ADAM 2009, Denver, Colorado, U.S.A.
13 July 2009

Presentations of the trivial semigroup.
R. Padmanabhan, University of Manitoba.

The trivial group, i.e. the group {e}, with just one element, is perhaps
the simplest possible group. Despite its simplicity (no pun intended), the
trivial group is still the subject of much interesting and deep research in
combinatorial group theory. One famous problem in this area is whether
there exists an algorithm to decide if a balanced presentation, that is a
presentation with the same finite number of generators and defining
relations, determines the trivial group (e.g. see [1]). Here we look at
this problem from the CS-conjecture point of view. In other words, take a
well-known such presentation of the trivial group, transform the defining
relations so that there are no more unary inverses and then prove that the
transformed presentation determines the trivial semigroup. Prover9 is an
ideal tool for doing such experimentations. Here we demonstrate some
transformed examples and proofs obtained by Prover9. In fact, I have given
some examples below so that each one of us can try to prove or disprove the
triviality of the corresponding semigroup. This will justify the concept of
an informal workshop (vis-a-vis a formal Conference). These problems (i.e.
the presentations of the trivial group) are related to the famous
Andrews-Curtis conjecture of group theory which in turn has connections
with algebraic topology.

[1] Ivanov, S.V. On balanced presentations of the trivial group.
Invent. Math. 165 (2006), no. 3, 525-549.