Geodesic
Generators and ranks in finite partial transformation semigroups
Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 237-248.

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We extend the concept of path-cycles, defined in [2], to the semigroup $\mathcal{P}_{n}$, of all partial maps on $X_{n}=\{1,2,\ldots,n\}$, and show that the classical decomposition of permutations into disjoint cycles can be extended to elements of $\mathcal{P}_{n}$ by means of path-cycles. The device is used to obtain information about generating sets for the semigroup $\mathcal{P}_{n}\setminus\mathcal{S}_{n}$, of all singular partial maps of $X_{n}$. Moreover, by analogy with [3], we give a definition for the ($m,r$)-rank of $\mathcal{P}_{n}\setminus\mathcal{S}_{n}$ and show that it is $\frac{n(n+1)}{2}$.
Keywords: path-cycle, $(m,r)$-path-cycle, $m$-path, generating set, $(m,r)$-rank. 
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Goje Uba Garba; Abdussamad Tanko Imam. Generators and ranks in finite partial transformation semigroups. Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 237-248. http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a6/

[1] Andre J. M., “Semigroups that contain all singular transformations”, Semigroup Forum, 68 (2004), 304–307 | DOI | MR | Zbl

[2] Ayik G., Ayik H., Howie H. M., “On factorisations and generators in transformations semigroups”, Semigroup Forum, 70:2 (2005), 225–237 | DOI | MR | Zbl

[3] Ayik G., Ayik H., Ünlü Y., Howie H. M., “Rank properties of the semigroup of singular transformations on a finite set”, Communications in Algebra, 36 (2008), 2581–2587 | DOI | MR | Zbl

[4] Clifford A. H., Preston G. B., The Algebraic Theory of Semigroups, v. 2, Mathematical Surveys of the American Mathematical Society, Providence, R.I., 1967 | MR

[5] Amer. Math. Soc. Transl., 139:2 (1988), 67–76 | MR | Zbl | Zbl

[6] Garba G. U., “Idempotents in partial transformation semigroup”, Proc. Roy. Soc. Edinburgh Sec. A, 116 (1990), 359–366 | DOI | MR | Zbl

[7] Howie J. M., “The subsemigroup generated by the idempotents of a full transformation semigroup”, J. London Math. Soc., 41 (1966), 707–716 | DOI | MR | Zbl

[8] Howie J. M., “Idempotent generators in finite full transformation semigroups”, Proc. Roy. Soc. Edinburgh Sec. A, 81 (1978), 317–323 | DOI | MR | Zbl

[9] Howie J. M., “Products of idempotents in a finite full transformation semigroup”, Proc. Roy. Soc. Edinburgh Sec. A, 86 (1980), 243–254 | DOI | MR | Zbl

[10] Howie J. M., McFadden R. B., “Idempotent rank in finite full transformation semigroups”, Proc. Roy. Soc. Edinburgh Sec. A, 116 (1990), 161–167 | DOI | MR

[11] Howie J. M., Lusk E. L., McFadden R. B., “Combinatorial results relating to products of idempotents in finite full transformation semigroups”, Proc. Roy. Soc. Edinburgh Sec. A, 115 (1990), 289–299 | DOI | MR | Zbl

[12] Howie J. M., Robertson R. B., Schein B. M., “A combinatorial property of finite full transformation semigroups”, Proc. Roy. Soc. Edinburgh Sec. A, 109 (1988), 319–328 | DOI | MR | Zbl

[13] Kearnes K. A., Szendrei A., Wood J., “Generating singular transformations”, Semigroup Forum, 63 (2001), 441–448 | DOI | MR | Zbl

[14] Lipscomb S., Symmetric Inverse Semigroups, Mathematical Surveys, 46, American Mathematical Society, Providence, R.I., 1996 | MR | Zbl

[15] Saito T., “Products of idempotents in finite full transformation semigroups”, Semigroup forum, 39 (1989), 295–309 | DOI | MR | Zbl

[16] Amer. Math. Soc. Transl., 36:2 (1964), 295–336 | MR | Zbl | Zbl