**Speaker**: James East

**Affiliation**: University of Western Sydney

**Time**: Monday 28/04/2014 from 14:00 to 15:00

**Venue**: Access Grid UWS. Presented from Parramatta (EB.1.32), accessible from Campbelltown (26.1.50) and Penrith (Y239).

**Abstract**:

Everyone knows that a permutation can be written as a product of cycles; this is the cycle decomposition. For partial permutations, we also need *trails*. For example, $[1,2,3]$ means that $1$ maps to $2$, $2$ maps to $3$, but $3$ is mapped nowhere, and nothing is mapped to $1$.

Everyone also knows that a permutation can be written as a product of $2$-cycles (also known as transpositions). A partial permutation can be written as a product of $2$-cycles and $2$-trails. The question of when two such products represent the same (partial) permutation leads to consideration of *presentations*.

We give a presentation for the semigroup of all partial permutations of a finite set with respect to the generating set of all $2$-cycles and $2$-trails. We also give a presentation for the singular part of this semigroup with respect to the generating set of all $2$-trails.

**Biography**: James is a member of the Centre for Research in Mathematics and a lecturer with the School of Computing, Engineering and Mathematics.

© University of Western Sydney 2010

Locked Bag 1797

Penrith South DC

NSW 1797

Tel: +61 2 9852 5222

ABN 53 014 069 881

CRICOS Provider No: 00917k