Publicação
Algebraic aspects of tiling semigroups
| Resumo: | This thesis is devoted to the algebraic study of tiling semigroups, in the context of inverse semigroup theory. Tiling semigroups were originally motivated by the work of Johannes Kellendonk [23], in connection with a problem in solid state physics, formulated in terms of almost-groupoids by the same author in [24], and established by Kellendonk and Mark V. Lawson in [26] and Lawson in [32]. Since then, quite a lot of research has been done on this subject, mainly regarding tilings of the real line. In this dissertation, we aimed at furthering the study of one-dimensional tiling semigroups and extending the theory concerning this class to a special type of n-dimensional tilings, named n-dimensional hypercubic tilings. Following the tradition initiated by Lawson in [33], we often conduct our investigations in the more general setting of an inverse semigroup associated with a factorial language. The first three chapters are essentially introductory. In Chapter 1, we recall some selected background material; in Chapter 2, we present and investigate a construction, called generalized Bruck-Reilly extension, which will clarify the connection between hypercubic tiling semigroups and Bruck-Reilly extensions; in Chapter 3, we define all the concepts involved in the construction of the tiling semigroup, give a complete review on the research conducted on this subject, and introduce the notion of n-dimensional hypercubic tiling. In Chapter 4, we introduce the notions of language of an n-dimensional hypercubic tiling and of n-dimensional factorial language with the purpose of generalizing to n-dimensional hypercubic tiling semigroups a convenient representation of one-dimensional tiling semigroups in terms of a language associated with the tiling introduced by Lawson in [32]. We also present another representation of an n-dimensional hypercubic tiling semigroup as a Rees factor semigroup of a subsemigroup of a generalized Bruck-Reilly extension. In Chapter 5, we develop a description of a tiling semigroup, both one-dimensional and hypercubic, as a P -semigroup; in Chapter 6, we compute a presentation for one-dimensional tilings semigroups and discuss some aspects of the presentability of n-dimensional hypercubic tiling semigroups; in Chapter 7, we provide a necessary and sufficient condition for hypercubic tilings to give rise to isomorphic tiling semigroups. |
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| Autores principais: | Soares, Filipa |
| Assunto: | Semigrupos Isomorfismos (matemática) Teses de doutoramento - 2010 |
| Ano: | 2010 |
| País: | Portugal |
| Tipo de documento: | tese de doutoramento |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade de Lisboa |
| Idioma: | inglês |
| Origem: | Repositório da Universidade de Lisboa |
| Resumo: | This thesis is devoted to the algebraic study of tiling semigroups, in the context of inverse semigroup theory. Tiling semigroups were originally motivated by the work of Johannes Kellendonk [23], in connection with a problem in solid state physics, formulated in terms of almost-groupoids by the same author in [24], and established by Kellendonk and Mark V. Lawson in [26] and Lawson in [32]. Since then, quite a lot of research has been done on this subject, mainly regarding tilings of the real line. In this dissertation, we aimed at furthering the study of one-dimensional tiling semigroups and extending the theory concerning this class to a special type of n-dimensional tilings, named n-dimensional hypercubic tilings. Following the tradition initiated by Lawson in [33], we often conduct our investigations in the more general setting of an inverse semigroup associated with a factorial language. The first three chapters are essentially introductory. In Chapter 1, we recall some selected background material; in Chapter 2, we present and investigate a construction, called generalized Bruck-Reilly extension, which will clarify the connection between hypercubic tiling semigroups and Bruck-Reilly extensions; in Chapter 3, we define all the concepts involved in the construction of the tiling semigroup, give a complete review on the research conducted on this subject, and introduce the notion of n-dimensional hypercubic tiling. In Chapter 4, we introduce the notions of language of an n-dimensional hypercubic tiling and of n-dimensional factorial language with the purpose of generalizing to n-dimensional hypercubic tiling semigroups a convenient representation of one-dimensional tiling semigroups in terms of a language associated with the tiling introduced by Lawson in [32]. We also present another representation of an n-dimensional hypercubic tiling semigroup as a Rees factor semigroup of a subsemigroup of a generalized Bruck-Reilly extension. In Chapter 5, we develop a description of a tiling semigroup, both one-dimensional and hypercubic, as a P -semigroup; in Chapter 6, we compute a presentation for one-dimensional tilings semigroups and discuss some aspects of the presentability of n-dimensional hypercubic tiling semigroups; in Chapter 7, we provide a necessary and sufficient condition for hypercubic tilings to give rise to isomorphic tiling semigroups. |
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