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

Ranks of ideals in inverse semigroups of difunctional binary relations

The set Dn of all difunctional relations on an n element set is an inverse semigroup under a variation of the usual composition operation. We solve an open problem of Kudryavtseva and Maltcev (Publ Math Debrecen 78(2):253–282, 2011), which asks: What is the rank (smallest size of a generating set) of Dn? Specifically, we show that the rank of Dn is B(n)+n, where B(n) is the nth Bell number. We also give the rank of an arbitrary ideal of Dn. Although Dn bears many similarities with families such as the full transformation semigroups and symmetric inverse semigroups (all contain the symmetric group and have a chain of J-classes), we note that the fast growth of rank(Dn) as a function of n is a property not shared with these other families

Patterned Microstructure Fabrication: Polyelectrolyte Complexes vs Polyelectrolyte Multilayers

The author would like to thank the following funding agencies which partly supported this work: National Nature Science Foundation of China (grant No 21503058) (J.F.), Chinese Scholarship Council (CSC) Grant No. 201406120038, Queen Mary University of London. Startup grant of HIT for J.F. The authors thank Dr. Dayo Addebayo for proofreading the manuscript. This work was supported by National Nature Science Foundation of China (grant No 21503058) (J.F.), Chinese Scholarship Council (CSC) Grant No. 201406120038 (M.G.), Queen Mary University of London (M.G.). Startup grant of HIT for J.F., Russian Governmental Program “Nauka”, № 1.1658.2016; 4002 (V.K.)

The principal bundles over an inverse semigroup

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra. For the topos of sets, we show that torsion-free functors on Loganathan's category $L(S)$ of an inverse semigroup $S$ are equivalent to a special class of non-strict representations of $S$, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of $S$. We describe the correspondence between directed and pullback preserving functors on $L(S)$ and transitive and effective representations of $S$, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation of an inverse semigroup in the topos of sheaves ${\mathsf{Sh}}(X)$ on a topological space $X$ as well as outline an approach on how to define such a representation in an arbitrary topos. We prove that the category of filtered functors from $L(S)$ to the topos ${\mathsf{Sh}}(X)$ is equivalent to the category of universal representations of $S$ in ${\mathsf{Sh}}(X)$