic logics and generalized Blok–Esakia theory

Authors: Rodrigo Nicolau Almeida

Journal of Logic and Computation, Volume 34, Issue 5, July 2024, Pages 887–935 • 2023

This paper presents a novel concept of a polyatomic logic and initiates its systematic study.

Accepted Articles

Unification with Simple Variable Restrictions and Admissibility of Pi2-rules

Authors: Rodrigo Nicolau Almeida, Silvio Ghilardi

• Accepted at AIML 2024 Proceedings

We develop a method to recognize admissibility of Π2-rules, relating this problem to a specific instance of the unification problem with linear constants restriction, called here unification with simple variable restriction. It is shown that for logical systems enjoying an appropriate algebraic semantics and a finite approximation of left uniform interpolation, this unification with simple variable restriction can be reduced to standard unification. As a corollary, we obtain the decidability of admissibility of Π2-rules for many logical systems.

A Coalgebraic Semantics for Intuitionistic Modal Logic

Authors: Rodrigo Nicolau Almeida, Nick Bezhanishvili

• Accepted at AIML 2024 Proceedings

We give a new coalgebraic semantics for intuitionistic modal logic. In particular, we provide a coalgebraic representation of intuitionistic descriptive modal frames and of intuitonistic modal Kripke frames based on image-finite posets.

Submitted Articles

Maximality Principles in Modal Logic and the Axiom of Choice

Authors: Rodrigo Nicolau Almeida, Guram Bezhanishvili

• Submitted on 18 Dec 2024

We investigate the set-theoretic strength of several maximality principles that play an important role in the study of modal and intuitionistic logics. We focus on the well-known Fine and Esakia maximality principles, present two formulations of each, and show that the stronger formulations are equivalent to the Axiom of Choice (AC), while the weaker ones to the Boolean Prime Ideal Theorem (BPI).

Colimits of Heyting Algebras through Esakia Duality

Authors: Rodrigo Nicolau Almeida

• Submitted on 12 Feb 2024 (v1), last revised 13 Nov 2024 (this version, v3)

We generalize the construction, due to Ghilardi, of the free Heyting algebra generated by a finite distributive lattice, to the case of arbitrary distributive lattices. Categorically, this provides an explicit construction of a left adjoint to the inclusion of Heyting algebras in the category of distributive lattices.