CAD and creativity: does the computer really help?

We are frequently told by its exponents that computeraided design (CAD) liberates designers and gives them new ways of envisioning their work, but is this really true? CAD in architecture is examined to see to what extent it has enhanced creativity in design. This is partly done by applying a test of creativity advanced by contemporary architect Herman Hertzberger. In this analysis, CAD is found somewhat wanting, and some suggestions are made as to why this might be so

Tarski monoids: Matui's spatial realization theorem

We introduce a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class of etale topological groupoids under a non-commutative generalization of classical Stone duality and, significantly, they arise naturally in the theory of dynamical systems as developed by Matui. We are thereby able to reinterpret a theorem of Matui on a class of \'etale groupoids as an equivalent theorem about a class of Tarski monoids: two simple Tarski monoids are isomorphic if and only if their groups of units are isomorphic. The inverse monoids in question may also be viewed as countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the classical Thompson groups.Comment: arXiv admin note: text overlap with arXiv:1407.147

Dequantisation of the Dirac Monopole

Using a sheaf-theoretic extension of conventional principal bundle theory, the Dirac monopole is formulated as a spherically symmetric model free of singularities outside the origin such that the charge may assume arbitrary real values. For integral charges, the construction effectively coincides with the usual model. Spin structures and Dirac operators are also generalised by the same technique.Comment: 22 pages. Version to appear in Proc. R. Soc. London

Distributive inverse semigroups and non-commutative Stone dualities

We develop the theory of distributive inverse semigroups as the analogue of distributive lattices without top element and prove that they are in a duality with those etale groupoids having a spectral space of identities, where our spectral spaces are not necessarily compact. We prove that Boolean inverse semigroups can be characterized as those distributive inverse semigroups in which every prime filter is an ultrafilter; we also provide a topological characterization in terms of Hausdorffness. We extend the notion of the patch topology to distributive inverse semigroups and prove that every distributive inverse semigroup has a Booleanization. As applications of this result, we give a new interpretation of Paterson's universal groupoid of an inverse semigroup and by developing the theory of what we call tight coverages, we also provide a conceptual foundation for Exel's tight groupoid.Comment: arXiv admin note: substantial text overlap with arXiv:1107.551