From radin@macs.biu.ac.il Sun Oct 22 09:58:33 2006 Date: Sun, 22 Oct 2006 09:58:25 +0200 (IST) From: Adin Ron To: Combinatorics Seminar -- Aharoni Ron: ; Subject: Bar-Ilan Combinatorics Seminar ============================== Bar-Ilan Combinatorics Seminar ============================== The first meeting of the seminar this year will take place, IYH, on (when) Tuesday, 2 Marcheshvan 5767 (Oct. 24) refreshments at 11:45 am, talk at 12 noon (where) Room 331 (Math Dept Seminar Room), Math and CS Building, Bar-Ilan University (who) Prof. Boris Schein (University of Arkansas) will talk about (what) "Restrictive semigroups and bi-semigroups" Abstract: --------- Everyone knows that V. V. Wagner (Vagner) considered inverse semigroups of one-to-one partial transformations, found an abstract characterization of these semigroups (as regular semigroups with commuting idempotents), proved a Representation Theorem for inverse semigroups, etc., etc., in 1952 and later. Wagner's motivation came from the foundations of differential geometry. Relatively few people know that the same motivation led Wagner to a different class of semigroups in 1962. Their elements are partial mappings from one set into another (the sets need not be equal), and their operation is a restriction of one partial mapping to the domain (or the range) of another partial mapping. Wagner was attracted to this model because, for him, partial mappings were coordinate systems of a differential-geometric space. There is a certain (incomplete) duality between domains and ranges of partial mappings. Wagner considered domains and characterized the resulting semigroups by a system of simple identities. His semigroups (he called them ``restrictive'') are idempotent and right normal (that is, satisfy the identities x^2=x and xyz=yxz). From a purely algebraic standpoint, such semigroups were studied by Japanese semigroup theorists in the sixties. However, Wagner's motivation was entirely new and led to a novel look at this class of semigroups. Wagner proved a Representation Theorem for restrictive semigroups and found numerous properties of them. Further research in this direction was done by the speaker and some of his students. Almost all results in this direction were originally published in Russian, and relatively few of them were translated to English. This is the first attempt to present some of the main results of this branch of semigroup theory in a more or less coherent form. As an application (and time permitting), I will present a complete solution to a problem raised by Karl Menger and investigated by his students in the 1940-ies--1960-ies. Forthcoming Events ================== * 9 Marcheshvan (Oct. 31) Avital Frumkin (Tel-Aviv University): "Longest increasing subsequences and asymmetric electron repulsion" ************************************************************************* You are all invited ! (Graduate students especially welcome) If you want to give a talk at the seminar, or know a prospective speaker, please contact Ron Adin . Seminar's homepage: *************************************************************************