Investigating Push-out Properties in the Category of Riesz Modules: A Study on Denotational Semantics in probabilistic Programming Languages

This paper explores the concept of Riesz spaces and their application in computer probabilistic programming languages. Riesz spaces, derived from functional analysis theory, are algebraic structures used to handle mathematical spaces such as Hilbert Spaces and Sobolev Spaces. The study investigates the relationship between Riesz spaces and probabilistic programming languages, focusing on their shared mathematical structures and the utilization of category theory. The research demonstrates how Riesz spaces, with their algebraic structures and lattice orders, are well-suited for representing and processing complex mathematical structures in probabilistic programming languages. The findings highlight the importance of Riesz modules, which are Riesz spaces on the left R-module, in constructing semantic models and facilitating mathematical proofs and logical reasoning in probabilistic programming languages. Additionally, the paper examines the properties of push-out in the category of Riesz modules and establishes its existence and uniqueness. Overall, this study sheds light on the role of Riesz spaces in addressing complex mathematical problems within functional programming languages.