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.
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Citkin, A. On Finitely-Generated Johansson Algebras. Order 40, 371–385 (2023). https://doi.org/10.1007/s11083-022-09616-4
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DOI: https://doi.org/10.1007/s11083-022-09616-4