Omission and Bases for Quasivarieties

Abstract

In this section we show that if $\mathcal{K}$ is a locally finite quasivariety of finite type, T is a finite algebra in $\mathcal{K}$, and α ≻ Δ in $\mathop{\mathrm{Con}}\nolimits _{\mathcal{K}}\,\mathbf{T}$, then the quasivariety $\langle \varepsilon _{\mathbf{T},\alpha }\rangle$ consists of all algebras in $\mathcal{K}$ that omit a finite list of forbidden subalgebras. A slight variation finds the quasivarieties that are minimal with respect to not being contained in $\langle \varepsilon _{\mathbf{T},\alpha }\rangle$. Both these results are in Theorem 3.4. As a consequence, subquasivarieties that are finitely based relative to $\mathcal{K}$ can be characterized by the exclusion of finitely many subalgebras (Theorem 3.10).