Eilenberg--Moore algebras for stochastic relations
type: Item
instance of: scholarly article
title: Eilenberg--Moore algebras for stochastic relations
zbMATH Open document ID: 1116.18002
published in: Information and Computation
publication date: 2007-01-22T00:00:00Z
review text: A stochastic relation between two Polish spaces is a Kleisli morphism for the so-called Giry monad [\textit{M. Giry}, ``A categorical approach to probability theory'', Lect. Notes Math. 915, 68--85 (1982; Zbl 0486.60034)]. Hence, the Kleisli algebras of this monad could give interesting informations on stochastic relations. For categories of algebras, the Kleisli algebras describe only the free algebras whereas the category of Eilenberg-Moore algebras of a monad contains all algebras defined by this monad and gives therefore more information about the algebraic component of stochastic relations on Polish spaces. \textit{M. Giry} (loc. cit.) investigated the functor assigning to a Polish space \(X\) the space \({\mathbf P}(X)\) of probability measures on the Borel sets \({\mathcal B}(X)\) of \(X\). Together with the mapping \(\mu_X:{\mathbf P}({\mathbf P}(X))\to{\mathbf P}(X)\) assigning to a Borel measure \(M\in{\mathbf P}({\mathbf P}(X))\) and a \(B\in{\mathcal B}(X)\) the integral \[ \mu_X(M)(B)= \int_{{\mathbf P}(X)} \tau(B)\,M(d\tau), \] the probability functor \({\mathbf P}\) defines the Giry monad. The author considers a modification of the Giry monad namely the analogous monad induced by the subprobability functor \({\mathbf S}(X)\), \(X\) a Polish space. He succeeds to characterize the Eilenberg-Moore algebras of this modified Giry monad completely; they are the positively convex structures [called ``positively convex modules'' in \textit{D. Pumplün}, J. Convex Anal. 10, 109--127 (2003; Zbl 1056.52002)] on Polish spaces with the additional property that the positively convex operations are continuous mappings. He also proves the analogous results for the original Giry monad induced by \({\mathbf P}(X)\) and gets as algebras the convex structures on Polish spaces with continuous convex operations. Since the introduction of the Giry monad in 1981 this is the first complete and canonical characterization of these algebras. With this representation of the Eilenb…
reviewed by: Dieter Pumplün
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 18C05
Mathematics Subject Classification ID: 68N30
Mathematics Subject Classification ID: 68Q65
zbMATH DE Number: 5117647
zbMATH Keywords: stochastic relations
zbMATH Keywords: stochastic powerdomain
zbMATH Keywords: Giry monad
zbMATH Keywords: Eilenberg--Moore algebras
zbMATH Keywords: computation through monads
zbMATH Keywords: convexity
zbMATH Keywords: positive convex structures
author: Ernst-Erich Doberkat
MaRDI profile type: Publication
full work available at URL: j.ic.2006.09.001
OpenAlex ID: W2092744709
cites work: Notions of computation and monads
cites work: Probability measures in topology
cites work: Q4402856
cites work: Q4955922
cites work: Q5533878
cites work: Q4836494
cites work: Q3946875
cites work: Q4422677
cites work: Q4396494
cites work: Bisimulation for labelled Markov processes
cites work: Stochastic Relations: Congruences, Bisimulations and the Hennessy--Milner Theorem
cites work: Q3203061
cites work: Q3994935
cites work: Q4954447
cites work: Nuclear and trace ideals in tensored \(^*\)-categories
cites work: Pipelines: modelling a software architecture through relations
cites work: Q4449236
cites work: Q5179962
cites work: Q3935671
DOI: 10.1016/J.IC.2006.09.001
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description: scientific article; zbMATH DE number 5117647