TY: THES
T1 - Algebraic aspects of tiling semigroups
A1 - Almeida, Filipa Soares de, 1978-
N2 - 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.
UR - http://repositorio.ul.pt/handle/10451/2299
Y1 - 2010
PB - No publisher defined