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.

令\({\mathcal {A}}\)是具有\(I, A \in {\mathcal {A}}\)的代数类。我们在 A的所有子代数\(S_{{\mathcal {A}}}(I,A)\)形式的格中解释格理论“严格满足不可约/覆盖”情况\(B < C\)包含I，我们称这样的\(B < C\)为最小适当扩展(mpe)，并表明这意味着B在\(S_{{\mathcal {A}}}(I,A)\)中是最大的因为不包含一些\(r \in A\)而C是由B和r生成的。对于班级\({\mathcal {G}}\)组，我们使用不变量确定\(S_{{\mathcal {G}}}(\{0\},{\mathbb {Q}})\)中的 mpe Beaumont 和 Zuckerman 的研究并表明这些（加上 Hamel 基的利用）决定了\(S_{{\mathcal {G}}}(\{0\},{\mathbb {R}})\)中的 mpe 。最后，我们表明后者在\(\mathbf {W}^{*}\)中产生了一些（不是全部）最小适当的基本扩展，这是具有强序单元的阿基米德\(\ell \)组的类别和保持单位的\(\ell \) -群同态。