The problem of determining whether the elementary theory of free algebras of a given variety is decidable has attracted significant attention. For instance, a major problem raised by Tarski asked whether any two free groups with two or more free generators are elementarily equivalent and whether the theory of free groups is decidable. Both problems were shown to have an affirmative answer. Other significant examples derive from the study of free Heyting algebras. On the one hand, the elementary theory of the countably-generated free Heyting algebra is undecidable [4]. On the other hand, its universal theory is decidable [5] and, although not finitely axiomatizable, was given an independent infinite axiomatization in [3].

In this talk, I will focus on pseudocomplemented distributive lattices, the implication-free subreducts of Heyting algebras, which can be viewed as the algebraic counterpart of the (∧,∨,¬,⊤)-fragment of the intuitionistic propositional calculus. While the elementary theory of all finitely generated free pseudocomplemented distributive lattices is undecidable [2], we show that their universal theory is decidable and give a recursive axiomatization. The proof relies on Priestley duality and a dual description of free pseudocomplemented distributive lattices, together with general principles applying to finitely axiomatizable and locally finite varieties of finite type. A key ingredient is the notion of free skeleton, which dually characterizes the finite pseudocomplemented distributive lattices that embed into a free algebra.

This talk is based on the paper [1], written in collaboration with T. Moraschini.

References:
[1] L. Carai and T. Moraschini, On the universal theory of the free pseudocomplemented distributive lattice, J. Algebra 682 (2025), 634-671
[2] P.M. Idziak, Undecidability of free pseudocomplemented distributive lattices, Rep. Math. Log. 21 (1987) 97--100.
[3] E. Jeřábek, Independent bases of admissible rules, Log. J. IGPL 16 (3) (2008) 249--267.
[4] V.V. Rybakov, Elementary theories of free topo-Boolean and pseudo-Boolean algebras, Math. Notes Acad. Sci. USSR 37 (1985) 435--438.
[5] V.V. Rybakov, The universal theory of the free pseudo-Boolean algebra F_ω(H) in the signature extended by constants for free generators, in: Proceedings of the International Conference on Algebra, Part 3, Novosibirsk, 1989, in: Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 645--656.