Generating "large" subgroups and subsemigroups

In this thesis we will be exclusively considering uncountable groups and semigroups. Roughly speaking the underlying problem is to find “large” subgroups (or subsemigroups) of the object in question, where we consider three different notions of “largeness”: (i) We classify all the subsemigroups of the set of all mapping from a countable set back to itself which contains a specific uncountable subsemigroup; (ii) We investigate topological “largeness”, in particular subgroups which are finitely generated and dense; (iii) We investigate if it is possible to find an integer r such that any countable collection of elements belongs to some r-generated subsemigroup, and more precisely can these elements be obtained by multiplying the generators in a prescribed fashion

An algebraic model for inversion and deletion in bacterial genome rearrangement

Reversals are a major contributor to variation among bacterial genomes, with studies suggesting that reversals involving small numbers of regions are more likely than larger reversals. Deletions may arise in bacterial genomes through the same biological mechanism as reversals, and hence a model that incorporates both is desirable. However, while reversal distances between genomes have been well studied, there has yet to be a model which accounts for the combination of deletions and short reversals. To account for both of these operations, we introduce an algebraic model that utilises partial permutations. This leads to an algorithm for calculating the minimum distance to the most recent common ancestor of two bacterial genomes evolving by short reversals and deletions. The algebraic model makes the existing short reversal models more complete and realistic by including deletions, and also introduces new algebraic tools into evolutionary distance problems.Comment: 19 pages, 10 figure

Random ubiquitous transformation semigroups

Refereed/Peer-reviewed(VLID)4517224Version of recor

An algebraic model for inversion and deletion in bacterial genome rearrangement

Inversions, also sometimes called reversals, are a major contributor to variation among bacterial genomes, with studies suggesting that those involving small numbers of regions are more likely than larger inversions. Deletions may arise in bacterial genomes through the same biological mechanism as inversions, and hence a model that incorporates both is desirable. However, while inversion distances between genomes have been well studied, there has yet to be a model which accounts for the combination of both deletions and inversions. To account for both of these operations, we introduce an algebraic model that utilises partial permutations. This leads to an algorithm for calculating the minimum distance to the most recent common ancestor of two bacterial genomes evolving by inversions (of adjacent regions) and deletions. The algebraic model makes the existing short inversion models more complete and realistic by including deletions, and also introduces new algebraic tools into evolutionary distance problems

Sets of universal sequences for the symmetric group and analogous semigroups

A universal sequence for a group or semigroup S is a sequence of words w1, w2,... such that for any sequence s1, s2,... ε S, the equations wn = sn, n ε ℕ, can be solved simultaneously in S. For example, Galvin showed that the sequence {a-1(anba-n)b-1(anb-1a-n)ba: n ε ℕ is universal for the symmetric group Sym(X) when X is infinite, and Sierpiński showed that (a2b3 (abab3)n+1 ab2 ab3)nεℕ is universal for the monoid XX of functions from the infinite set X to itself. In this paper, we show that under some conditions, the set of universal sequences for the symmetric group on an infinite set X is independent of the cardinality of X. More precisely, we show that if Y is any set such that |Y| ≥ |X|, then every universal sequence for Sym(X) is also universal for Sym(Y). If |X| > 2ℵ0, then the converse also holds. It is shown that an analogue of this theorem holds in the context of inverse semigroups, where the role of the symmetric group is played by the symmetric inverse monoid. In the general context of semigroups, the full transformation monoid XX is the natural analogue of the symmetric group and the symmetric inverse monoid. If X and Y are arbitrary infinite sets, then it is an open question as to whether or not every sequence that is universal for XX is also universal for YY. However, we obtain a sufficient condition for a sequence to be universal for XX which does not depend on the cardinality of X. A large class of sequences satisfy this condition, and hence are universal for XX for every infinite set X