Generalised states: a multi-sorted algebraic approach to probability

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• 作者：Tomáš Kroupa ; Vincenzo Marra
• 关键词：Probability ; State ; Multi ; sorted algebra ; Heterogeneous algebra ; Boolean algebra ; MV ; algebra ; Lattice ; ordered Abelian group
• 刊名：Soft Computing
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：21
• 期：1
• 页码：57-67
• 全文大小：
• 刊物类别：Engineering
• 刊物主题：Computational Intelligence; Artificial Intelligence (incl. Robotics); Mathematical Logic and Foundations; Control, Robotics, Mechatronics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1433-7479
• 卷排序：21

文摘

We introduce a generalised notion of state as an additive map from a Boolean algebra of events to an arbitrary MV-algebra. Generalised states become unary operations in two-sorted algebraic structures that we call state algebras. Since these, as we show, form an equationally defined class of algebras, universal-algebraic techniques apply. We discuss free state algebras, their geometric representation, and their connection with the theory of affine representations of lattice groups.