2026/07/21 by Tomasz Kowalski, Katarzyna Słomczyńska
Mathematics · #math.LO
We give a generic construction of the n-generated free Hilbert algebras and Brouwerian semilattices by a uniform method applicable to any variety of Hilbert algebras or Brouwerian semilattices, as long as a manageable description of subdirectly irreducible algebras in that variety is available. A slight modification yields analogous constructions for varieties of Hilbert algebras and Brouwerian semilattices with zero. As examples we construct free algebras in varieties of bounded height and of bounded width; we also find a closed formula for the free spectrum of linear Hilbert algebras. Finally, we obtain a few results on structural completeness of varieties of Hilbert algebras with zero, in particular, we give a sufficient condition for a quasi-equation to be equivalent to an equation.