Products of Nilpotents in Partial Transformation Semigroups using Digraphic Paths and Chains
DOI :
https://doi.org/10.62050/fjst2025.v9n1.511Mots-clés :
Depth formula, Full transformation, Idempotents, Nilpotents, SemigroupsRésumé
In this paper, we investigate the factorization of singular partial self-maps on a finite set into products of the least number of nilpotent elements. This research demonstrates that the semigroup of such maps can be expressed within a union of nilpotent-generated sets, specifically up to the third power. Some of our key findings include the determination of the nilpotent rank and the nilpotent depth for these maps, which vary based on whether the set size is even or odd. Additionally, this study surveys the relationship between these results and Stirling numbers, leveraging the Vagner Theorem and digraphic representations. We also examine stable quasi-idempotents, which correspond to specific digraphic paths and chains, providing further insights into the structure of partial
transformation semigroups.
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Références
Ayik G., Ayik H. and Howie J. M. (2005). On factorizations and generators in transformation semigroups, Semigroup Forum, 70(2), 225-237.
Erdos J. A. (1967). On products of idempotent matrices, Glasgow Math. J., 8, 118-122.
Garba G. U. (1990). Idempotents in partial transformation semigroup, Proc. Roy. Soc. Edinburgh, 116A, 359-366.
Garba G. U. (1994a). Nilpotents in semigroups of partial one-to-one order preserving mappings, Semigroup Forum, 48, 37-49.
Garba G. U. (1994b). Nilpotents in semigroups of partial order-preserving transformation, Proc. Eding. Math. Soc., 37, 361-377.
Howie J. M (1966). The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc., 41, 707-716.
Howie J. M. (1978). Idempotent generators in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh, 81A, 317-323.
Howie J. M. (1980). Products of idempotents in a finite full transformation semigroup, Proc. Roy. Soc. Edinburgh, 86A, 243-254.
Howie J. M. (2002). Semigroups, past, present and future, proceedings of the International Conference on Algebra and Its Applications.
Imam A. T. and Ibrahim M. J. (2022). On products of 3-paths in finite full transformation Semigroups, Algr. and Discr. Math., 33(2), 60-77.
Imam A. T., Usman L., Idris A. and Ibrahim S. (2024). Products of Quasi-Idempotents in Finite Semigroup of Partial Order-Preserving Transformations, International Journal of Mathematical Sciences and Optimization, 10(1), 53-59.
Madu B. A. (1999). Quasi-idempotents and quasi-nilpotents in finite transformation Semigroups, PhD Thesis, Ahmadu Bello University Zaria-Nigeria.
Saito T. (1989). Products of idempotents in finite full transformation semigroups, Semigroup Forum, 39, 295-309.
Schein B. M. (1996). Finite semigroup and universal algebra, Semigroup Forum, 52, 393-396.
Vagner V. V. (1964). Representations of ordered semigroups, Amer. Math. Soc. Transl., 36(2), 295-336.
Yang X (1998). Non-group ranks in finite full transformation semigroups, Semigroup Forum, 57, 42-47.
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