A categorical description of simple Beth companions

Abstract: A pp expansion of a quasivariety $\mathsf{K}$ is said to be simple when it is of the form $\mathsf{K}[\mathscr{L}_\mathcal{F}]$. For instance, when $\mathsf{K}$ has the amalgamation property, all its pp expansions are simple. It is shown that the simple pp expansions of a quasivariety $\mathsf{K}$ coincide with the quasivarieties $\mathsf{M}$ for which the forgetful functor $U \colon \mathsf{M} \to \mathsf{K}$ is well defined and induces an isomorphism from $\mathsf{M}$ to a mono-reflective subcategory of $\mathsf{K}$. As a consequence, if a quasivariety $\mathsf{K}$ possesses a simple Beth companion $\mathsf{M}$, then $\mathsf{M}$ is the unique (up to term equivalence) quasivariety whose monomorphisms are regular that, moreover, satisfy the categorical description of simple pp expansions of $\mathsf{K}$ given above.