On Elements of Finite Semigroups of Order-Preserving and Decreasing Transformations
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On Elements of Finite Semigroups of Order-Preserving and Decreasing Transformations Melek Yagcı ˘ 1
· Emrah Korkmaz1
Received: 27 September 2019 / Revised: 27 September 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract For n ∈ N, let Cn be the semigroup of all order-preserving and decreasing transformations on X n = {1, . . . , n}, under its natural order, and let N (Cn ) be the set of all nilpotent elements of Cn and let Fix (α) = {x ∈ X n : xα = x} for any transformation α. An element a of a finite semigroup is called m-potent (m-nilpotent) element if a m+1 = a m (a m = 0) and a, a 2 , . . . , a m are distinct. In this paper, we obtain a formulae for the number of m-nilpotent elements and so the number of m-potent elements in N (Cn ) for 1 ≤ m ≤ n − 1. Moreover, for any subset Y of X n , we obtain a formulae for the number of m-potent elements of Cn,Y = {α ∈ Cn : Fix (α) = Y }. Keywords Order-preserving/decreasing transformation · m-nilpotent · m-potent Mathematics Subject Classification 20M20 · 05A15
1 Introduction For n ∈ N, let Tn be the (full) transformations semigroup (under composition) on X n = {1, . . . , n} under its natural order. A transformation α ∈ Tn is called order-preserving if x ≤ y implies xα ≤ yα for all x, y ∈ X n and decreasing (increasing) if xα ≤ x (xα ≥ x) for all x ∈ X n . The semigroup of all order-preserving transformations in Tn is denoted by On and the semigroup of all decreasing (increasing) transformations in On is denoted by Cn (Cn+ ). It is a well-known fact from [13, Corollary 2.7] that Cn and Cn+ are isomorphic semigroups. The problem of finding certain combinatorial
Communicated by Peyman Niroomand.
B
Melek Ya˘gcı [email protected] Emrah Korkmaz [email protected]
1
Department of Mathematics, Çukurova University, 01330 Adana, Turkey
123
M. Yagcı, ˘ E. Korkmaz
properties of On and Cn , has been an important research area in semigroup theory; see, for example, [1,3,4,6,8–10,12]. The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that a m+r = a m . An element with index m and period 1 is called an m-potent element. In [2], the authors obtained a formulae for the number of m-potent and (m, r )-potent elements in Tn . An element a of a finite semigroup S with a zero, denoted by 0, is called nilpotent if a m = 0 for some positive integer m, and moreover, if a m−1 = 0 then a is called an m-nilpotent element of S. The set of all nilpotent elements of S is denoted by N (S). For any α ∈ Cn , it is clear that α is an m-potent element for some m ∈ Z+ . Recall that nth Catalan number Cn is defined by C0 = 1
2n 1 2n 1 = Cn = n+1 n n n−1
for n ≥ 1,
(see, for example, p. 38 of [5]). From [8, Theorem 2.1 and Proposition 2.3], we know that |Cn | = |Cn+ | = Cn , |N (Cn )| = |N (Cn+ )| = Cn−1 . In general, a transformation α ∈ Tn is represented by the following tabular form: α=
1 ··· n 1α · · · nα
.
For any α ∈ Tn and for any non-empty subset Y of X n , we define
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