DOCSLIB.ORG

FAITHFUL and TRANSITIVE MONOID ACTIONS AMS - EMS - SPM International Meeting

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
FAITHFUL and TRANSITIVE MONOID ACTIONS AMS - EMS - SPM International Meeting FAITHFUL AND TRANSITIVE MONOID ACTIONS AMS - EMS - SPM International Meeting Porto, 2015 Enric Cosme-Llópez Departament d'Àlgebra Universitat de València • Generalisation of group actions • Automata Theory Introduction Related Work Our Work Bibliography ACTIONS An action of a monoid M on a nonempty set Ω is a monoid homomorphism : M −! TΩ = ff j f :Ω ! Ωg 2 Introduction Related Work Our Work Bibliography ACTIONS An action of a monoid M on a nonempty set Ω is a monoid homomorphism : M −! TΩ = ff j f :Ω ! Ωg • Generalisation of group actions • Automata Theory 2 Objective Give a description of transitive transformation monoids Introduction Related Work Our Work Bibliography ACTIONS The action is faithful if the corresponding monoid homomorphism is injective. We refer to the pair (M; Ω) as transformation monoid. The action is transitive if for each α; β 2 Ω, the cyclic sets coincide αM = βM. 3 Introduction Related Work Our Work Bibliography ACTIONS The action is faithful if the corresponding monoid homomorphism is injective. We refer to the pair (M; Ω) as transformation monoid. The action is transitive if for each α; β 2 Ω, the cyclic sets coincide αM = βM. Objective Give a description of transitive transformation monoids 3 It is based on Module theory - Green Semigroup representation - Clifford, Munn, Ponizovsky Introduction Related Work Our Work Bibliography STEINBERG Related Work [Ste10] B. Steinberg. A Theory of Transformation Monoids: Combinatorics and Representation Theory. Electron. J. Combin., 17(1): Research Paper 164, 56 pp., 2010. 4 Introduction Related Work Our Work Bibliography STEINBERG Related Work [Ste10] B. Steinberg. A Theory of Transformation Monoids: Combinatorics and Representation Theory. Electron. J. Combin., 17(1): Research Paper 164, 56 pp., 2010. It is based on Module theory - Green Semigroup representation - Clifford, Munn, Ponizovsky 4 ] ] ] eM = Gej = Hxiej ej2E(eM) xi2T ej2E(eM) Introduction Related Work Our Work Bibliography PRELIMINARIES Proposition Any transitive and faithful action of a monoid M on a set Ω is equivalent to the action of M on eM/≡ where e is an idempo- tent in I(M) and ≡ is a right-congruence on eM. 5 Introduction Related Work Our Work Bibliography PRELIMINARIES Proposition Any transitive and faithful action of a monoid M on a set Ω is equivalent to the action of M on eM/≡ where e is an idempo- tent in I(M) and ≡ is a right-congruence on eM. ] ] ] eM = Gej = Hxiej ej2E(eM) xi2T ej2E(eM) 5 For each element xi in a right-transversal T, we define an equivalence relation 'i in the set n = f1; ··· ; ng as follows i 0 0 j ' j , Hxi(j)ρm = Hxi(j )ρm; for all m 2 M Introduction Related Work Our Work Bibliography PRELIMINARIES M embeds in Ge on M via the map m 7! (ρm; m), where (j)ρm = g such that ejm = gej0 6 Introduction Related Work Our Work Bibliography PRELIMINARIES M embeds in Ge on M via the map m 7! (ρm; m), where (j)ρm = g such that ejm = gej0 For each element xi in a right-transversal T, we define an equivalence relation 'i in the set n = f1; ··· ; ng as follows i 0 0 j ' j , Hxi(j)ρm = Hxi(j )ρm; for all m 2 M 6 Introduction Related Work Our Work Bibliography ON TRANSITIVE TRANSFORMATION MONOIDS Theorem If (M; Ω) is a transitive transformation monoid, there exists a core-free subgroup H of Ge such that every member of the partition fHxiej j xi 2 T; ej 2 E(eM)g is contained in a ≡-class. Furthermore, 0 i 0 i) If Hxiej≡Hxi0 ej0 then i = i and j ' j ; ii) If Hxiej≡Hxiej0 then (Hxiej)m≡(Hxiej0 )m for all m 2 M; iii) For different idempotents ej; ej0 there exists some xi 2 T with Hxiej and Hxiej0 not ≡-related. The converse of this Theorem also holds. 7 Introduction Related Work Our Work Bibliography ON TRANSITIVE TRANSFORMATION MONOIDS Consequences • We can construct all transitive and faithful actions of a monoid that restrict to a given transitive and faithful group action. • There exist non-equivalent transitive and faithful monoid actions that restrict to the same group action. • These theorems extend the classical theorems on the description of transitive permutation groups. 8 A Theory of Transformations Monoids [Ste10] ''We hope that the theory of primitive permutation groups can be used to understand primitive transformation monoids in the case the maximal subgroups of I(M) are non-trivial'' Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS An action of a monoid M on a set Ω is said to be primitive if it admits no non-trivial proper congruences. 9 Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS An action of a monoid M on a set Ω is said to be primitive if it admits no non-trivial proper congruences. A Theory of Transformations Monoids [Ste10] ''We hope that the theory of primitive permutation groups can be used to understand primitive transformation monoids in the case the maximal subgroups of I(M) are non-trivial'' 9 Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS Theorem Let M be a transitive transformation monoid such that Ge is non-trivial. Then M is primitive if and only if there exists a maximal core-free subgroup H of Ge such that the relations ' defined by H in E(eM) satisfy the following conditions i) For different idempotents ej; ej0 there exist finite s s+1 sequences (xit )t=1 and (ejt )t=1 of elements of T and E(eM), respectively, such that i1 i2 is−1 is 0 j = j1 ' j2 ' ···' js ' js+1 = j ii) For different idempotents ej; ej0 there exists some xi 2 T with j 6'i j0. 10 Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS Let (M; Ω) be a transitive transformation monoid. Consequences • If Ge has a primitive action, there exists a unique primitive action of M that restricts to the primitive action of Ge. • If Ge is a non-trivial nilpotent group. Then M has a primitive action if and only if M is a cyclic group of prime order. • If jΩej = 2, then the action of M on Ω cannot be primitive. 11 Introduction Related Work Our Work Bibliography BIBLIOGRAPHY A. Ballester-Bolinches, E. Cosme-Llópez, P. Jiménez-Seral, Some Contributions to the Theory of Transformation Monoids, Preprint, 2015. J. Rhodes and B. Steinberg, The q-theory of finite semigroups, Springer Monographs in Mathematics. Springer, 2009. B. Steinberg, A theory of transformation monoids: Combinatorics and representation theory, The Electronic Journal of Combinatorics, 17:1–56, 2010. HSRM Beamer Theme, by Benjamin Weiss. 12.
Load more

Recommended publications
  • Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19Th, 2020
    Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19th, 2020 Group Isomorphism Theorems + Group Actions The Isomorphism Theorems for Groups Group Actions Polynomial Invariants and An This material represents x3.2.3-3.3.2 from the course notes. Quotients and Homomorphisms, I Like with rings, we also have various natural connections between normal subgroups and group homomorphisms. To begin, observe that if ' : G ! H is a group homomorphism, then ker ' is a normal subgroup of G. In fact, I proved this fact earlier when I introduced the kernel, but let me remark again: if g 2 ker ', then for any a 2 G, then '(aga−1) = '(a)'(g)'(a−1) = '(a)'(a−1) = e. Thus, aga−1 2 ker ' as well, and so by our equivalent properties of normality, this means ker ' is a normal subgroup. Thus, we can use homomorphisms to construct new normal subgroups. Quotients and Homomorphisms, II Equally importantly, we can also do the reverse: we can use normal subgroups to construct homomorphisms. The key observation in this direction is that the map ' : G ! G=N associating a group element to its residue class / left coset (i.e., with '(a) = a) is a ring homomorphism. Indeed, the homomorphism property is precisely what we arranged for the left cosets of N to satisfy: '(a · b) = a · b = a · b = '(a) · '(b). Furthermore, the kernel of this map ' is, by definition, the set of elements in G with '(g) = e, which is to say, the set of elements g 2 N. Thus, kernels of homomorphisms and normal subgroups are precisely the same things.
    [Show full text]
  • The Fundamental Groupoid of the Quotient of a Hausdorff
    The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action Ronald Brown,∗ Philip J. Higgins,† Mathematics Division, Department of Mathematical Sciences, School of Informatics, Science Laboratories, University of Wales, Bangor South Rd., Gwynedd LL57 1UT, U.K. Durham, DH1 3LE, U.K. October 23, 2018 University of Wales, Bangor, Maths Preprint 02.25 Abstract The main result is that the fundamental groupoidof the orbit space of a discontinuousaction of a discrete groupon a Hausdorffspace which admits a universal coveris the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author’s book on Topology [3]. This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these results, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader.
    [Show full text]
  • GROUP ACTIONS 1. Introduction the Groups Sn, An, and (For N ≥ 3)
    GROUP ACTIONS KEITH CONRAD 1. Introduction The groups Sn, An, and (for n ≥ 3) Dn behave, by their definitions, as permutations on certain sets. The groups Sn and An both permute the set f1; 2; : : : ; ng and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If we label the vertices of the n-gon in a definite manner by the numbers from 1 to n then we can view Dn as a subgroup of Sn. For instance, the labeling of the square below lets us regard the 90 degree counterclockwise rotation r in D4 as (1234) and the reflection s across the horizontal line bisecting the square as (24). The rest of the elements of D4, as permutations of the vertices, are in the table below the square. 2 3 1 4 1 r r2 r3 s rs r2s r3s (1) (1234) (13)(24) (1432) (24) (12)(34) (13) (14)(23) If we label the vertices in a different way (e.g., swap the labels 1 and 2), we turn the elements of D4 into a different subgroup of S4. More abstractly, if we are given a set X (not necessarily the set of vertices of a square), then the set Sym(X) of all permutations of X is a group under composition, and the subgroup Alt(X) of even permutations of X is a group under composition. If we list the elements of X in a definite order, say as X = fx1; : : : ; xng, then we can think about Sym(X) as Sn and Alt(X) as An, but a listing in a different order leads to different identifications 1 of Sym(X) with Sn and Alt(X) with An.
    [Show full text]
  • Lie Group and Geometry on the Lie Group SL2(R)
    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear
    [Show full text]
  • Definition 1.1. a Group Is a Quadruple (G, E, ⋆, Ι)
    GROUPS AND GROUP ACTIONS. 1. GROUPS We begin by giving a definition of a group: Definition 1.1. A group is a quadruple (G, e, ?, ι) consisting of a set G, an element e G, a binary operation ?: G G G and a map ι: G G such that ∈ × → → (1) The operation ? is associative: (g ? h) ? k = g ? (h ? k), (2) e ? g = g ? e = g, for all g G. ∈ (3) For every g G we have g ? ι(g) = ι(g) ? g = e. ∈ It is standard to suppress the operation ? and write gh or at most g.h for g ? h. The element e is known as the identity element. For clarity, we may also write eG instead of e to emphasize which group we are considering, but may also write 1 for e where this is more conventional (for the group such as C∗ for example). Finally, 1 ι(g) is usually written as g− . Remark 1.2. Let us note a couple of things about the above definition. Firstly clo- sure is not an axiom for a group whatever anyone has ever told you1. (The reason people get confused about this is related to the notion of subgroups – see Example 1.8 later in this section.) The axioms used here are not “minimal”: the exercises give a different set of axioms which assume only the existence of a map ι without specifying it. We leave it to those who like that kind of thing to check that associa- tivity of triple multiplications implies that for any k N, bracketing a k-tuple of ∈ group elements (g1, g2, .
    [Show full text]
  • Ideal Classes and SL 2
    IDEAL CLASSES AND SL2 KEITH CONRAD 1. Introduction A standard group action in complex analysis is the action of GL2(C) on the Riemann sphere C [ f1g by linear fractional transformations (M¨obiustransformations): a b az + b (1.1) z = : c d cz + d We need to allow the value 1 since cz + d might be 0. (If that happens, az + b 6= 0 since a b ( c d ) is invertible.) When z = 1, the value of (1.1) is a=c 2 C [ f1g. It is easy to see this action of GL2(C) on the Riemann sphere is transitive (that is, there is one orbit): for every a 2 C, a a − 1 (1.2) 1 = a; 1 1 a a−1 so the orbit of 1 passes through all points. In fact, since ( 1 1 ) has determinant 1, the action of SL2(C) on C [ f1g is transitive. However, the action of SL2(R) on the Riemann sphere is not transitive. The reason is the formula for imaginary parts under a real linear fractional transformation: az + b (ad − bc) Im(z) Im = cz + d jcz + dj2 a b a b when ( c d ) 2 GL2(R). Thus the imaginary parts of z and ( c d )z have the same sign when a b ( c d ) has determinant 1. The action of SL2(R) on the Riemann sphere has three orbits: R [ f1g, the upper half-plane h = fx + iy : y > 0g, and the lower half-plane. To see that the action of SL2(R) on h is transitive, pick x + iy with y > 0.
    [Show full text]
  • Chapter 9: Group Actions
    Chapter 9: Group actions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Spring 2014 M. Macauley (Clemson) Chapter 9: Group actions Math 4120, Spring 2014 1 / 27 Overview Intuitively, a group action occurs when a group G \naturally permutes" a set S of states. For example: The \Rubik's cube group" consists of the 4:3 × 1019 actions that permutated the 4:3 × 1019 configurations of the cube. The group D4 consists of the 8 symmetries of the square. These symmetries are actions that permuted the 8 configurations of the square. Group actions help us understand the interplay between the actual group of actions and sets of objects that they \rearrange." There are many other examples of groups that \act on" sets of objects. We will see examples when the group and the set have different sizes. There is a rich theory of group actions, and it can be used to prove many deep results in group theory. M. Macauley (Clemson) Chapter 9: Group actions Math 4120, Spring 2014 2 / 27 Actions vs. configurations 1 2 The group D can be thought of as the 8 symmetries of the square: 4 4 3 There is a subtle but important distinction to make, between the actual 8 symmetries of the square, and the 8 configurations. For example, the 8 symmetries (alternatively, \actions") can be thought of as e; r; r 2; r 3; f ; rf ; r 2f ; r 3f : The 8 configurations (or states) of the square are the following: 1 2 4 1 3 4 2 3 2 1 3 2 4 3 1 4 4 3 3 2 2 1 1 4 3 4 4 1 1 2 2 3 When we were just learning about groups, we made an action diagram.
    [Show full text]
  • Group Action on Quantum Field Theory
    International Mathematical Forum, Vol. 14, 2019, no. 4, 169 - 179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.9728 Group Action on Quantum Field Theory Ahmad M. Alghamdi and Roa M. Makki Department of Mathematical Sciences Faculty of Applied Sciences Umm Alqura University P.O. Box 14035, Makkah 21955, Saudi Arabia This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd. Abstract The main aim of this paper is to develop some tools for understand- ing and explaining the concept of group action on a space and group action on algebra by using subgroups with finite index and both re- striction and transfer maps. We extend this concept on quantum field theory. 0 Introduction Group action is a very important tool in mathematics. Namely, if a certain group acts on a certain object, then many results and phenomena can be resolved. Let G be a group, the object can be a set and then we have the notion of a group acts on a set see [7]. Such object can be a vector space which results the so called G-space or G-module see for instance [6]. In some cases, the object can be a group and then we have the notion of group acts on a group see [3]. The topic of a group acting on an algebra can be seen in [5, 9]. This approaches open many gates in the research area called the G-algebras over a field. If we have an action from the group G to an object, then there is a fixed point under this action.
    [Show full text]
  • Proper Groupoids and Momentum Maps: Linearization, Affinity, and Convexity
    PROPER GROUPOIDS AND MOMENTUM MAPS: LINEARIZATION, AFFINITY, AND CONVEXITY NGUYEN TIEN ZUNG Abstract. We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural locally flat integral affine structure, which makes it into an affine manifold with locally convex polyhedral bound- ary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps. Dedicated to Alan Weinstein on the occasion of his 60th birthday Contents 1. Introduction 2 2. Proper groupoids 2 2.1. Linearization of proper groupoids 2 2.2. The averaging process 7 2.3. Spaces of maps and Ck-norms 9 2.4. C0 estimates 11 2.5. Ck estimates 13 2.6. Proof of Theorem 2.5 15 3. Momentum maps 19 3.1. Affine structure on base spaces 19 3.2. Affinity and local convexity of momentum maps 21 3.3. Global convexity of momentum maps 24 References 28 Date: third version, 30/Aug/2004.
    [Show full text]
  • The General Linear Group Over the field Fq, Is the Group of Invertible N × N Matrices with Coefficients in Fq
    ALGEBRA - LECTURE II 1. General Linear Group Let Fq be a finite field of order q. Then GLn(q), the general linear group over the field Fq, is the group of invertible n × n matrices with coefficients in Fq. We shall now compute n the order of this group. Note that columns of an invertible matrix give a basis of V = Fq . Conversely, if v1, v2, . vn is a basis of V then there is an invertible matrix with these vectors as columns. Thus we need to count the bases of V . This is easy, as v1 can be any non-zero vector, then v2 any vector in V not on the line spanned by v1, then v3 any vector in V not n n on the plane spanned by v1 and v2, etc... Thus, there are q − 1 choices for v1, then q − q n 2 choices for v2, then q − q choices for v3, etc. Hence the order of GLn(q) is n−1 Y n i |GLn(q)| = (q − q ). i=0 2. Group action Let X be a set and G a group. A group action of G on X is a map G × X → X (g, x) 7→ gx such that (1) 1 · x = x for every x in X. (Here 1 is the identity in G.) (2) (gh)x = g(hx) for every x in X and g, h in G. Examples: n (1) G = GLn(q) and X = Fq . The action is given by matrix multiplication. (2) X = G and the action is the conjugation action of G.
    [Show full text]
  • The Modular Group and the Fundamental Domain Seminar on Modular Forms Spring 2019
    The modular group and the fundamental domain Seminar on Modular Forms Spring 2019 Johannes Hruza and Manuel Trachsler March 13, 2019 1 The Group SL2(Z) and the fundamental do- main Definition 1. For a commutative Ring R we define GL2(R) as the following set: a b GL (R) := A = for which det(A) = ad − bc 2 R∗ : (1) 2 c d We define SL2(R) to be the set of all B 2 GL2(R) for which det(B) = 1. Lemma 1. SL2(R) is a subgroup of GL2(R). Proof. Recall that the kernel of a group homomorphism is a subgroup. Observe ∗ that det is a group homomorphism det : GL2(R) ! R and thus SL2(R) is its kernel by definition. ¯ Let R = R. Then we can define an action of SL2(R) on C ( = C [ f1g ) by az + b a a b A:z := and A:1 := ;A = 2 SL (R); z 2 : (2) cz + d c c d 2 C Definition 2. The upper half-plane of C is given by H := fz 2 C j Im(z) > 0g. Restricting this action to H gives us another well defined action ":" : SL2(R)× H 7! H called the fractional linear transformation. Indeed, for any z 2 H the imaginary part of A:z is positive: az + b (az + b)(cz¯ + d) Im(z) Im(A:z) = Im = Im = > 0: (3) cz + d jcz + dj2 jcz + dj2 a b Lemma 2. For A = c d 2 SL2(R) the map µA : H ! H defined by z 7! A:z is the identity if and only if A = ±I.
    [Show full text]
  • Groups Acting on Sets
    Groups acting on sets 1 Definition and examples Most of the groups we encounter are related to some other structure. For example, groups arising in geometry or physics are often symmetry groups of a geometric object (such as Dn) or transformation groups of space (such as SO3). The idea underlying this relationship is that of a group action: Definition 1.1. Let G be a group and X a set. Then an action of G on X is a function F : G × X ! X, where we write F (g; x) = g · x, satisfying: 1. For all g1; g2 2 G and x 2 X, g1 · (g2 · x) = (g1g2) · x. 2. For all x 2 X, 1 · x = x. When the action F is understood, we say that X is a G-set. Note that a group action is not the same thing as a binary structure. In a binary structure, we combine two elements of X to get a third element of X (we combine two apples and get an apple). In a group action, we combine an element of G with an element of X to get an element of X (we combine an apple and an orange and get another orange). Example 1.2. 1. The trivial action: g · x = x for all g 2 G and x 2 X. 2. Sn acts on f1; : : : ; ng via σ · k = σ(k) (here we use the definition of multiplication in Sn as function composition). More generally, if X is any set, SX acts on X, by the same formula: given σ 2 SX and x 2 X, define σ · x = σ(x).
    [Show full text]