Semigroups of order-decreasing transformations (original) (raw)

On the semigroups of order-decreasing finite full transformations

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1992

SynopsisLet Singn be the subsemigroup of singular elements of the full transformation semigroup on a totally ordered finite set with n elements. Let be the subsemigroup of all decreasing maps of Singn. In this paper it is shown that is a non-regular abundant semigroup with n − 1 -classes and . Moreover, is idempotent-generated and it is generated by the n(n − 1)/2 idempotents in J*n−1. LetandSome recurrence relations satisfied by J*(n, r) and sh (n, r) are obtained. Further, it is shown that sh (n, r) is the complementary signless (or absolute) Stirling number of the first kind.

On Certain Finite Semigroups of Order-decreasing Transformations I

Semigroup Forum, 2004

Let Dn (On) be the semigroup of all finite order-decreasing (orderpreserving) full transformations of an n-element chain, and let D(n, r) = {α ∈ Dn : |Im α| ≤ r} (C(n, r) = D(n, r) ∩ On) be the two-sided ideal of Dn (Dn ∩ On). Then it is shown that for r ≥ 2 , the Rees quotient semigroup

ON CERTAIN INFINITE SEMIGROUPS OF ORDER INCREASING TRANSFORMATIONS II

2003

Let X be an arbitrary poset. The semigroup I + (X ) of all order-increasing partial oneone mappings of X is shown to be an ample semigroup with some interesting properties. Moreover, a necessary and sufficient condition (on the totally ordered sets X and Y ) for I + (X ) and I + (Y ) to be isomorphic has been established.

On the ranks of certain finite semigroups of order-decreasing transformations

Portugaliae Mathematica, 1996

Stirling number of the second kind, and for 1 ≤ r ≤ n − 1 ... The rank of a finite semigroup S is usually defined by ... Received: June 28, 1993; Revised: July 21, 1995. * The results of this paper form part of the author's Ph.D. dissertation submitted to the University of St. Andrews.

On the semigroups of partial one-to-one order-decreasing finite transformations

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1993

SynopsisLet In be the symmetric inverse semigroup on Xn = {1,…, n}, let Sln be the subsemigroup of strictly partial one-to-one self-maps of Xn and let = { α ∊ SIn: x} ≦ x = U = ∅= be the semigroup of all partial one-to-one decreasing maps including the empty or zero map of Xn. In this paper it is shown that is an (irregular, for n ≧ 2) type A semigroup with n D*-classes and D* = I*. Further, it is shown that is generated by the n(n + l)/2 quasi-idempotents in

Combinatorial Results for Semigroups of Order-Preserving and A-Decreasing Finite Transformations

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.

On Relative Ranks of Finite Transformation Semigroups with Restricted Range

Ukrainian Mathematical Journal, 2021

In this paper, we determine the relative rank of the semigroup T (X, Y) of all transformations on a finite chain X with restricted range Y ⊆ X modulo the set OP(X, Y) of all orientation-preserving transformation in T (X, Y). Moreover, we state the relative rank of the semigroup OP(X, Y) modulo the set O(X, Y) of all order-preserving transformations in OP(X, Y). In both cases we characterize the minimal relative generating sets.

On the rank of generalized order-preserving transformation semigroups

Turkish Journal of Mathematics

For any two non-empty (disjoint) chains X and Y , and for a fixed order-preserving transformation θ : Y → X , let GO(X, Y ; θ) be the generalized order-preserving transformation semigroup. Let O(Z) be the orderpreserving transformation semigroup on the set Z = X ∪Y with a defined order. In this paper, we show that GO(X, Y ; θ) can be embedded in O(Z, Y) = { α ∈ O(Z) : Zα ⊆ Y } , the semigroup of order-preserving transformations with restricted range. If θ ∈ GO(Y, X) is one-to-one, then we show that GO(X, Y ; θ) and O(X, im(θ)) are isomorphic semigroups. If we suppose that |X| = m , |Y | = n , and |im(θ)| = r where m, n, r ∈ N , then we find the rank of GO(X, Y ; θ) when θ is one-to-one but not onto. Moreover, we find lower bounds for rank(GO(X, Y ; θ)) when θ is neither one-to-one nor onto and when θ is onto but not one-to-one.