Search

AbstractFor n∈Z+ , let Xn={1,…,n} be a finite chain and Tn be the full transformation semigroup on Xn . If OCTn is the set of all order-preserving and full contraction mappings on Xn , then it is well-known that OCTn is a submonoid of Tn . In this paper we give a monoid presentation of OCTn . [ABSTRACT FROM AUTHOR]

The purpose of this paper is the study of congruences on semigroups of transformations on a countably infinite fence. We consider the monoid O ℕ of all full transformations on the set ℕ of all natural numbers preserving the zig-zag order on ℕ , as well as the monoid E (O ℕ ∗) of all idempotent transformations in O ℕ additionally preserving the usual linear order on ℕ. We show that there are uncountably many congruences on O ℕ and determine seven maximal congruences on O ℕ , which are all the maximal congruences containing a particular congruence on O ℕ. Moreover, we characterize all congruences on the monoid of all transformations in E (O ℕ ∗) with infinite rank. For the semigroup of all transformations in E (O ℕ ∗) with finite rank, we determine the Rees congruences. [ABSTRACT FROM AUTHOR]

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup FN of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that FN has no minimal generating sets and present two particular infinite decreasing chains of generating sets of FN. [ABSTRACT FROM AUTHOR]

Let be the semigroup of all singular order-preserving mappings on the finite set . It is known that is generated by its set of idempotents of rank , and its rank and idempotent rank are and , respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank in . We also calculate its rank and idempotent rank. [ABSTRACT FROM AUTHOR]

In this paper we consider various classes of monoids of transformations of a finite chain, including those of transformations that preserve or reverse either the order or the orientation. In line with Howie and McFadden [24], we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals. [ABSTRACT FROM AUTHOR]

For n ∈ ℕ, let On be the semigroup of all singular order-preserving mappings on [n]= {1,2,...,n}. For each nonempty subset A of [n], let On(A) = {α ∈ On: (∀ k ∈A) kα ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that On(A) is an abundant semigroup with n-1풟*-classes. Moreover, On(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. Further, it is shown that the rank of On(A) is equal to n-1 if 1 ∈ A, and it is equal to n otherwise. [ABSTRACT FROM AUTHOR]

In this note we consider various classes of monoids of transformations on a finite chain, in particular of transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested in computing both their cardinals and their idempotent numbers. Fibonacci and Lucas numbers play an essential role in the last computations. [ABSTRACT FROM AUTHOR]

In this paper we calculate presentations for same natural monoids of transformations on a chain Xn &lcub;1 < 2 < ... < n&rcub;. First we consider OD, [PODn], the monoid of all full [partial] transformations on Xn that preserve or reverse the order. Two other monoids of partial transformations on Xn we took at are PODn and PORn—the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation. [ABSTRACT FROM AUTHOR]

In this paper we consider three natural extensions of the inverse monoid POIn of all order preserving injective partial transformations of a chain with n elements. Namely, the inverse monoid PODIn consisting of all injective partial transformations on a chain with n elements which are either order preserving or order reversing, the inverse monoid POPIn of all orientation preserving injective partial transformations on a chain with n elements and the inverse monoid PORIn whose elements are all orientation preserving together with all orientation reversing injective partial transformations on a chain with n elements. Our main aim is to exhibit a presentation for POPIn in terms of n generators and ½(n² + 7n - 2) relations, a presentation for POPIn in terms of 2 generators and 2n relations and, finally, a presentation for PORIn in terms of 3 generators and 2n + 4 relations. [ABSTRACT FROM AUTHOR]