Principal congruences in weak Heyting algebras

Abstract

Let A be a weak Heyting algebra and let $${a, b \in A}$$ . We give a description for the congruence generated by the pair (a, b), and we use it in order to give a necessary and sufficient condition for a function $${f : A^{k} \rightarrow A}$$ to be compatible with every congruence of A. We also find conditions on a not necessarily polynomial function g(a, b) in A that imply that the function $${a \mapsto {\rm min}\{b \in A : g(a, b) \leq b}\}$$ is compatible when defined.