On Kiselman quotients of 0-Hecke monoids

Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify these monoids up to isomorphism, determine their idempotents and show that they are $\mathcal{J}$-trivial. For type $A$ we show that Catalan numbers appear as the maximal cardinality of our monoids, in which case the corresponding monoid is isomorphic to the monoid of all order-preserving and order-decreasing total transformations on a finite chain. We construct various representations of these monoids by matrices, total transformations and binary relations. Motivated by these results, with a mixed graph we associate a monoid, which we call a Hecke-Kiselman monoid, and classify such monoids up to isomorphism. Both Kiselman semigroups and Kiselman quotients of 0-Hecke monoids are natural examples of Hecke-Kiselman monoids.Comment: 14 pages; International Electronic Journal of Algebra, 201

L\'evy Processes on Quantum Permutation Groups

We describe basic motivations behind quantum or noncommutative probability, introduce quantum L\'evy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups. The first half of this paper is a survey on quantum probability, compact quantum groups, and L\'evy processes on compact quantum groups. In the second half the theory is applied to quantum permutations groups. Explicit examples are constructed and certain classes of such L\'evy processes are classified.Comment: 60 page

Minsky machines and algorithmic problems

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.Comment: 19 page

Characterizing groupoid C*-algebras of non-Hausdorff \'etale groupoids

Given a non-necessarily Hausdorff, topologically free, twisted etale groupoid $(G, L)$, we consider its "essential groupoid C*-algebra", denoted $C^*_{ess}(G, L)$, obtained by completing $C_c(G, L)$ with the smallest among all C*-seminorms coinciding with the uniform norm on $C_c(G^0)$. The inclusion of C*-algebras $(C_0(G^0), C^*_{ess}(G, L))$ is then proven to satisfy a list of properties characterizing it as what we call a "weak Cartan inclusion". We then prove that every weak Cartan inclusion $(A, B)$, with $B$ separable, is modeled by a topologically free, twisted etale groupoid, as above. In another main result we give a necessary and sufficient condition for an inclusion of C*-algebras $(A, B)$ to be modeled by a twisted etale groupoid based on the notion of "canonical states". A simplicity criterion for $C^*_{ess}(G, L)$ is proven and many examples are provided.Comment: New references and a new main result characterizing arbitrary twisted etale groupoid C*-algebras were added. The title was changed to account for the inclusion of the new main result. Still a preliminary versio