论自然二重性理论的某些发展
On selected developments in the theory of natural dualities
摘要：
这是一个关于自然二元性理论的选择性发展的综述，作者有机会与他的外国同事一起取得了一些突破，并推动了理论的发展。它的目的是作为作者的反思他的作品自然对偶在牛津和墨尔本在二十年期间1993年至2012年（之前他的注意力与同事在泛代数和格理论已完全集中在理论的典型扩展和理论的bilattices）。这也意味着，作为一个剩余的主要问题的理论的自然对偶性，对偶性问题和决定性问题的对偶性，仍然是开放的。自然对偶理论（Theory of natural dualities）是代数拟簇的一般理论，它推广了“经典”对偶，如布尔代数的Stone对偶、阿贝尔群的Pontryagin对偶、分配格的Priestley对偶和半格的Hofmann-Mislove-Stralka对偶。我们介绍了该理论的背景，然后说明它的应用，我们的研究蕴涵问题，问题的Endodualisability与Endoprimality，然后一个著名的充分与强问题的相关发展。
英文摘要：
This is a survey on selected developments in the theory of natural dualities where the author had the opportunity to make with his foreign colleagues several breakthroughs and move the theory forward. It is aimed as author's reflection on his works on the natural dualities in Oxford and Melbourne over the period of twenty years 1993-2012 (before his attention with the colleagues in universal algebra and lattice theory has been fully focused on the theory of canonical extensions and the theory of bilattices). It is also meant as a remainder that the main problems of the theory of natural dualities, Dualisability Problem and Decidability Problem for Dualisability, remain still open. Theory of natural dualities is a general theory for quasi-varieties of algebras that generalizes `classical' dualities such as Stone duality for Boolean algebras, Pontryagin duality for abelian groups, Priestley duality for distributive lattices, and Hofmann-Mislove-Stralka duality for semilattices. We present a brief background of the theory and then illustrate its applications on our study of Entailment Problem, Problem of Endodualisability versus Endoprimality and then a famous Full versus Strong Problem with related developments.
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