Show simple item record

Files in this item


Item metadata

dc.contributor.advisorHowie, John M. (John Mackintosh)
dc.contributor.authorGarba, Goje Uba
dc.coverage.spatial135 p.en_US
dc.description.abstractLet 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 ∈ Eįµā½įµƒā¾, 𝑎 ∉ Eįµā½įµƒā¾ā»¹, 𝑎 ∈ Eā‚įµ¹ā½įµƒā¾, 𝑎 ∉ Eā‚įµ¹ā½įµƒā¾ā»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.publisherUniversity of St Andrewsen
dc.subject.lcshGroup theoryen
dc.titleIdempotents, nilpotents, rank and order in finite transformation semigroupsen_US
dc.contributor.sponsorNigeria. Federal Governmenten_US
dc.contributor.sponsorAhmadu Bello Universityen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US

This item appears in the following Collection(s)

Show simple item record