To each discrete left cancellative semigroup S one may associate an inverse semigroup I l (S), often called the left inverse hull of S. We show how the full and reduced C * -algebras of I l (S) are related to the full and reduced semigroup C * -algebras for S, recently introduced by Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.

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Abstract. We use a recent result by Cuntz, Echterhoff and Li about the K-theory of certain reduced C * -crossed products to describe the K-theory of C * r (S) when S is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that C * r (S) is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

With widespread, long-term historical use of plastics and the presence of microplastics in a range of new and existing products, there is rising concern about their potential impacts on freshwater ecosystems. Understanding how microplastics are transported and distributed along river systems is key to assessing impacts. Modelling the main flow dynamics, mixing, sedimentation and resuspension processes is essential for an understanding of the transport processes. We use the new, processed based, dynamic, integrated catchments (INCA) microplastics model and apply this to the whole of the freshwater catchment of the River Thames, UK, to evaluate inputs, loads and concentrations along the river system. Recent data from UK water industry studies on microplastics in effluent discharges and sewage sludge disposal has been utilised to drive the INCA microplastics model. Predicted concentrations and microplastic loads moving along the river system are shown to be significant, with a build-up of concentrations along the river, with increasing deposition on the riverbed. The potential impacts on aquatic ecosystems are evaluated and a review of policy implications is explored.

We develop the notion of independent resolutions for crossed products attached to totally disconnected dynamical systems. If such a crossed product admits an independent resolution of finite length, then its K-theory can be computed (at least in principle) by analysing the corresponding six-term exact sequences. Building on our previous paper on algebraic independent resolutions, we give a criterion for the existence of finite length independent resolutions. Moreover, we illustrate our ideas in various concrete examples.