Lattice paths and order-preserving partial transformations

arXiv: Rings and Algebras, 2021

Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X, Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X, Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size.

Bulletin of the Malaysian Mathematical Sciences Society, 2017

Let On and Cn be the semigroup of all order-preserving transformations and of all order-preserving and order-decreasing transformations on the finite set Xn = {1, 2,. .. , n} , respectively. Let Fix(α) = {x ∈ Xn : xα = x} for any transformation α. In this paper, for any Y ⊆ Xn , we find the cardinalities of the sets On,Y = {α ∈ On : Fix(α) = Y } and Cn,Y = {α ∈ Cn : Fix(α) = Y }. Moreover, we find the numbers of transformations of On and Cn with r fixed points.

2015

A b s t r ac t. It is well-known [16] that the semigroup T n of all total transformations of a given n-element set X n is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain X n is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of X n is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain X n are covered by their inverse subsemigroups precisely when n 3.

2018

A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup $$\mathscr {TF}_{n}$$TFn of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of $$\mathscr {TF}_{n}$$TFn and provide a minimal generating set for $$ \mathscr {TF}_{n}$$TFn. Moreover, a formula for the number of idempotents in $$\mathscr {TF}_{n}$$TFn is given.

2012

For n ∈ IN, let Xn = {1 < 2 < · · · < n} be a finite chain with n elements. As usual, we denote by PTn the semigroup of all partial transformations α : Xn → Xn under composition. A transformation α ∈ PTn is called order-preserving if x ≤ y implies xα ≤ yα for all x, y from the domain of α. As usual, POn denotes the subsemigroup of PTn of all partial order-preserving transformations of Xn. This semigroup has been extensively studied. In recent years, interest in maximal subsemigroups of the transformation semigroups arises. In particular, Xiuliang Yang [6] characterized the maximal subsemigroups of the semigroup On of all full order-preserving transformations. Dimitrova and Koppitz [1] classified the maximal subsemigroups of the ideals of On. Ganyushkin and Mazorchuk [3] gave a description of the maximal subsemigroups of the semigroup POIn of all partial order-preserving injections. In [2], Dimitrova and Koppitz characterized the maximal subsemigroups of the ideals of the se...

Turkish Journal of Mathematics

In the present paper, we consider the semigroup On,p of all order-preserving full transformations α on an n-elements chain Xn , where p ∈ Xn is the only fixed point of α. The nilpotent semigroup On,p was first studied by Ayik et al. in 2011. Moreover, On,1 is the maximal nilpotent subsemigroup of the Catalan Monoid Cn. Its rank is the difference of the (n − 1) th and the (n − 2) th Catalan number. The aim of the present paper is to provide further fundamental information about the nilpotent semigroup On,p. We will calculate the rank of On,p for p > 1 and provide a semigroup presentation for On,1 .

2017

We extend the concept of path-cycle, to the semigroup \(\mathcal{P}_{n}\), of all partial maps on \(X_{n}=\{1,2,\ldots,n\}\), and show that the classical decomposition of permutations into disjoint cycles can be extended to elements of \(\mathcal{P}_{n}\) by means of path-cycles. The device is used to obtain information about generating sets for the semigroup \(\mathcal{P}_{n}\setminus\mathcal{S}_{n}\), of all singular partial maps of \(X_{n}\). Moreover, we give a definition for the (\(m,r\))-rank of \(\mathcal{P}_{n}\setminus\mathcal{S}_{n}\) and show that it is \(\frac{n(n+1)}{2}\).

Mathematics and Statistics, 2021

Let S be a semigroup and let G be a subset of S. A set G is a generating set G of S which is denoted by G = S. The rank of S is the minimal size or the minimal cardinality of a generating set of S, i.e. rankS := min{|G| : G ⊆ S, G = S}. In last twenty years, the rank of semigroups is worldwide studied by many researchers. Then it lead to a new definition o f r ank that is called the relative rank of S modulo U is the minimal size of a subset G ⊆ S such that G ∪ U generates S, i.e. rank(S : U) := min{|G| : G ⊆ S, G ∪ U = S}. A set G ⊆ S with G ∪ U = S is called generating set of S modulo U. The idea of the relative rank was generalized from the concept of the rank of a semigroup and it was firstly i ntroduced b y H owie, R uškuc a nd H iggins i n 1998. Let X be a finite c hain a nd l et Y b e a s ubchain o f X. We denote T (X) the semigroup of full transformations on X under the composition of functions. Let T (X, Y) be the set of all transformations from X to Y which is so-called the transformation semigroup with restricted range Y. It was firstly i ntroduced a nd s tudied b y S ymons i n 1 975. Many results in T (X) were extended to results in T (X, Y). In this paper, we focus on the relative rank of semigroup T (X, Y) and the semigroup OP(X, Y) of all orientation-preserving transformations in T (X, Y). In Section 2.1, we determine the relative rank of T (X, Y) modulo the semigroup OD(X, Y) of all order-preserving or order-reversing transformations. In Section 2.2, we describe the results of the relative rank of T (X, Y) modulo the semigroup OP(X, Y). In Section 2.3, we determine the relative rank of T (X, Y) modulo the semigroup OPR(X, Y) of all orientation-preserving or orientation-reversing transformations. Moreover, we obtain that the relative rank T (X, Y) modulo OP(X, Y) and modulo OPR(X, Y) are equal.

Bayero Journal of Pure and Applied Sciences

INTRODUCTION Let 1,2, … , be a finite chain. A map which has its domain and range both subset of is said to be a transformation and to be the domain and image of a transformation , respectively. If then is said to be a partial and if then is said to be a transformation. The collection of all partial (resp., full) transformation is known to be the semigroup of partial(resp., full) transformation of a finite chain. We adopt as in the literature, the notations and to be the semigroups of partial and full transformations of Algebraic and combinatorial properties of these semigroup where extensively studied in the literature, for example see Prest and Howie (Clifford and Preston Ganyushkin and Mazorchuk, 1995). For proper understanding of the concept of semigroup, we refer the reader to any o

Mathematical Proceedings of the Cambridge Philosophical Society, 1993

Consider the finite set Xn = {1,2, …,n} ordered in the standard way. Let Tn denote the full transformation semigroup on Xn, that is, the semigroup of all mappings α: Xn→Xn under composition. We shall call α order-preserving if i ≤ j implies iα ≤ jα for i,j∈Xn, and α is decreasing if iα ≤ i for all i∈Xn. This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings on Xn, and of its subsemigroup C which consists of all decreasing and order-preserving mappings.