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Nilpotents in semigroups of partial order-preserving transformations

Published online by Cambridge University Press:  20 January 2009

G. U. Garba
G. U. Garba
Affiliation:
Department of Mathematical and Computational Sciences, University of St Andrews, Scotland and Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria
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Abstract

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In this paper we extend the results of Garba [1] on IOn, the semigroup of all partial one-one order-preserving maps on Xn = {1,…, n}, to POn, the semigroup of all partial order-preserving maps on Xn, A description of the subsemigroup of POn generated by the set N of all its nilpotent elements is given. The set {α∈POn:lim α/≦r and |Xn /dom α|≧r} is shown to be contained in 〈N〉 if and only if r≦½n. The depth of 〈N〉, which is the unique k for which 〈N〉 = N ∪ N2 ∪…∪ Nk and 〈N〉 ≠ N ∪ N2 ∪…∪Nk−1 is shown to be equal to 3 for all n≧3. The rank of the subsemigroup {α∈POπ|imα|≦/n − 2 and α∈〈N〉} is shown to be equal to 6(n − 2), and its nilpotent rank to be equal to 7n−15.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 3 , October 1994 , pp. 361 - 377
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

REFERENCES

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