Information algebras arise from the idea that information comes in pieces which can be aggregated or combined into new pieces, that information refers to questions and that from any piece of information, the part relevant to a given question can be extracted. This leads to a certain type of algebraic structures, basically semilattices endowed with with additional unary operations. These operations essentially are (dual) existential quantifiers on the underlying semilattice. The archetypical instances of such algebras are semilattices of subsets of some universe, together with the saturation operators associated with a family of equivalence relations on this universe. Such algebras will be called {\em set algebras} in our context. Our first result is a basic representation theorem: Every abstract information algebra is isomorphic to a set algebra. When it comes to combine pieces of information, the idea to model the logical connectives {\em and}, {\em or} or {\em not} is quite natural. Accordingly, we are especially interested in information algebras where the underlying semilattice is a lattice, typically distributive or even Boolean. A major part of this paper is therefore devoted to developing explicitly a full-fledged natural duality theory extending Stone resp. Priestley duality in a suitable way in order to take into account the additional operations.

中文翻译：

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可交换信息代数：表示和对偶理论

信息代数的产生源于以下思想：信息分成多个部分，这些部分可以聚合或组合成新的部分；信息是指问题；从任何信息中，都可以提取与给定问题相关的部分。这导致了某种类型的代数结构，基本上是半格具有附加的一元运算。这些操作本质上是底层半格上的（双）存在量词。这种代数的原型实例是某些宇宙的子集的半格，以及与该宇宙上的等价关系族相关的饱和算符。在我们的上下文中，此类代数将称为{\ em set algebras}。我们的第一个结果是一个基本的表示定理：每个抽象信息代数与集合代数是同构的。在组合信息时，对逻辑连接词{\ em和}，{\ em or}或{\ em not}建模的想法是很自然的。因此，我们对信息代数特别感兴趣，在信息代数中，下半格是一个格，通常是分布型甚至布尔型。因此，本文的主要内容致力于显式地发展一种完善的自然对偶理论，以扩展Stone的观点。以适当的方式进行Priestley对偶，以考虑到其他操作。因此，本文的主要内容致力于显式地发展一种完善的自然对偶理论，以扩展Stone的观点。以适当的方式进行Priestley对偶，以考虑到其他操作。因此，本文的主要内容致力于显式地发展一种完善的自然对偶理论，以扩展Stone的观点。以适当的方式进行Priestley对偶，以考虑到其他操作。