Geodesic
The Cardinal and the Idempotent Number of Various Monoids of Transformations on a Finite Chain
Bulletin of the Malaysian Mathematical Society, Tome 34 (2011) no. 1 Cet article a éte moissonné depuis la source Bulletin of the Malaysian Mathematical Society website

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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.
Classification : Primary: 20M17; Secondary: 20M20, 05A10. 
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     author = {V{\'\i}tor H. Fernandes and Gracinda M. S. Gomes and Manuel M. Jesus},
     title = {The {Cardinal} and the {Idempotent} {Number} of {Various} {Monoids} of {Transformations} on a {Finite} {Chain}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2011},
     volume = {34},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2011_34_1_a6/}
}
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Vítor H. Fernandes; Gracinda M. S. Gomes; Manuel M. Jesus. The Cardinal and the Idempotent Number of Various Monoids of Transformations on a Finite Chain. Bulletin of the Malaysian Mathematical Society, Tome 34 (2011) no. 1. http://geodesic.mathdoc.fr/item/BMMS_2011_34_1_a6/