Yann Peresse

Topological graph inverse semigroups

Abstract To every directed graph E one can associate a graph inverse semigroup G ( E ) , where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C ⁎ -algebras, and Toeplitz C ⁎ -algebras. We investigate topologies that turn G ( E ) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G ( E ) ∖ { 0 } must be discrete for any directed graph E. On the other hand, G ( E ) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E, G ( E ) admits a T 1 semigroup topology in which G ( E ) ∖ { 0 } is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G ( E ) in larger topological semigroups.

Maximal subsemigroups of the semigroup of all mappings on an infinite set

This is the accepted manuscript of the following article: J. East, J. D. Mitchell and Y. Peresse, “Maximal subsemigroupsof the semigroup of all mappings on an infinite set”, Transactions of the American Mathematical Society, Vol. 367(3), November 2014. The final published version is available online at: http://www.ams.org/journals/tran/2015-367-03/S0002-9947-2014-06110-2/

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Abstract Let Ω Ω be the semigroup of all mappings of a countably infinite set Ω . If U and V are subsemigroups of Ω Ω , then we write U ≈ V if there exists a finite subset F of Ω Ω such that the subsemigroup generated by U and F equals that generated by V and F . The relative rank of U in Ω Ω is the least cardinality of a subset A of Ω Ω such that the union of U and A generates Ω Ω . In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω . The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈ . Moreover such semigroups have relative rank 0 , 1 , 2 , or d in Ω Ω where d is the minimum cardinality of a dominating family for N N . We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in Ω Ω are 0 , 1 , 2 , and d . We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2 ℵ 0 .

Computing finite semigroups

Abstract Using a variant of Schreiers Theorem, and the theory of Greens relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and 0-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Greens relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the D -classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Greens classes of an element of such a subsemigroup without determining the global structure of the semigroup.

Chains of subsemigroups.

We investigate the maximum length of a chain of subsemigroups in various classes of semigroups, such as the full transformation semigroups, the general linear semigroups, and the semigroups of order-preserving transformations of finite chains. In some cases, we give lower bounds for the total number of subsemigroups of these semigroups. We give general results for finite completely regular and finite inverse semigroups. Wherever possible, we state our results in the greatest generality; in particular, we include infinite semigroups where the result is true for these.The length of a subgroup chain in a group is bounded by the logarithm of the group order. This fails for semigroups, but it is perhaps surprising that there is a lower bound for the length of a subsemigroup chain in the full transformation semigroup which is a constant multiple of the semigroup order.

Universal sequences for the order-automorphisms of the rationals

This is the accepted version of the following article: J. Hyde, J. Jonusas, J. D. Mitchell, and Y. Peresse, Universal sequences for the order-automorphisms of the rationals, J. London Math. Soc., first published online May 13, 2016 which has been published in final form at doi:10.1112/jlms/jdw015

A classification of disjoint unions of two or three copies of the free monogenic semigroup

We prove that, up to isomorphism and anti-isomorphism, there are only two types of semigroups which are the union of two copies of the free monogenic semigroup. Similarly, there are only nine types of semigroups which are the union of three copies of the free monogenic semigroup. We provide finite presentations for semigroups of each of these types.

Generating the infinite symmetric group using a closed subgroup and the least number of other elements

Let denote the symmetric group on the natural numbers . Then is a Polish group with the topology inherited from with the product topology and the discrete topology on . Let denote the least cardinality of a dominating family for and let denote the continuum. Using theorems of Galvin, and Bergman and Shelah we prove that if is any subgroup of that is closed in the above topology and is a subset of with least cardinality such that generates , then .