On the ranks of certain finite semigroups of transformations

Abstract

It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutationsgenerate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.

On the nilpotent ranks of certain semigroups of transformations

“…,m}), where 5 r is the symmetric group on X r and m = ("). Hence, since the rank of S r is known to be 2, it follows by Theorem 3.3 in [3] that P r has inverse semigroup rank (") + 1.…”

“…"_[, followed by the subsets in /? n _ 2 , followed by the subsets in R n - 3 , and so on, until # , is reached. LEMMA 3.8.…”

Idempotent-generated semigroups and pseudovarieties

“…Recall that the rank of a finite semigroup is the minimum number of generators, and the idempotent rank of an idempotentgenerated finite semigroup is the minimum number of idempotent generators. Gomes and Howie [5] showed that the rank and idempotent rank of the subsemigroup of T n consisting of all full transformations with range less than n are both equal to n(n − 1)/2. The rank and idempotent rank of the subsemigroup of all contractive finite full transformations are both equal to n + 1, as showed by Umar [19].…”

Abelian Kernels, Solvable Monoids and the Abelian Kernel Length of a Finite Monoid

“…In fact, presentations for O n and PO n were established respectively by Aȋzenštat [1] in 1962 and by Popova [46] in the same year. Some years later (1971) Howie [42] studied some combinatorial and algebraic properties of O n and, in 1992, Gomes and Howie [37] established some more properties of O n , namely its rank and idempotent rank. Also in [37] the monoid PO n was studied.…”

“…Some years later (1971) Howie [42] studied some combinatorial and algebraic properties of O n and, in 1992, Gomes and Howie [37] established some more properties of O n , namely its rank and idempotent rank. Also in [37] the monoid PO n was studied. The monoid O n played also a main role in several other papers [40,51,9,28,47,34] where the central topic concerns the problem of the decidability of the pseudovariety generated by the family {O n | n ∈ N}.…”