Topics in Semigroups

14/09 - "Free inverse semigroups" We consider the following questions: 1. Existence of free inverse semigroups. 2. The original construction of free inverse semigroups found by H. E. Scheiblich in 1972–1973. 3. Subsequent constructions of W. D. Munn (1973–1974) and B. M. Schein (1975). 4. Final construction of O. Poliakova and B. M. Schein (2005). 5. Comparison of these constructions, their advantages and disadvantages. 6. Some properties of free inverse semigroups. 15/09 - "Restrictive semigroups and bisemigroups" Everyone knows that V. V. Wagner (Vagner) considered inverse semigroups of one-to-one partial transformations, found an abstract characterizations 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 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 on 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 x2 = 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 their numerous properties. 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 1940-ies–1960-ies. 19/09 - "Automorphisms of transformation semigroups" Attempts to consider the structure of various mathematical objects by means of certain morphisms of these structures are as old as abstract algebra. For example, E. Galois characterized rational polynomials with roots expressible in radicals as the polynomials with solvable (soluble) Galois group. There were similar attempts of considering semigroups and monoids of transformations (for example, of "endomorphisms") of various mathematical structures. One of these direction was developed by Marc Krasner who called it "Endothéorie de Galois". I do not plan to discuss it in my lectures. Another direction was originally discussed by C. J. Everett and Stanislaw Ulam in the early 1940-ies and considered by many researchers. We will consider various achievements of numerous mathematicians in this direction.