Files in this item
Semigroups of singular endomorphisms of vector space
Item metadata
dc.contributor.advisor | Howie, John M. (John Mackintosh) | |
dc.contributor.author | Dawlings, Robert J. H. | |
dc.coverage.spatial | iii, 143 p. | en_US |
dc.date.accessioned | 2018-06-05T11:35:11Z | |
dc.date.available | 2018-06-05T11:35:11Z | |
dc.date.issued | 1980 | |
dc.identifier.uri | https://hdl.handle.net/10023/13725 | |
dc.description.abstract | In 1967, J. A. Erdős showed, using a matrix theory approach that the semigroup Sing[sub]n of singular endomorphisms of an n-dimensional vector space is generated by the set E of idempotent endomorphisms of rank n - 1. This thesis gives an alternative proof using a linear algebra and semigroup theory approach. It is also shown that not all the elements of E are needed to generate Sing[sub]n. Necessary conditions for a subset of E to generate found; these conditions are shown to be sufficient if the vector space is defined over a finite field. In this case, the minimum order of all subsets of E that generate Sing[sub]n is found. The problem of determining the number of subsets of E that generate Sing[sub]n and have this minimum order is considered; it is completely solved when the vector space is two-dimensional. From the proof given by Erdős, it could be deduced that any element of Sing[sub]n could be expressed as the product of, at most, 2n elements of E. It is shown here that this bound may be reduced to n, and that this is best possible. It is also shown that, if E⁺ is the set of all idempotent of Singn, then (E⁺)ⁿ⁻¹ is strictly contained in Sing[sub]n. Finally, it is shown that Erdős's result cannot be extended to the semigroup Sing of continuous singular endomorphisms of a separable Hilbert space. The sub semigroup of Sing generated by the idempotent of Sing is determined and is, clearly, strictly contained in Sing. | en_US |
dc.language.iso | en | en_US |
dc.publisher | University of St Andrews | en |
dc.subject.lcc | QA186.D2 | |
dc.subject.lcsh | Vector spaces | en |
dc.title | Semigroups of singular endomorphisms of vector space | en_US |
dc.type | Thesis | en_US |
dc.contributor.sponsor | Science Research Council (Great Britain) | en_US |
dc.contributor.sponsor | Forrest and Grinsell Foundation | en_US |
dc.type.qualificationlevel | Doctoral | en_US |
dc.type.qualificationname | PhD Doctor of Philosophy | en_US |
dc.publisher.institution | The University of St Andrews | en_US |
dc.publisher.department | Monash University, Australia | en_US |
This item appears in the following Collection(s)
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.