Non-associative binary systems


Ales< Drápal (Prague)


This intensive course will be held 23 May - 28 May 2002 (Thu-Sat, Mon-Tue) at the University of Helsinki in the Department of Mathematics. The lectures will take place in seminar room SIV.

The daily schedule is (tentatively)

except for the Saturday when the lectures start four hours earlier.

This intensive course is one in the series of algebra courses arranged in connection to the Finite Model Theory project:

Description of the course

While most of the algebraic structures that are considered in mathematics are associative, quite a few important innovative steps in the development of mathematics were done using nonassociative operations. These events are not well known since the discovery stage was in most cases later superseded by other descriptions using more conventional or more general structures. It is the purpose of the course to open a window into the world of nonassociative binary structures and to point out some of those innovative connections.

The larger part of the course is devoted to loops. A loop can be described as a group without associativity. Some of loops capture symmetries in very compact ways, and their discovery was of extreme importance in the process of classification of finite simple groups. The Parker loop is still being used as one of the vehicles how to access the Monster (the largest sporadic simple group). We shall start by describing basic properties of loops, in order to get a feeling for what one should look when encountering a loop, and then we shall deal with individual interesting classes of loop (symplectic loop - which include the Parker loop, Moufang loops - which have strong geometric connections, and Conjugacy closed loops). The latter can be of interest to logicians since there are recent attempts, partially successfull, to solve some of the open problems by means of automatic reasoning. On the last day of the course I would like to touch briefly left distributive operations which appear in surprising connections in the theory of large cardinals and in braids.

Course plan

    Thursday 23 May

  1. (1/3): Binary systems: loops, quasigroups, groupoids (Definitions, motivations from combinatorics, geometry, logic)
  2. (2/3): Basic notions of the loop theory (Multiplication group, inner mapping group, congruences, centre, nucleus, commutator, associator, central nilpotency)

    Friday 24 May

  3. (2/3): Symplectic Moufang loops - an example of a nonassociative structure with many interactions (Definition, combinatorial degree, connection to doubly even codes, Parker loop)
  4. (1/3): Isotopies (Albert's theorem, connection to the inverse multiplication group problem, loops isomorphic to its principal isotopes)

    Saturday 25 May

  5. Conjugacy closed loops

    Monday 27 May

  6. Moufang loops (Equivalent identities, diassociativity, commutative Moufang loops, octonions, simple Moufang loops)

    Tuesday 28 May

  7. Left distributive groupoids

Course material

Some reading material by Drapal is available:

This page as a Postscript-file

Department of Mathematics Finite model theory in Finland Teaching of mathematical logic