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**Journal:** *J. Pure Appl. Algebra* **94** (1994), 49-57

**Abstract:**
In this paper we consider semigroups S(r,n) defined by the presentations for Fibonacci
groups F(r,n). We prove that S(r,n) is a union
of a finite number of copies of F(r,n).
We also consider semigroups S(r,n,k) defined by the presentations
for generalised Fibonacci groups F(r,n,k).
We show that if gcd(n,k)=1 or gcd(n,r+k-1)=1 then again
S(r,n,k) is a union of a finite number of copies of F(r,n,k).
Finally we show that S(2,6,3) is infinite although
F(2,6,3) is finite.

**Journal:** *Computational Support for Discrete Mathematics, DIMACS Series
in Discrete Mathematics and Theoretical Computer Science*
**15** (1994), 29-39

**Abstract:**
Computer based techniques for recognizing finitely presented groups are
quite powerful. Tools available for this purpose are outlined. A general computational
approach for investigating finitely presented groups by way of quotients and subgroups
is described and examples are presented. The techniques can provide detailed
information about group structure. Under suitable circumstances a finitely
presented group can be shown to be soluble and its complete derived series
can be determined, using what is in effect a soluble quotient algorithm.

**Journal:** *Glasgow Math. J.* **35** (1994), 363-371

**Abstract:**
This paper considers a number of questions posed by John Leech thirty
years ago regarding quotients of the triangle group (2, 3, 7) which have remained
unanswered, We provide answers to three of these and throw some light on a fourth
one, which appears to be quite difficult. We examine a few related results. Our
approach is mostly computational, using machine implementations of coset
enumeration techniques.

**Proceedings:** "The Proceedings of the ICMS Workshop on Geometric
and Combinatorial Methods in Group Theory" (A.J. Duncan, N.D. Gilbert
and J. Howie (eds.), Cambridge University Press, 1994), 29-42

**Abstract:** The purpose of this paper is first to give a survey of some
recent results concerning semigroup presentations, and then
to prove a new result which enables us to describe the structure
of semigroups defined by certain presentations.
The main theme is to relate the semigroup S defined by a
presentation P to the group G defined by P. After
mentioning a result of Adjan's giving a sufficient condition for
S to embed in G, we consider some cases where S maps
surjectively (but not necessarily injectively) onto G. In these
examples, we find that S has minimal left and right ideals, and
it turns out that this is a sufficient condition for S to map onto
G. In this case, the kernel of S (i.e. the unique minimal two-sided
ideal of S) is a disjoint union of pairwise isomorphic groups, and
we describe a necessary and sufficient condition for these groups to
be isomorphic to G.
We then move on and expand on these results by proving a new result, which is a sort of rewriting theorem,
enabling us to determine the presentations of the groups in the kernel
in certain cases. We finish off by applying this new result to certain
semigroup presentations and by pointing out its limitations.

**Journal:** *Bull. London Math. Soc.* ** 27** (1995), 46-50

**Abstract:** Let P be a semigroup presentation, let S be the semigroup
defined by P, and let G be the group defined by P.
We prove that if S has both minimal left ideals and minimal right ideal, then
the natural homomorphism f:S->G is onto. The restriction of f
to a maximal subgroup H of the minimal two-sided ideal I of S
is a group epimorphism. We prove that this restriction is an isomorphism if
and only if the idempotents of I are closed under multiplication.
Finally, we apply the obtained results to describe the structure
of the semigroups defined by a semigroup variant of (l,m,n)-presentations.

**Journal:** *Internat. J. Algebra Comput.* **5** (1995), 81-103

**Abstract:** Let S be a finitely presented semigroup having a minimal
left ideal L and a minimal right ideal R.
The main result gives a presentation for the group R intersection L.
It is obtained by rewriting the relations of S,
using the actions of S on its minimal left and
minimal right ideals.
This allows the structure of the minimal two-sided ideal
of S to be described explicitly in terms of a Rees matrix semigroup.
These results are applied to the Fibonacci semigroups,
proving the conjecture that S(r,n,k) is infinite
if g.c.d.(n,k)>1 and g.c.d.(n,r+k-1)>1.
Two enumeration procedures, related to rewriting the presentation of
S into a presentation for R intersection L, are described.
The first enumerates the minimal left and minimal right ideals
of S, and gives the actions of S on these ideals.
The second enumerates the idempotents of the minimal two-sided
ideal of S.

**Journal:** *Semigroup Forum* ** 51** (1995), 47-62

**Abstract:** In this paper we develop a general method for
finding presentations for subsemigroups of semigroups defined
by presentations. This method is based on the idea of
rewriting, akin to the Reidemeister-Schreier method for groups.
We also give two applications of this method.
The first gives a presentation for a two-sided ideal of a semigroup, and implies
that a two-sided ideal of finite index in a finitely presented semigroup is
itself finitely presented. The second gives a presentation for the Schutzenberger group
of a 0-minimal two-sided ideal, and implies that this group is finitely
presented if the ideal contains either finitely many 0-minimal left ideals
or finitely many 0-minimal right ideals.

**Journal:** *Proc. Royal Soc. Edinburgh*, **125** (1995), 1063-1075

**Abstract:** Presentations of Coxeter type are defined for semigroups.
Minimal right ideals of a semigroup defined by such a presentation are
proved to be isomorphic to the group with the same presentation.
A necessary and sufficient condition for
these semigroups to be finite is found.
The structure of semigroups defined by Coxeter type presentations for the
symmetric and alternating groups is examined in detail.

**Journal:** * J. Algebra* **180** (1996), 1-21

**Abstract:** In this paper we investigate subsemigroups of
finitely presented semigroups with respect to the properties of
being finitely generated or finitely presented.
We prove that a right ideal of finite index in a finitely presented semigroup
is itself finitely presented. We also prove that in a free semigroup
of finite rank
a subsemigroup of finite index is finitely presented, and that
any ideal which is finitely generated as a subsemigroup is finitely presented.

**Journal:** *Comm. Algebra* **23** (1995), 5207-5219

**Abstract:** We apply some recent results to
investigate finiteness and structure of some
(semigroup) one-relator
products of two cyclic groups.

**Journal:** *Comm. Algebra* **24** (1996), 3483-3487.

**Abstract:**
We answer a question of some twenty years standing: are the central factors
of nilpotent groups of deficiency zero 3-generated? We prove a negative answer by
giving an explicit presentation for a 3-generator, 3-relator group of order 131072
and class 5 which has central factors which are 4-generated but not 3-generated.
We outline the computational techniques which lead to this result.

**Journal:** * Internat. J. Algebra Comput.* , to appear

**Abstract:** Subsemigroups and ideals of free products of
semigroups are studied with respect to the properties
of being finitely generated or finitely presented.
It is proved that the free product
of any two semigroups, at least one of which is
non-trivial, contains a two-sided ideal which is not
finitely generated as a semigroup, and also contains
a subsemigroup which is finitely generated but not finitely
presented. By way of contrast, in the free
product of two trivial semigroups, every
subsemigroup is finitely generated and finitely presented.
Further, it is proved
that an ideal of a free product of finitely presented
semigroups, which is finitely generated as a semigroup,
is also finitely presented. It is not known whether
one-sided ideals of free products have the same
property, but it is shown that they do
when the free factors are free commutative.

**Journal:** *Proc. ATLAS 10 years on, Cambridge University Press*, to appear

**Abstract:** We examine series of finite presentations which are invariant under the
full symmetric group acting on the set of generators. Evidence from computational
experiments reveals a remarkable tendency for the groups in these series to be
closely related to the orthogonal groups. We examine cases of finite groups in
such series and look in detail at an infinite group with such a presentation. We
prove some theoretical results about 3-generator symmetric presentations and
make a number of conjectures regarding n-generator symmetric presentations.

**Journal:** , submitted

**Abstract:** Let S and T be two infinite semigroups. It is shown that S x T is finitely
generated if and only if S and T are finitely generated and S^2 = S and T^2 = T.
Further, necessary and sufficient conditions are given on S and T for S x T to be
finitely presented. The conditions are applied to find a finite semigroup S with 11
elements and the property that given any infinite finitely presented semigroup T
with T^2 = T then S x T is finitely generated but not finitely presented.

**Journal:** , submitted

**Abstract:** Let S be a finite semigroup. Consider the set p(S) of all elements of S
which can be represented as a product of all the elements of S in some order.
It is shown that p(S) is contained in the minimal ideal M of S and intersects
each maximal subgroup H of M in essentially the same way. It is also shown
that p(S) intersects H in a union of cosets of H'.

**Journal:** , in preparation

**Abstract:** We examine some properties of finitely presented semigroups and investigate whether they are recursively enumerable and whether they are recursive.

**Journal:** , in preparation

**Abstract:** Efficient computational methods are available for computing with finite
groups of permutations. In this paper we utilise such methods in developing an
algorithm to compute the order and algebraic structure of finite transformation
monoids. We start from an algorithm due to Lallement and McFadden and
develop some theoretical improvements. The underlying strategy is to translate
computations in the transformation monoid into computations in some
permutation groups.

**Journal:** *Mathematische Zeitschrift *, to appear

**Abstract: **

**Journal:** *J. Pure Appl. Algebra*, to appear

**Abstract:** A method for finding presentations for subsemigroups
of semigroups defined by presentations is
used to investigate when ideals are finitely presented. It is shown that
an ideal of a finitely presented semigroup
is not necessarily finitely presented, even if it is finitely generated as a semigroup.
By way of contrast,
it is then proved that in a free product of two,
or indeed of finitely many, finite semigroups, every
right ideal which is finitely generated as a
semigroup is finitely presented.

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