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Idempotents, nilpotents, rank and order in finite transformation semigroups
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dc.contributor.advisor | Howie, John M. (John Mackintosh) | |
dc.contributor.author | Garba, Goje Uba | |
dc.coverage.spatial | 135 p. | en_US |
dc.date.accessioned | 2018-06-04T14:41:33Z | |
dc.date.available | 2018-06-04T14:41:33Z | |
dc.date.issued | 1992 | |
dc.identifier.uri | https://hdl.handle.net/10023/13703 | |
dc.description.abstract | Let E, Eā denote, respectively, the set of singular idempotents in T[sub]n (the semigroup of all full transformations on a finite set X[sub]n = {1,..., n}) and the set of idempotents of defect 1. For a singular element 𝑎 in Tn, let k(𝑎),kā (𝑎) be defined by the properties 𝑎 ∈ Eįµā½įµā¾, 𝑎 ∉ Eįµā½įµā¾ā»¹, 𝑎 ∈ Eāįµ¹ā½įµā¾, 𝑎 ∉ Eāįµ¹ā½įµā¾ā»¹. In this Thesis, we obtain results analogous to those of Iwahori (1977), Howie (1980), Saito (1989) and Howie, Lusk and McFadden (1990) concerning the values of k(𝑎) and kā(𝑎) for the partial transformation semigroup P[sub]n. The analogue of Howie and McFadden's (1990) result on the rank of the semigroup K(n,r) = { 𝑎 ∈ T [sub]n: |im 𝑎 | ≤ r,2 ≤ r ≤ n-1} is also obtained. The nilpotent-generated subsemigroup of P[sub]n was characterised by Sullivan in 1987. In this work, we have obtained its depth and rank. Nilpotents in IO[sub]n and PO[sub]n (the semigroup of all partial one-one order-preserving maps, and all partial order-preserving maps) are studied. A characterisation of their nilpotent-generated subsemigroups is obtained. So also are their depth and rank. We have also characterised their nilpotent-generated subsemigroup for the infinite set X = {1,2,...}. The rank of the semigroup L(n,r) = {a ∈ S : |im 𝑎 | ≤r, 1 ≤ r ≤ n - 2} is investigated for S = O[sub]n,PO[sub]n,SPO[sub]n and I[sub]n (where O[sub]n is the semigroup of all order-preserving full transformations, SPO[sub]n the semigroup of all strictly partial order- preserving maps, and In the semigroup of one-one partial transformation). | en_US |
dc.language.iso | en | en_US |
dc.publisher | University of St Andrews | en |
dc.subject.lcc | QA171.S3G2 | |
dc.subject.lcsh | Group theory | en |
dc.title | Idempotents, nilpotents, rank and order in finite transformation semigroups | en_US |
dc.type | Thesis | en_US |
dc.contributor.sponsor | Nigeria. Federal Government | en_US |
dc.contributor.sponsor | Ahmadu Bello University | 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 |
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