On nilpotent subsemigroups of the order-preserving and decreasing transformation semigroups
- PDF / 2,076,031 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 25 Downloads / 223 Views
On nilpotent subsemigroups of the order‑preserving and decreasing transformation semigroups Melek Yağcı1 · Emrah Korkmaz1 Received: 23 September 2019 / Accepted: 29 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let Cn be the semigroup of all order-preserving and decreasing transformations on Xn = {1, … , n} under its natural order, and let N(Cn ) be the subsemigroup of all nilpotent elements of Cn . In this paper we determine the minimum generating set of N(Cn ) , and so the rank of N(Cn ) . Moreover, we investigate the nilpotent subsemigroups of Cn. Keywords Order-preserving/decreasing transformation · Minimum generating set · Rank · Nilpotent subsemigroup
1 Introduction For n ∈ ℕ , let Tn be the (full) transformation semigroup (under composition) on Xn = {1, … , n} under its natural order. A transformation 𝛼 ∈ Tn is called orderpreserving if x ≤ y implies x𝛼 ≤ y𝛼 for all x, y ∈ Xn and decreasing (increasing) if x𝛼 ≤ x ( x𝛼 ≥ x ) for all x ∈ Xn . The subsemigroup of all order-preserving transformations in Tn is denoted by On , and the subsemigroup of all order-preserving and decreasing (increasing) transformations in Tn is denoted by Cn ( C+n ). It is a well known fact [13, Corollary 2.7] that Cn and C+n are isomorphic semigroups. Let S be a semigroup, and let A be any non-empty subset of S. Then the subsemigroup generated by A is denoted by ⟨A⟩ . The rank of a finitely generated semigroup S is defined by
rank (S) = min{ �A� ∶ ⟨A⟩ = S}. Communicated by Pascal Weil. * Melek Yağcı [email protected] Emrah Korkmaz [email protected] 1
Department of Mathematics, Çukurova University, 01330 Adana, Turkey
13
Vol.:(0123456789)
M. Yağcı, E. Korkmaz
An element a of a semigroup S is called indecomposable if a ≠ xy for all x, y ∈ S that is, a ∈ S ⧵ S2 . It is clear that every generating set of S must contain all indecomposable elements of S. Thus, if S = ⟨A⟩ and A consists entirely of indecomposable elements of S, then it is clear that A is the minimum generating set of S. The image, the kernel and the fix of any transformation 𝛼 ∈ Tn are defined by
im (𝛼) = {x𝛼 ∶ x ∈ Xn }, ker(𝛼) = {(x, y) ∶ x𝛼 = y𝛼 for all x, y ∈ Xn } and fix (𝛼) = {x ∈ Xn ∶ x𝛼 = x}, respectively. An element e of a semigroup S is called idempotent if e2 = e , and the set of all idempotents in S is denoted by E(S). It is literally immediate that for 𝛼 ∈ Tn , 𝛼 is an idempotent if and only if x𝛼 = x for all x ∈ im (𝛼) . Equivalently, 𝛼 is an idempotent if and only if fix (𝛼) = im (𝛼) . Moreover, an element a of a semigroup S with zero, denoted by 0, is called a nilpotent element if am = 0 for some positive integer m. The set of all nilpotent elements of S is denoted by N(S). A semigroup S with 0 is called nilpotent if there exists m ∈ ℕ such that Sm = 0 . It is shown in [12] that if S is a finite semigroup, then the following statements are equivalent: (i) S is nilpotent, (ii) every element a ∈ S is nilpotent, and (iii) the unique idempotent of S is the zero element. It is well known that 𝜀 , wh
Data Loading...
