Locally finite varieties of Heyting algebras of width 2
Thijs Benjamins
Abstract:
In this thesis we investigate locally finite varieties of Heyting algebras of width 2. We show that a variety of width 2 is locally finite if and only if its 2-generated members are finite. This confirms a conjecture of G. Bezhanishvili and R. Grigolia (2005) for varieties of width 2. We prove this result by showing that non-locally finite varieties of width 2 contain the Rieger-Nishimura lattice with a new bottom element, which is a 2-generated infinite Heyting algebra. We also prove that this characterisation does not carry through to the case of varieties of width 3.
Using this characterisation we show that the variety generated by the Rieger-Nishimura lattice with a new bottom element is the only pre-locally finite variety of Heyting algebras of width 2. As a consequence, we obtain that local finiteness is decidable for finitely axiomatisable varieties of width 2. Finally, we show that there are continua of both locally finite and non-locally finite varieties of width 2.