On Finitely-Generated Johansson Algebras

Abstract

Kuznetsov’s Theorem about finitely-generated Heyting algebras has been extended to Johansson algebras in the following way: if $\mathbf {A} = (\mathsf {A}; \wedge ,\vee ,\rightarrow ,\mathbf {1},\mathbf {f})$ is a Johansson algebra, by the rank of element $\mathsf {a} \in \mathsf {A}$, we understand the cardinality of the set $\{\mathsf {b} \in \mathbf {A} | \mathsf {b} \le \mathsf {a} \}$, and we prove that if $\mathbf {A}$ is finitely-generated, then for each element $\mathsf {a} \in \mathsf {A}$ of a finite rank, the algebra $(\{\mathsf {b} \in \mathbf {A} | \mathsf {a} \le \mathsf {b}\}; \wedge , \vee , \rightarrow , \mathbf {1}, \mathbf {f}')$, where $\mathbf {f}' = \mathsf {a} \vee \mathbf {f}$, is finitely generated as well.