In Section 4.4 we will calculate rank(ΩΩ : U) for any subset U of ΩΩ that lies in a
known ≈-equivalence class (other than the≈-equivalence class of Sym(Ω)).
Once this is done, any result concerning the position of a given a subset U of ΩΩ
in the preorder 4, immediately implies something about rank(ΩΩ :U).
1.5
Outlook
The following is a brief overview of the structure of the rest of the thesis.
In Chapter 2 we will recall some well-known results from set theory, algebra and topology and state some definitions and conventions that will be used and followed in the later chapters.
In Chapter 3 we will calculate the Sierpi´nski rank of several naturally occurring transformation semigroups such as semigroups of injections and surjections, and semigroups of order endomorphisms, continuous functions or differentiable functions. In Chapter 4 we will study the Bergman-Shelah preorder on subsets of ΩΩ. We will
summarise the results by Mesyan from [25], then construct some previously unknown
≈-equivalence classes and calculate the relative ranks of ΩΩ modulo subsets that lie
in known equivalence classes of ≈.
In Chapter 5 we will consider discrete metrics defined on the countable set Ω and the semigroups of Lipschitz functions on these metric spaces. We will investigate the question of where the semigroups of Lipschitz functions on Ω lie in the Bergman- Shelah preorder on ΩΩ and what their relative rank in ΩΩ is. We will prove several
results that answer these questions for a wide range of countable, discrete metric spaces.
Analogously to Chapter 5 we will investigate where such semigroups of endomor- phisms can lie in the Bergman-Shelah preorder and what their relative rank in ΩΩ
is. We will completely answer these questions in the case where the binary relation is a preorder, bipartite graph or a tolerance (a reflexive and symmetric relation).
Finally, in Chapter 7 we will give a short summary of some of the open problems and questions that arise from the results in the other chapters.
Chapter 2
Preliminaries
2.1
Functions
Throughout, Ω will be a non-empty set. As mentioned earlier ΩΩ denotes the set of
all functions with range and domain equal to Ω. We will also call elements of ΩΩ
maps (ormappings)on Ω. Elements of Ω will be denoted by lower case Greek letters like α, β, γ and elements of ΩΩ by lower case roman letters like f, g, h. Denote by
dom(f) the domain off and by im(f) the image of f.
We will write functions on the right of their argument so that composition of functions is done from left to right. That is, the function value of an element α∈Ω under a map f ∈ΩΩ is denoted byαf and αf g = (αf)g.
Apartial map on Ω is a functionf whose domain and range are subsets of Ω. Iff
is a partial map on Ω andA is a subset of the domain off, thenf restricted to Aor
f onAis the partial mapf′ :A−→Ω such thatαf′ =αf for allα∈A. Conversely, if f′ is a partial map with domain A, then a partial map f is an extension of f′, if
A is a subset of the domain of f and f restricted to A isf′.
Since partial maps on a set Ω are a special kind of subset of Ω×Ω we may form unions of partial maps. Unions of partial maps are not, in general, partial maps themselves. In fact, if {gi : i ∈ I } is a set of partial maps on Ω, then Si∈Igi is
a partial map if and only if gi restricted to dom(gi)∩dom(gj) equals gj restricted
to dom(gi)∩dom(gj) for all i, j ∈ I. Two special cases in which Si∈Igi is a partial
map are if
(i) dom(gi)∩dom(gj) = ∅ whenever i6=j; or
(ii) the set of functions is {g0, g1, g2, . . .} and everygi extends gi−1.
In the chapters to come we will often construct functions on Ω from partial maps on Ω. Sometimes this is done implicitly. For example if A and B are subsets of Ω and
f ∈ ΩΩ, then a sentence like “f maps A bijectively (injectively, etc.) to B” means
that f restricted toA is a bijection (injection etc.) from A to B.
IfA ⊆ Ω and f ∈ΩΩ we will denote by Af the image of the set A under f, i.e.
the set {αf : α ∈ A}. Similarly Af−1 ={α ∈ Ω : αf ∈ A} is the pre-image of A under f. A kernel class of f is the pre-image {α}f−1 of a singleton subset of Ω.
Note that A⊆(Af)f−1 but that (Af)f−1 is not equal toA in general. Also observe
that A(f g) = (Af)g and A(f g)−1 = (Ag−1)f−1 for any f, g ∈ ΩΩ and so we will
simply write these as Af g and Ag−1f−1 respectively.
Ifg is a (partial) bijection, we will still writeg−1 to denote the inverse of g. Note
that, in this case, {α}g−1 ={αg−1} for any α in the image ofg.