On Semigroups Amalgams
| dc.contributor.advisor | Sohail, Nasir, juhendaja | |
| dc.contributor.author | Rahkema, Kristiina | |
| dc.contributor.other | Tartu Ülikool. Matemaatika-informaatikateaduskond | et |
| dc.contributor.other | Tartu Ülikool. Matemaatika instituut | et |
| dc.date.accessioned | 2013-06-19T07:29:31Z | |
| dc.date.available | 2013-06-19T07:29:31Z | |
| dc.date.issued | 2013 | |
| dc.description | Antud lõputöö käsitleb poolrühma amalgaame. Poolrühma amalgaamide all mõistame listi [U;S1, S2; !1,!2], kus U, S1 ja S2 on poolr¨uhmad ning !i ∶ U → Si (i ∈ {1, 2}) on monomorfismid. ¨ Oeldakse, et poolr¨uhma amalgaam [U;S1, S2; !1,!2] on sisestatav poolr¨uhma W, kui esiteks leiduvad monomorfismid μi ∶ S1 → W (i ∈ {1, 2}) nii, et diagramm U !2 ✏ !1 /S1 μ1 ✏ S2 μ2 /W kommuteerub. Ja teiseks juhul, kui mingid kaks elementi poolr¨uhmadest S1 ja S2 kujutatakse poolr¨uhmas W samaks elemendiks, siis leidub neil ¨uhine originaal poolr¨uhmas U. O. Schreier p¨ustitas esimesena k¨usimuse, kas r¨uhma amalgaamid on sisestatavad r¨uhma. Aastal 1927 t˜oestas ta, et see t˜oesti nii on. Seega oli loomulik k¨usida, kas sellist omadust oleks v˜oimalik edasi kanda ka teistele klassidele. Aastal 1957 konstrueeris N. Kimura oma doktorit¨o¨os kontran¨aite, mis t˜oestas, et k˜oik poolr¨uhma amalgaamid ei ole sisestatavad poolr¨uhma. J. M. Howie t˜oestas aastal 1962, et poolr¨uhma amalgaam [U;S1, S2], kus S1 ja S2 on r¨uhmad, on sisestatav siis ja ainult siis kui U on samuti r¨uhm. See tulemus oli eriti huvitav, kuna t¨anu sellele tekkis l˜opmatu hulk poolr¨uhma amalgaame, mis ei ole sisestatavad. T. E. Hall t˜oestas aastal 1975, et Schreieri tulemust on v˜oimalik ¨uldistada ning et analoogne tulemus kehtib ka inverssete poolr¨uhmade korral. 20 Antud l˜oput¨o¨o eesm¨argiks oli l¨ahemalt tutvuda poolr¨uhma amalgaamidega ning t˜oestada m˜oned tulemused poolr¨uhma amalgaamide sisestusest. Viimases peat¨ukis on t˜oestatud, et poolr¨uhma amalgaam [U;S1, S2], kus S1 ja S2 on kommutatiivsed regulaarsed poolr¨uhmad, on sisestatav parajasti siis, kui U on regulaarne. See tulemus ¨uldistab teatud m¨a¨aral varem Howie poolt t˜oestatud tulemust, et kommutatiivne regulaarne amalgaam, kus ka U on regulaarne, on sisestatav. Samuti on ka t˜oestatud, et poolr¨uhma amalgaam, kus S1 ja S2 on t¨aielikult regulaarsed ei ole sisestatav, kui U ei ole t¨aielikult regulaarne. Sama tulemus t˜oestatakse ka Cli↵ordi poolr¨uhmade jaoks. 21 | et |
| dc.description.abstract | Consider two groups G1 and G2 intersecting in a common subgroup U. Can we find a group W such that G1 and G2 are contained in it and their intersection is still U? More formally, one asks if the group amalgam [U;G1,G2] is embeddable. This question (about the embeddability of group amalgams) was first posed by Otto Schreier. In 1927 he proved that all group amalgams are embeddable. It was then natural to ask if this result could be expanded to the class of semigorups. In 1957 N. Kimura constructed a counter example in his doctoral thesis, showing that this is not the case for semigroups. J. M. Howie proved in 1962 that a semigroup amalgam [U;S1, S2], in which S1 and S2 are groups, is embeddable if and only if U is also a group. Howie’s result was particularly interesting as it provided an infinite class of non-embeddable semigroup amalgams. In 1975 T. E. Hall generalized Schreier’s result to the class of inverse semigroups (Journal of Algebra 34, 375–385 (1975)). The aim of this thesis is to consider some situations where semigroup amalgams fail to embed. In the first chapter frequently used terminology and some needed lemmas are introduced. In the second chapter we discuss the existence of pushouts, a categorical notion that is linked with amalgamation. The last chapter, which concentrates on semigroup amalgams, provides some results regarding non-embeddable semigroup amalgams and establishes a link between pushouts and amalgamation. | en |
| dc.identifier.uri | http://hdl.handle.net/10062/31051 | |
| dc.language.iso | en | en |
| dc.publisher | Tartu Ülikool | et |
| dc.subject | semigroups | en |
| dc.subject | semigroup amalgams | en |
| dc.subject | pushouts | en |
| dc.subject | embedding | en |
| dc.subject | bachelor thesis | en |
| dc.title | On Semigroups Amalgams | en |
| dc.type | Thesis | en |