MMI bakalaureusetööd – Bachelor's theses. Kuni 2015

Selle kollektsiooni püsiv URIhttps://hdl.handle.net/10062/30415

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Sirvi MMI bakalaureusetööd – Bachelor's theses. Kuni 2015 Märksõna "bachelor thesis" järgi

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  • Pisipilt
    Kirje
    On Semigroups Amalgams
    (Tartu Ülikool, 2013) Rahkema, Kristiina; Sohail, Nasir, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
    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.