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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
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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