Relational Semigroups and Object-Free Categories

Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories

This paper is a sequel to arXiv:0902.2355 and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance.

Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories

This paper is a sequel to arXiv:0902.2355 and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance.

Demonic Lattices and Semilattices in Relational Semigroups with Ordinary Composition

Relation algebra and its reducts provide us with a strong tool for reasoning about nondeterministic programs and their partial correctness. Demonic calculus, introduced to model the behaviour of a machine where the demon is in control of nondeterminism, has also provided us with an extension of that reasoning to total correctness. We formalise the framework for relational reasoning about total correctness in nondeterministic programs using semigroups with ordinary composition and demonic lattice operations. We show that the class of representable demonic join semigroups is not finitely axiomat…

A Variety Theorem for Relational Universal Algebra

We develop an analogue of universal algebra in which generating symbols are interpreted as relations. We prove a variety theorem for these relational algebraic theories, in which we find that their categories of models are precisely the definable categories. The syntax of our relational algebraic theories is string-diagrammatic, and can be seen as an extension of the usual term syntax for algebraic theories.

从分配幺半群范畴推导程序逻辑

Deriving Program Logics from Distributive Monoidal Categories

该研究从命令式范畴的公理推导多种程序逻辑，其规则遵循该范畴公理，且推导的防护命令代数推广了带测试的防护Kleene代数。

Cubical Categories for Higher-Dimensional Parametricity

Reynolds' theory of relational parametricity formalizes parametric polymorphism for System F, thus capturing the idea that polymorphically typed System F programs always map related inputs to related results. This paper shows that Reynolds' theory can be seen as the instantiation at dimension 1 of a theory of relational parametricity for System F that holds at all higher dimensions, including infinite dimension. This theory is formulated in terms of the new notion of a p-dimensional cubical category, which we use to define a p-dimensional parametric model of System F for any p, where p is a na…