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Infinite transformation semigroups
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dc.contributor.advisor | Howie, John M. (John Mackintosh) | |
dc.contributor.author | Marques, Maria Paula | |
dc.coverage.spatial | 115 p. | en_US |
dc.date.accessioned | 2018-06-04T15:41:49Z | |
dc.date.available | 2018-06-04T15:41:49Z | |
dc.date.issued | 1983 | |
dc.identifier.uri | https://hdl.handle.net/10023/13705 | |
dc.description.abstract | In this thesis some topics in the field of Infinite Transformation Semigroups are investigated. In 1966 Howie considered the full transformation semigroup 𝓣 (x) on an infinite set x of cardinality m. For each 𝝰 in 𝓣 (x) he defined defect of 𝝰 = def 𝝰 and collapse of 𝝰= C(a) to be the sets X \ X 𝝰 and { 𝓍 ∊ x : (∃∊ x, y ≠ 𝓍) X𝝰 = Y𝝰 }, respectively. Later, in 1981 he introduced the set S[sub]m̱ = {𝝰 ∊ 𝓣(x): |def 𝝰 | = | c(𝝰) | = | ran 𝝰 | = m, |y 𝝰 [super]-1 | <m, (∀ y ∊ ran 𝝰) } which is a subsemigroup of 𝓣 (x) provided the cardinal m is regular. Taking m to be a regular cardinal number, Howie proved that S[sub]m̱ is then a bisimple, idempotent-generated semigroup of depth 4. Next he considered the congruence defined in S[sub]m̱ by △[sub]m̱ = {(𝝰, β) ∊ S[sub]m̱ x S[sub]m̱ : max (|D(𝝰, β) 𝝰| , | D((𝝰, β) β | ) < m̱ } where D(𝝰, β) = { 𝓍 ∊ X : 𝓍 𝝰 ≠ 𝓍β } and showed that S[sub]m̱* = S[sub]m̱/ △[sub]m̱ is a bisimple, congruence-free and idempotent-generated semigroup of depth 4. In this thesis comparable results are obtained for the semigroup P[sub]m̱ which is the top principal factor of the semigroup 𝓠[sub]m̱ = {𝝰 ∊ 𝓣(x): |def 𝝰 | = | c(𝝰) | = m̱} Here it is no longer necessary to restrict to a regular cardinal m̱. The set S[sub]m̱ considered by Howie fails to be a subsemigroup of 𝓣 (𝓍) if m̱ is not regular. It is shown that in this case <S[sub]m̱ > = O[sub]m̱ . In the case where m̱ = 𝓍₀ (a regular cardinal) it is shown that △[sub]𝓍₀ is the only proper congruence on S[sub]m̱. Within the symmetric inverse semigroup 𝓣(𝓍), the Baer-Levi semigroup B of type (m̱, m̱) on X is considered and a dual B* found. The products BB* and B*B are investigated and the semigroup Km̱ = <B*B> is described. The top principal factor of Km̱ is denoted by Lm̱ and it is shown that Lm̱ = B*B ⋃ {O}. On the set Lm̱ a congruence δ[sub]m̱, closely analogous to the congruence △[sub]m̱ defined above, is considered, and it is shown that Lm̱ / δ[sub]m̱ is a o-bisimple, inverse and nilpotent-generated semigroup. Finally, two embedding theorems for inverse semigroups and semigroups in general are presented. The cardinalities of some of the semigroups introduced in this thesis are studied. | en_US |
dc.language.iso | en | en_US |
dc.publisher | University of St Andrews | en |
dc.subject.lcc | QA171.M28 | |
dc.subject.lcsh | Group theory | en |
dc.title | Infinite transformation semigroups | en_US |
dc.type | Thesis | en_US |
dc.contributor.sponsor | Universidade do Minho | en_US |
dc.contributor.sponsor | Committee of Vice-Chancellors and Principals of the Universities of the United Kingdom | en_US |
dc.contributor.sponsor | University of St Andrews | 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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