Quasivarieties of algebras whose compact relative congruences are principal

A quasivariety \(\mathfrak N\) is called relative congruence principal if, for every algebra \(A\in \mathfrak N\), every compact \(\mathfrak N\)-congruence on A is a principal \(\mathfrak N\)-congruence. We characterize relative congruence principal quasivarieties in terms of one identity and two quasi-identities. We will use the characterization to show that there exists a continuum of relative congruence principal quasivarieties of algebras of a signature \(\sigma \), provided \(\sigma \) contains at least one operation of arity greater than 1. Several examples are provided.