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

Variants of finite full transformation semigroups

“…In John Howie's famous 1966 paper [39], it was shown that the semigroup Sing X of all singular transformations on a finite set X (i.e., all non-invertible functions X → X) is generated by it idempotents. In subsequent works, and with other authors, Howie calculated the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of Sing X [27,41]; classified the idempotent generating sets of Sing X of minimal size [41]; calculated the rank and idempotent rank of the ideals of Sing X [44]; investigated the length function on Sing X with respect to the generating set consisting of all idempotents of defect 1 [43]; and extended these results to various other kinds of transformation semigroups and generating sets [6,7,27,28,40]. These works have been enormously influential, and have led to the development of several vibrant areas of research covering semigroups of (partial) transformations, matrices, partitions, endomorphisms of various algebraic structures, and more; see for example [5, 16-19, 21-23, 29, 30, 54, 64] and references therein.…”

Quasi-idempotent ranks of the proper ideals in finite symmetric inverse semigroups

“…There are studies of various ranks of semigroups, such as idempotent-rank, (m, r) rank, nilpotent rank, etc., as well as of minimal generating sets of elements of a given kind (see, for example, [2,3,6,8,13,17,18]).…”

Quasi-idempotent ranks of some permutation groups and transformation semigroups

“…The important problem of finding the rank of a semigroup has long been studied in the literature and there are some studies that examined some special kinds of ranks such as idempotent-rank, nilpotent-rank, or ( m, r )rank (see, for example, [1,3,7]). In this paper we restrict our attention to another special kind of rank, the quasi-idempotent rank of S , which is defined by qrank (S) = min{ |W | : ⟨W ⟩ = S, W ⊆ Q(S)}.…”