We study the singular part of the partition monoid ; that is, the ideal , where is the symmetric group. Our main results are presentations in terms of generators and relations. We also show that is idempotent generated, and that its rank and idempotent-rank are both equal to . One of our presentations uses an idempotent generating set of this minimal cardinality.

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On the Singular Part of the Partition Monoid

J.M. Howie proved that \(\operatorname {Sing}_{n}\), the semigroup of all singular mappings of {1,…,n} into itself, is generated by its idempotents of defect 1 (in J. London Math. Soc. 41, 707–716, 1966). He also proved that if n≥3 then a minimal generating set for \(\operatorname {Sing}_{n}\) contains n(n−1)/2 transformations of defect 1 (in Gomes and Howie, Math. Proc. Camb. Philos. Soc. 101. 395–403, 1987). In this paper we find necessary and sufficient conditions for any set for transformations of defect 1 in \(\operatorname {Sing}_{n}\) to be a (minimal) generating set for \(\operatorname {Sing}_{n}\).

Generating sets of finite singular transformation semigroups

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 \(\mathcal{T}_{n}\).

A presentation for the singular part of the full transformation semigroup

It is known that the semigroup Sing n of all singular self-maps of X n  = {1,2,…, n} has rank n(n − 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing n , has the same value as the rank. (See Gomes and Howie, 1987.) Idempotents generating Sing n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing n , defined as the smallest number of (m, r)-path-cycles generating Sing n , is once again n(n − 1)/2.

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

Given a transformation semigroup S over a finite set X we determine necessary and sufficient conditions on the generators of S in order that it contains all singular transformations.

Semigroups That Contain All Singular Transformations

We compute the singular rank and the idempotent rank of those subsemigroups of the full transformation semigroup that contain all singular transformations.

J. Wood

Papers

Generating singular transformations