We prove that the category of Dedekind \(\sigma \)-complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that \({\mathbb {R}}\), regarded as a Dedekind \(\sigma \)-complete Riesz space, generates this category as a variety; further, we use this fact to obtain the even stronger result that \({\mathbb {R}}\) generates this category as a quasi-variety. Analogous results are established for the categories of (i) Dedekind \(\sigma \)-complete Riesz spaces with a weak order unit, (ii) Dedekind \(\sigma \)-complete lattice-ordered groups, and (iii) Dedekind \(\sigma \)-complete lattice-ordered groups with a weak order unit.

中文翻译：

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Dedekind $$ \ sigma $$σ-完全$$ \ ell $$ℓ-组和Riesz空间作为变体

我们证明Dedekind \（\ sigma \）-完全Riesz空间的类别是一个不定式，并且提供了一个明确的方程式公理化。实际上，我们证明了有限的许多公理足以满足Riesz空间通常的方程式公理化。我们的主要结果是\（{\ mathbb {R}} \）被视为Dedekind \（\ sigma \）完全Riesz空间，生成了这个类别。此外，我们利用这一事实获得更强的结果，即\（{\ mathbb {R}} \）将此类别生成为准品种。对于（i）具有弱阶单位的Dedekind \（\ sigma \）-完全Riesz空间，（ii）Dedekind \（\ sigma \）类别建立了类似结果-完整的晶格有序组，以及（iii）Dedekind \（\ sigma \） -具有弱序单元的完整晶格有序组。