A Representation Theorem and Applications to Topological Groups

A Representation Theorem and Applications

We introduce a set of transformations on the set of all probability distributions over a finite state space, and show that these transformations are the only ones that preserve certain elementary probabilistic relationships. This result provides a new perspective on a variety of probabilistic inference problems in which invariance considerations play a role. Two particular applications we consi...

A Stone’s Representation Theorem and Some Applications

In this paper, we prove the following form of Stone’s representation theorem: Let ∑ be a σ -algebra of subsets of a set X . Then there exists a totally disconnected compact Hausdorff space K for which ( ∑ ,∪,∩) and (C(K),∪,∩) , where C(K) denotes the set of all clopen subsets of K , are isomorphic as Boolean algebras. Furthermore, by defining appropriate joins and meets of countable families in...

Representation Theory of Central Topological Groups

A note on quasi irresolute topological groups

In this study, we investigate the further properties of quasi irresolute topological groups defined in [20]. We show that if a group homomorphism f between quasi irresolute topological groups is irresolute at $e_G$, then $f$ is irresolute on $G$. Later we prove that in a semi-connected quasi irresolute topological group $(G,*,tau )$, if $V$ is any symmetric semi-open neighborhood of $e_G$, then...

A Topological Representation Theorem for tropical oriented matroids

Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes – in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Not every oriented matroi...