Minimum proper extensions in some lattices of subalgebras

Abstract

Let \({\mathcal {A}}\) be a class of algebras with \(I, A \in {\mathcal {A}}\). We interpret the lattice-theoretic “strictly meet irreducible/cover” situation \(B < C\) in lattices of the form \(S_{{\mathcal {A}}}(I,A)\) of all subalgebras of A containing I, where we call such \(B < C\) a minimum proper extension (mpe), and show that this means B is maximal in \(S_{{\mathcal {A}}}(I,A)\) for not containing some \(r \in A\) and C is generated by B and r. For the class \({\mathcal {G}}\) of groups, we determine the mpe’s in \(S_{{\mathcal {G}}}(\{0\},{\mathbb {Q}})\) using invariants of Beaumont and Zuckerman and show that these (plus utilization of a Hamel basis) determine the mpe’s in \(S_{{\mathcal {G}}}(\{0\},{\mathbb {R}})\). Finally, we show that the latter yield some (not all) of the minimum proper essential extensions in \(\mathbf {W}^{*}\), the category of Archimedean \(\ell \)-groups with strong order unit and unit-preserving \(\ell \)-group homomorphisms.