http://hdl.handle.net/10023/98 Tue, 23 Apr 2013 18:49:00 GMT 2013-04-23T18:49:00Z http://hdl.handle.net/10023/3383 Abstract: Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0. Wed, 01 Sep 2010 00:00:00 GMT http://hdl.handle.net/10023/3383 2010-09-01T00:00:00Z Mitchell, James David Morayne, Michal Peresse, Yann Hamon Quick, Martyn Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0. http://hdl.handle.net/10023/3342 Abstract: Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/10023/3342 2013-01-01T00:00:00Z Dolinka, I Ruskuc, Nik Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area. http://hdl.handle.net/10023/3341 Abstract: Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/10023/3341 2013-01-01T00:00:00Z Abughazalah, Nabilah Ruskuc, Nik Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite. http://hdl.handle.net/10023/3335 Abstract: In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/10023/3335 2013-01-01T00:00:00Z Gray, Robert Duncan Maltcev, Victor D. Mitchell, J. Ruskuc, N. In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals. http://hdl.handle.net/10023/3237 Abstract: For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/10023/3237 2013-01-01T00:00:00Z Boore, Graeme Falconer, Kenneth John For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. http://hdl.handle.net/10023/3058 Abstract: For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties. Sun, 01 Aug 2010 00:00:00 GMT http://hdl.handle.net/10023/3058 2010-08-01T00:00:00Z Quick, Martyn Ruskuc, Nikola For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties. http://hdl.handle.net/10023/2760 Abstract: The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into T-relative H -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups). Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10023/2760 2012-01-01T00:00:00Z Cain, A.J. Gray, R Ruskuc, Nik The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into T-relative H -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups). http://hdl.handle.net/10023/2756 Abstract: Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1 then dimH VθF = 1 for almost all θ , where dimH denotes Hausdorff dimension. We conrm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected. Description: JMF was supported by an EPSRC grant whilst undertaking this work. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/10023/2756 2013-01-01T00:00:00Z Falconer, Kenneth John Fraser, Jonathan Macdonald Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1 then dimH VθF = 1 for almost all θ , where dimH denotes Hausdorff dimension. We conrm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected. http://hdl.handle.net/10023/2149 Abstract: The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families. Tue, 01 Feb 2011 00:00:00 GMT http://hdl.handle.net/10023/2149 2011-02-01T00:00:00Z Atkinson, M.D. Ruskuc, Nik Smith, R The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families. http://hdl.handle.net/10023/2148 Abstract: An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FA-presentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, Bruck-Reilly extensions, zero-direct unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FA-presentable semigroups under that construction is considered, as is the question of whether the FA-presentability of the semigroup obtained from such a construction implies the FA-presentability of the original semigroup[s]. Classifications are also given of the FA-presentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0-simple semigroups. Sun, 01 Aug 2010 00:00:00 GMT http://hdl.handle.net/10023/2148 2010-08-01T00:00:00Z Cain, Alan J. Oliver, Graham Ruskuc, Nik Thomas, Richard M. An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FA-presentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, Bruck-Reilly extensions, zero-direct unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FA-presentable semigroups under that construction is considered, as is the question of whether the FA-presentability of the semigroup obtained from such a construction implies the FA-presentability of the original semigroup[s]. Classifications are also given of the FA-presentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0-simple semigroups. http://hdl.handle.net/10023/2147 Abstract: This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved. Description: Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008) Sun, 01 Nov 2009 00:00:00 GMT http://hdl.handle.net/10023/2147 2009-11-01T00:00:00Z Cain, Alan J. Oliver, Graham Ruskuc, Nik Thomas, Richard M. This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved. http://hdl.handle.net/10023/2146 Abstract: It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general. Tue, 01 Sep 2009 00:00:00 GMT http://hdl.handle.net/10023/2146 2009-09-01T00:00:00Z Gray, R. Ruskuc, N. It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general. http://hdl.handle.net/10023/2145 Abstract: In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property. Sat, 01 Aug 2009 00:00:00 GMT http://hdl.handle.net/10023/2145 2009-08-01T00:00:00Z Maltcev, V. Mitchell, J. D. Ruskuc, N. In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property. http://hdl.handle.net/10023/2144 Abstract: Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved. Wed, 15 Oct 2008 00:00:00 GMT http://hdl.handle.net/10023/2144 2008-10-15T00:00:00Z Gray, R. Ruskuc, Nik Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved. http://hdl.handle.net/10023/2142 Abstract: In this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B. Sun, 01 Jun 2008 00:00:00 GMT http://hdl.handle.net/10023/2142 2008-06-01T00:00:00Z Descalco, L. Ruskuc, N. In this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B. http://hdl.handle.net/10023/2140 Abstract: A pattern class is a set of permutations closed under pattern involvement or, equivalently, defined by certain subsequence avoidance conditions. Any pattern class X which is atomic, i.e. indecomposable as a union of proper subclasses, has a representation as the set of subpermutations of a bijection between two countable (or finite) linearly ordered sets A and B. Concentrating on the situation where A is arbitrary and B = N, we demonstrate how the order-theoretic properties of A determine the structure of X and we establish results about independence, contiguousness and subrepresentations for classes admitting multiple representations of this form. Tue, 01 Jan 2008 00:00:00 GMT http://hdl.handle.net/10023/2140 2008-01-01T00:00:00Z Huczynska, Sophie Ruskuc, Nikola A pattern class is a set of permutations closed under pattern involvement or, equivalently, defined by certain subsequence avoidance conditions. Any pattern class X which is atomic, i.e. indecomposable as a union of proper subclasses, has a representation as the set of subpermutations of a bijection between two countable (or finite) linearly ordered sets A and B. Concentrating on the situation where A is arbitrary and B = N, we demonstrate how the order-theoretic properties of A determine the structure of X and we establish results about independence, contiguousness and subrepresentations for classes admitting multiple representations of this form. http://hdl.handle.net/10023/2138 Abstract: It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.). Fri, 01 Feb 2008 00:00:00 GMT http://hdl.handle.net/10023/2138 2008-02-01T00:00:00Z Cain, Alan James Robertson, Edmund E. Ruskuc, Nik It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.). http://hdl.handle.net/10023/2137 Abstract: Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates. Fri, 22 Oct 2010 00:00:00 GMT http://hdl.handle.net/10023/2137 2010-10-22T00:00:00Z Albert, M.H. Atkinson, M.D. Brignall, R Ruskuc, Nik Smith, R West, J Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates. http://hdl.handle.net/10023/2136 Abstract: In this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented. Mon, 01 Jun 2009 00:00:00 GMT http://hdl.handle.net/10023/2136 2009-06-01T00:00:00Z Dombi, E. R. Ruskuc, N. In this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented. http://hdl.handle.net/10023/2134 Abstract: Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying |im (e)| = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10023/2134 2012-01-01T00:00:00Z Gray, R Ruskuc, Nikola Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying |im (e)| = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr. http://hdl.handle.net/10023/2131 Abstract: Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given. Tue, 01 Nov 2011 00:00:00 GMT http://hdl.handle.net/10023/2131 2011-11-01T00:00:00Z Gray, R Ruskuc, Nik Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given. http://hdl.handle.net/10023/2129 Abstract: For a semigroup S its d-sequence is d(S) = (d1, d2, d3, . . .), where di is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10023/2129 2012-01-01T00:00:00Z Hyde, James Thomas Loughlin, Nicholas Quick, Martyn Ruskuc, Nik Wallis, Alistair For a semigroup S its d-sequence is d(S) = (d1, d2, d3, . . .), where di is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups. http://hdl.handle.net/10023/2128 Abstract: We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister–Schreier type rewriting. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10023/2128 2012-01-01T00:00:00Z Gray, R Ruskuc, Nik We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister–Schreier type rewriting. http://hdl.handle.net/10023/2000 Abstract: A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem. Sun, 01 May 2011 00:00:00 GMT http://hdl.handle.net/10023/2000 2011-05-01T00:00:00Z Albert, M.H. Linton, Stephen Alexander Ruskuc, Nik Vatter, V Waton, S A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem. http://hdl.handle.net/10023/1998 Abstract: Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups. Wed, 01 Jun 2011 00:00:00 GMT http://hdl.handle.net/10023/1998 2011-06-01T00:00:00Z Carvalho, C.A. Gray, R Ruskuc, Nik Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups. http://hdl.handle.net/10023/1997 Abstract: An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible. Fri, 01 Jul 2011 00:00:00 GMT http://hdl.handle.net/10023/1997 2011-07-01T00:00:00Z Brignall, R Ruskuc, Nik Vatter, V An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible. http://hdl.handle.net/10023/1956 Abstract: We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most $\alpha$, for $\alpha \in [2,3)$. In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions. Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10023/1956 2011-01-01T00:00:00Z Falconer, Kenneth John Fraser, Jonathan Macdonald We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most $\alpha$, for $\alpha \in [2,3)$. In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions. http://hdl.handle.net/10023/1615 Abstract: The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed. Fri, 01 Aug 2003 00:00:00 GMT http://hdl.handle.net/10023/1615 2003-08-01T00:00:00Z Huczynska, Sophie Cohen, SD The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed. http://hdl.handle.net/10023/1561 Abstract: This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set. Sat, 01 Jul 2006 00:00:00 GMT http://hdl.handle.net/10023/1561 2006-07-01T00:00:00Z Cain, AJ Robertson, Edmund Frederick Ruskuc, Nikola This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set. http://hdl.handle.net/10023/1553 Abstract: For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O-X on X modulo the full transformation semigroup Ex. In other words, we ask what is the smallest cardinality of a set A of mappings such that <O-X boolean OR A> = T-X. When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set of real numbers) it is uncountable. Mon, 01 Sep 2003 00:00:00 GMT http://hdl.handle.net/10023/1553 2003-09-01T00:00:00Z Higgins, PM Mitchell, James David Ruskuc, Nikola For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O-X on X modulo the full transformation semigroup Ex. In other words, we ask what is the smallest cardinality of a set A of mappings such that <O-X boolean OR A> = T-X. When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set of real numbers) it is uncountable. http://hdl.handle.net/10023/1442 Abstract: Let G be a group, and let <A \ R> be a finite group presentation for G with \R\ greater than or equal to \A\. Then there exists a, finite semigroup, presentation <B \ Q> for G such that \Q\ - \B\ = \R\ - \A\. Moreover, B is either the same generating set or else it contains one additional generator. Mon, 01 Jul 2002 00:00:00 GMT http://hdl.handle.net/10023/1442 2002-07-01T00:00:00Z Campbell, Colin Matthew Mitchell, James David Ruskuc, Nikola Let G be a group, and let <A \ R> be a finite group presentation for G with \R\ greater than or equal to \A\. Then there exists a, finite semigroup, presentation <B \ Q> for G such that \Q\ - \B\ = \R\ - \A\. Moreover, B is either the same generating set or else it contains one additional generator.