Abstract
The (full) transformation semigroup \(\mathcal{T}_{n}\) is the semigroup of all functions from the finite set {1,…,n} to itself, under the operation of composition. The symmetric group \({\mathcal{S}_{n}\subseteq \mathcal{T}_{n}}\) is the group of all permutations on {1,…,n} and is the group of units of \(\mathcal{T}_{n}\). The complement \(\mathcal{T}_{n}\setminus \mathcal{S}_{n}\) is a subsemigroup (indeed an ideal) of \(\mathcal{T}_{n}\). In this article we give a presentation, in terms of generators and relations, for \(\mathcal{T}_{n}\setminus \mathcal{S}_{n}\), the so-called singular part of \(\mathcal{T}_{n}\).
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Communicated by Thomas E. Hall.
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East, J. A presentation for the singular part of the full transformation semigroup. Semigroup Forum 81, 357–379 (2010). https://doi.org/10.1007/s00233-010-9250-1
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DOI: https://doi.org/10.1007/s00233-010-9250-1