Nilpotents in semigroups of partial one-to-one order-preserving mappings.

where {i} = A'nXdom a and {/'} = A^Xim a. For re even, the subsemigroup SL, of Pn consisting of all strictly partial one-one transformations was proved to be nilpotent-generated by Gomes and Howie [2]. If re is odd, they showed that the nilpotents in SL, generate SL^W,,..,, where Wn_l consists of all ae[n — l,n — 1] whose completions are odd permutations. Simultaneously and independently, Sullivan [7] showed that the subsemigroup SPn of Pn consisting of all strictly partial transformations of Xn is nilpotent-generated if re is even. If re is odd, the nilpotents in SPn generate SPn\\Wn_!. The rank of a semigroup 5 is the cardinality of any subset A of minimal order in 5 such that (A) = S. If the generating set A consists of nilpotent elements only, then we shall refer to the cardinality of A as the nilpotent rank of 5. Since one of the semigroups we will be considering is an inverse semigroup, we would like to clarify the notion of a generating set in an inverse semigroup. Given a subset A in an inverse semigroup 5, we shall always want to consider the smallest inverse subsemigroup containing A. In effect this is the set of all finite products of elements of A and their inverses. Following [3], we shall use the notation ((A)} for this inverse subsemigroup. Accordingly, by the rank of an inverse semigroup 5 we shall mean the cardinality of any subset A of minimal order in S such that ((A))=S. Let N and M be the sets of all nilpotent elements in SL, and SPn respectively. In [3], Gomes and Howie proved that the rank and the nilpotent rank of ((N)) are both equal to re + 1 for all n, and in [1], Garba showed that the rank and the nilpotent rank of (A/) are both equal to re + 2 for all re. In Section 2 we generalize the results of Gomes and Howie [3] (in line with Howie and McFadden [6]) by showing that if 1 < r < re 2 then the rank and the nilpotent rank of the inverse semigroup

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0017089500030482

On the nilpotent ranks of certain semigroups of transformations

A semigroup of N=1 superconformal transformations is introduced and analyzed. Noninvertible ones can describe transitions from body to soul and form a proper ideal containing a set of nilpotent transformations. The projective superspace is also considered. Transformations twisting the parity of a tangent space are brought in. They can be a nonsuperconformal ‘‘square root’’ of noninvertible superconformal transformations and the analogs of the Poincare metric and conformal invariance are suggested for them.

On the semigroup nature of superconformal symmetry

Let a,b be singular endomorphisms of a. finite dimensional vector space V and denote by S a the semigroup generated by all the elements g -1ag, where gϵAut(V).The aim of this paper is to prove that bϵS a if and only if rank(b) ≥ rank(a).

Semigroups of linear endomorphisms closed under conjugation

In this note, we obtain and discuss formulae for the total number of nilpotent partial and nilpotent partial one–one transformations of a finite set.

R.P Sullivan

Papers

Semigroups generated by nilpotent transformations