### Abstract

Let X_{n} = {1, 2,..., n}. On a partial transformation α : Domα ⊆ X_{n} → Im α ⊆ X_{n} of X_{n} the following parameters are defined: the breadth or width of α is | Domα |, the collapse of α is c(α) =| ∪_{t∈Imα}{tα−1 :| tα−1 |≥ 2} |, fix of α is f(α) =| {x ∈ X_{n} : xα = x} |, the height of α is | Im α |, and the right [left] waist of α is max(Im α) [min(Im α)]. The cardinalities of some equivalences defined by equalities of these parameters on T_{n}, the semigroup of full transformations of X_{n}, and P_{n} the semigroup of partial transformations of X_{n} and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.

Original language | English |
---|---|

Pages (from-to) | 110-134 |

Number of pages | 25 |

Journal | Algebra and Discrete Mathematics |

Volume | 17 |

Issue number | 1 |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- Breadth
- Collapse
- Fix
- Full transformation
- Height and right (left) waist of a transformation
- Idempotents and nilpotents
- Partial transformation

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

### Cite this

*Algebra and Discrete Mathematics*,

*17*(1), 110-134.

**Some combinatorial problems in the theory of partial transformation semigroups.** / Umar, A.

Research output: Contribution to journal › Article

*Algebra and Discrete Mathematics*, vol. 17, no. 1, pp. 110-134.

}

TY - JOUR

T1 - Some combinatorial problems in the theory of partial transformation semigroups

AU - Umar, A.

PY - 2014

Y1 - 2014

N2 - Let Xn = {1, 2,..., n}. On a partial transformation α : Domα ⊆ Xn → Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is | Domα |, the collapse of α is c(α) =| ∪t∈Imα{tα−1 :| tα−1 |≥ 2} |, fix of α is f(α) =| {x ∈ Xn : xα = x} |, the height of α is | Im α |, and the right [left] waist of α is max(Im α) [min(Im α)]. The cardinalities of some equivalences defined by equalities of these parameters on Tn, the semigroup of full transformations of Xn, and Pn the semigroup of partial transformations of Xn and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.

AB - Let Xn = {1, 2,..., n}. On a partial transformation α : Domα ⊆ Xn → Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is | Domα |, the collapse of α is c(α) =| ∪t∈Imα{tα−1 :| tα−1 |≥ 2} |, fix of α is f(α) =| {x ∈ Xn : xα = x} |, the height of α is | Im α |, and the right [left] waist of α is max(Im α) [min(Im α)]. The cardinalities of some equivalences defined by equalities of these parameters on Tn, the semigroup of full transformations of Xn, and Pn the semigroup of partial transformations of Xn and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.

KW - Breadth

KW - Collapse

KW - Fix

KW - Full transformation

KW - Height and right (left) waist of a transformation

KW - Idempotents and nilpotents

KW - Partial transformation

UR - http://www.scopus.com/inward/record.url?scp=84919731604&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919731604&partnerID=8YFLogxK

M3 - Article

VL - 17

SP - 110

EP - 134

JO - Algebra and Discrete Mathematics

JF - Algebra and Discrete Mathematics

SN - 1726-3255

IS - 1

ER -