We introduce compactly finite MV-algebras and continuous MV-algebras. We also investigate pro-compactly finite MV-algebras, which are the MV-algebras that are inverse limits of systems of compactly finite MV-algebras. We obtain that continuous MV-algebras as well as pro-compactly finite MV-algebras coincide with compact Hausdorff MV-algebras. In addition, further categorical properties of compact Hausdorff MV-algebras such as co-completeness, injective objects, (co)-Hopfian objects are considered and studied.

中文翻译：

---

亲紧有限MV-代数

我们介绍了紧凑有限MV-代数和连续MV-代数。我们还研究了亲紧有限 MV 代数，它们是紧有限 MV 代数系统的逆极限的 MV 代数。我们获得连续 MV 代数以及亲紧有限 MV 代数与紧致 Hausdorff MV 代数一致。此外，还考虑并研究了紧致 Hausdorff MV-代数的进一步分类属性，例如共完备性、单射对象、(co)-Hopfian 对象。