Given a finite state space E, we build a universal dilation for all possible discrete time Markov chains on E, homogeneous or not: we introduce a second system (an “environment”) and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that any Markov evolution of E can be realized by a proper choice of the initial (random) state of the environ...

Goes to the advanced topics rather fast. Contains up-to-date proofs for free inverse semigroups, topics on biordered sets, Isbell's zigzags , and some combinatorics of transformation semigroups.] Fundamentals of semigroup theory, 1995. [Probably the best introduction. Contains more advanced material such as Isbell's zigzags , and the free inverse semi-groups]. G. Lallement, Semigroups and combi...

Here we focus on the geometry of P d,g, the compactification of the universal Picard variety constructed by L. Caporaso. In particular, we show that the moduli space of spin curves constructed by M. Cornalba naturally injects into P d,g and we give generators and relations of the rational divisor class group of P d,g, extending previous work by A. Kouvidakis.

We study the complexity classes P and NP through a semigroup fP (“polynomial-time functions”), consisting of all polynomially balanced polynomial-time computable partial functions. Then P 6= NP iff fP is a non-regular semigroup. The one-way functions considered here are based on worstcase complexity (they are not cryptographic); they are the non-regular elements of fP. We prove various properti...

We prove that every semigroup S whose quasivariety contains a 3-nilpotent semigroup or a semigroup of index more than 2 has no finite basis for its quasi-identities provided that one of the following properties holds: • S is finite; • S has a faithful representation by injective partial maps on a set; • S has a faithful representation by order preserving maps on a chain. As a corollary it is sh...

To Professor Jǐŕı Rosický on his 60th birthday Abstra t. A category K is called α-determined if every set of non-isomorphic Kobjects such that their endomorphism monoids are isomorphic has a cardinality less than α. A quasivariety Q is called Q-universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subqua...

Let S-Top be the category of topological S-acts over a topological semigroup S and S-SemiTop be the category of semitopological S-acts over a semitopological semigroup S. It is obvious that any topological S-act is a semitopological S-act, however we will see that the converse is not true in general. In this note, we study the universal objects in the category of semitopological S-acts and intr...

This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let Prob∞(R) denote the convolution semigroup of all probability measures on R with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence κ = (κ : ∈ N), regarded as a function from Prob∞(R) into the additive topological group R∞ ...

We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the Bost-Connes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expan...