#### Year

2012

#### Degree Name

Doctor of Philosophy

#### Department

School of Mathematics and Applied Statistics

#### Recommended Citation

Elfard, Ali Sayed R., Free paratopological groups, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2012. https://ro.uow.edu.au/theses/3590

#### Abstract

In 2003, Romaguera, Sanchis and Tkachenko proved the existence of the free paratopological group and the free abelian paratopological group on any topological space.

This thesis is a study of some of fundamental properties of the topologies of the free paratopological group FP(X) and the free abelian paratopological group AP(X) on any topological space X (Romaguera, Sanchis and Tkachenko studied Graev free paratopological groups, we study Markov free paratopological groups).

The result often known as Joiner's lemma is fundamental in understanding the topology of the free topological group F(X) on a Tychono space X. In this thesis, analogues of Joiner's lemma for free paratopological groups FP(X) and AP(X) on a T_{1} space X are proved. For each n ∈ N, let FP_{n}(X) be the subspace of FP(X) consisting of all words of length at most n. The analogue of Joiner's lemma for FP(X) takes the following form: Let X be a T_{1} space and let w = x_{1}^{e1}x_{2}^{e2}...x_{n}^{en} be a reduced word in FP_{n}(X), where xi ∈ X and ∈i = ±1 for i = 1,2,…,n and if x_{i} = x_{i}+1 for some i = 1,2,…,n - 1 then ∈_{i} = ∈_{i}+1. Let B denote the collection of all sets of the form U_{1}^{e1}U_{2}^{e2}…U_{n}^{en}, where for i = 1,2,…,n the set U_{i} is a neighbourhood of x_{i} in X when ∈_{i} = 1 and U_{i} = {x_{i}} when ∈_{i} = -1. Then B is a base for the neighborhood system at w in the subspace FP_{n}(X) of FP(X). Using this, it is shown that the following conditions are equivalent for a space X: (1) X is T_{1}; (2) FP(X) is T_{1}; (3) the subspace X of FP(X) is closed; (4) the subspace X^{-1} of FP(X) is discrete; (5) the subspace X^{-1} is T_{1}; (6) the subspace X^{-1} is closed; and (7) the subspace FP_{n}(X) is closed for all n ∈ N.

In this thesis, a neighborhood base at the identity e in FP_{2}(X) is found. Furthermore, we describe neighborhood bases at the identity for the topologies of free paratopological groups FP(X) and AP(X) on any topological space X. In particular, we describe simple neighborhood bases for free paratopological groups FP(X) and AP(X) on any Alexandroff space X. Then as applications of this, we prove that if the space X is T_{0}, then the free paratopological group FP(X) is T_{0} and we characterize the space X for which free paratopological groups FP(X) and AP(X) are topological groups.

For each n ∈ N, denote by i_{n} the natural mapping from (X ⊕ X^{-1} ⊕ {e})^{n} to FP_{n}(X). In this thesis, a number of characterizations are given of the circumstances under which i_{2} : (X ⊕ X_{d}^{-1} ⊕ {e})^{2} → FP_{2}(X) is a quotient map, where X is a T_{1} space and X_{d}^{-1} denotes the set X^{-1} equipped with the discrete topology. Further characterizations are given in the case where X is a transitive T_{1} space. Several specific spaces and classes of spaces are also examined. For example, i_{2} is a quotient for every countable subspace of R, i_{2} is not a quotient for any uncountable compact subspace of R, and it is undecidable in ZFC whether an uncountable subspace of R exists for which i_{2} is a quotient.