## Refined classification of the degree 4 transformation semigroups

Previously we classified the degree 4 transformation semigroups by size and by the numbers of Green’s classes. This turned out to be insufficient for finding isomorphic semigroups, so we needed a finer classification. The new property is the number of idempotents, for instance, I016 in the filename means the semigroups in the file all contain 16 idempotents. Furthermore, letter b indicates bands, letter c commutative semigroups, and r stands for regular semigroups.

The size of the archive is around the same, but there are more files in it. 287401 to be precise.

T4SubSgpsConjClassesClassifiedV2.tar.xz

## Cycle-trail decomposition for partial permutations

Speaker: James East, Centre for Research in Mathematics, University of Western Sydney

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.