The main aim of this article is to develop a categorical duality between the category of semilattices with homomorphisms and a category of certain topological spaces with certain morphisms. The principal tool to achieve this goal is the notion of irreducible filter. Then, we apply this dual equivalence to obtain a topological duality for the category of bounded lattices and lattice homomorphism. We show that our topological dualities for semilattices and lattices are natural generalizations of the duality developed by Stone for distributive lattices through spectral spaces. Finally, we obtain directly the categorical equivalence between our topological spaces and those presented for Moshier and Jipsen (Algebra Univers 71(2):109–126, 2014).

中文翻译：

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半格和格的绝对对偶

本文的主要目的是在具有同态的半格范畴和具有某些态射的某些拓扑空间范畴之间建立范畴对偶。实现这一目标的主要工具是不可约滤波器的概念。然后，我们应用这个对偶等价来获得有界格和格同态范畴的拓扑对偶。我们证明了我们的半格和格的拓扑对偶性是 Stone 为通过光谱空间的分布格子开发的对偶性的自然推广。最后，我们直接获得了我们的拓扑空间与为 Moshier 和 Jipsen 呈现的拓扑空间之间的分类等价性（代数大学 71（2）：109-126，2014）。