Abstract
Given a simple digraph D on n vertices (with nge 2), there is a natural construction of a semigroup of transformations langle Drangle . For any edge (a, b) of D, let arightarrow b be the idempotent of rank n-1 mapping a to b and fixing all vertices other than a; then, define langle Drangle to be the semigroup generated by a rightarrow b for all (a,b) in E(D). For alpha in langle Drangle , let ell (D,alpha ) be the minimal length of a word in E(D) expressing alpha . It is well known that the semigroup mathrm {Sing}_n of all transformations of rank at most n-1 is generated by its idempotents of rank n-1. When D=K_n is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ell (K_n,alpha ), for any alpha in langle K_nrangle = mathrm {Sing}_n; however, no analogous non-trivial results are known when D ne K_n. In this paper, we characterise all simple digraphs D such that either ell (D,alpha ) is equal to Howie–Iwahori’s formula for all alpha in langle Drangle , or ell (D,alpha ) = n - mathrm {fix}(alpha ) for all alpha in langle Drangle , or ell (D,alpha ) = n - mathrm {rk}(alpha ) for all alpha in langle Drangle . We also obtain bounds for ell (D,alpha ) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n-1 of mathrm {Sing}_n). We finish the paper with a list of conjectures and open problems.
Highlights
For any n ∈ N, n ≥ 2, let Singn be the semigroup of all singular transformations on [n] := {1, . . . , n}
We explore the natural connections between simple digraphs on [n] and subsemigroups of Singn
For any subset U ⊆ Singn, denote by U the semigroup generated by U
Summary
For any n ∈ N, n ≥ 2, let Singn be the semigroup of all singular (i.e. non-invertible) transformations on [n] := {1, . . . , n}. For any subset U ⊆ Singn, denote by U the semigroup generated by U. We say that a subsemigroup S of Singn is arc-generated by a simple digraph D if S= D. The semigroup of non-decreasing transformations OIn := {α ∈ Singn : v ≤ vα} is arc-generated by the transitive tournament Tn on [n] (Fig. 1 illustrates T5). Connections between subsemigroups of Singn and digraphs have been studied before (see [9,10,11,12]). Recall that the orbits of α ∈ Singn are the connected components of the digraph on [n] with edges {(x, xα) : x ∈ [n]}. Denote by cycl(α) and fix(α) the number of cyclic orbits and fixed points of α, respectively.
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