## 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 | British English |
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Pages (from-to) | 110-134 |

Number of pages | 25 |

Journal | Algebra and Discrete Mathematics |

Volume | 17 |

Issue number | 1 |

State | Published - 2014 |

## Keywords

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