Predavanje prof. Vesne Kilibarde 20. 04. 2017.

Prof. Vesna Kilibarda sa Indiana University Northwest (SAD) će održati predavanje sa naslovom Ends of Cayley graphs for semigroups u četvrtak, 20. 04. u učionici 012 sa početkom u 19:00 časova. Pozivaju se zainteresovani profesori, saradnici i studenti.

Title: Ends of Cayley graphs for semigroups

Abstract: We give two graph-theoretic definitions for the number of ends of Cayley digraphs for finitely generated semigroups which both extend group definitions. For semigroups, left Cayley digraphs can be very different from right Cayley digraphs. First we define the ends of the Cayley graphs of finitely generated semigroups. We prove that the number of such ends for the Cayley graph does not depend upon set of generators of the semigroup. For natural numbers m and n, we exhibit finitely generated monoids for which the left Cayley graphs have m ends while the right Cayley graphs have n ends. For direct products and for many other semidirect products of a pair of finitely generated infinite monoids, the right Cayley graph of the semidirect product has only one end. A finitely generated subsemigroup of a free semigroup has either one end or else has infinitely many ends. The number of ends is preserved for subsemigroups of finite Rees index, and the same holds for finite Green index subsemigroups of cancellative semigroups. Aiming to obtain a better interrelation between algebraic structure and ends of a semigroup, we, alternatively, define the notion of the partial order of (directed) ends of the Cayley graph of a semigroup . We prove that the structure of the (directed) ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf’s Theorem, stating that an infinite group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.