Attempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator.

Size: px
Start display at page:

Download "Attempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator."

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination SEMIGROUP THEORY WITH ADVANCED TOPICS MTHE7011A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHE7011A Module Contact: Dr Robert Gray, MTH Copyright of the University of East Anglia Version: 1

2 - 2 - Note: Throughout, for an element x of a semigroup S, we use L x to denote the L- class of x, and R x to denote the R-class of x. We use ǫ to denote the empty word. Any theorem you use must be clearly stated. A theorem can be used without proof unless you are required to prove it. 1. (i) Define what it means for a semigroup S to be residually finite. (a) Prove that every finite semigroup is residually finite. (b) Prove that if S is a residually finite semigroup, and T is a subsemigroup of S, then T is also residually finite. (ii) Let A be a finite alphabet and let A be the free monoid over A. Prove that A is residually finite. (Hint: Consider Rees quotients.) (iii) Determine which of the following semigroups are residually finite. In each case justify your answer. (a) The bicyclic monoid B = b,c bc = 1. (b) The full transformation monoid T N of all mappings from N to N. (c) The semigroup defined by the following presentation a,b,x,y ax = xa,ay = ya,bx = xb,by = yb. MTHE7011A Version: 1

3 Let S be the semigroup defined by the presentation a,b a 5 = a, b 2 = b, ba = a 4 b. (i) Show that any word over the alphabet {a, b} can be transformed using the relations into a word from the set W = {a i b j : 0 i 4,0 j 1}\{ǫ}. By using standard methods it is possible to show that all of the words in W represent distinct elements of S, so that S = 9. You may use this fact without proof below. (ii) Draw the right and left Cayley graphs of S with respect to the generators {a,b}. Determine the R- and L-classes of S. (iii) Show that S has six H-classes, and that they are all groups. MTHE7011A PLEASE TURN OVER Version: 1

4 Let S be a semigroup. Recall that a two-sided congruence on S is an equivalence relation ρ satisfying x ρ y & z ρ t xz ρ yt for all x,y,z,t S. (i) Define the notions of left congruence and right congruence on a semigroup. Prove that a relation ρ on S is a two-sided congruence if and only if it is both a left and a right congruence. (ii) Define Green s relations R, L on a semigroup S. Prove that R is a left congruence and L is a right congruence. (iii) Let T = I Λ be a rectangular band. (a) Show that if I = 1 then every equivalance relation on T is a congruence. (b) Show that if I = Λ = 2 then there is at least one equivalence relation on T that is not a congruence. 4. (i) Define what it means for an element e of a semigroup S to be an idempotent. Show that for any pair of idempotents e,f S we have erf ef = f and fe = e. (ii) Let S be a semigroup, and let a,b S be two R-related elements with as = b and bt = a. Define mappings ρ s : L a L b and ρ t : L b L a by xρ s = xs and xρ t = xt. Prove that ρ s and ρ t are mutually inverse bijections which map H-classes onto H-classes. (This result is known as Green s Lemma.) [8 marks] (iii) Let S be a semigroup in which every H-class is a group and let D be a D-class of S. Show that D is a subsemigroup of S. [7 marks] MTHE7011A Version: 1

5 (i) Define what it means for a semigroup S to be simple. Define Green s relation J on a semigroup S. Prove that a semigroup S is simple if and only if it has a single J -class. [6 marks] (ii) Recall that the bicyclic monoid B is defined by the presentation b,c bc = 1 and that B = {c i b j : i,j 0} with all these elements being distinct. Consider the following two subsets of B: D 1 = {c 3i+7 b 3j+7 : i,j 0}, D 2 = {c 3i+8 b 3j+8 : i,j 0} and let T = D 1 D 2. (a) Prove that T is a subsemigroup of B. (b) Prove that D 1 and D 2 are the D-classes of T. (c) Prove that T is simple. Is J = D in T? Justify your answer. [14 marks] 6. (i) Define what it means for a semigroup S to be inverse. Let S be an inverse semigroup and let D be a D-class of S. Show that the number of R-classes in D is equal to the number of L-classes in D. (ii) Let S be a semigroup and let ρ be a congruence on S. Show that if S is finitely generated then S/ρ is finitely generated. (iii) Show that any finitely generated Clifford semigroup has finitely many idempotents. Does there exist a finitely generated inverse semigroup which has infinitely many idempotents? Justify your answer. END OF PAPER MTHE7011A Version: 1

6 Q1 (advanced topic) MTHE6011A and MTHE7011A Semigroup Theory Feedback on the Exam All of questions (i), (ii) and (iii)(a) and (iii)(b) were material seen in the advanced topic, in the sense that the answers to all of these questions may be found in the lecture notes for the advanced topic. The last part (iii)(c) was new. Students performed less well on this question than expected. Most did reasonably well on parts (i) and (ii). Marks were lost in (iii)(a) for not giving enough detail. Very few students correctly answered (iii)(b) even though this was covered in the module. Only one student noticed that the semigroup given in (iii)(c) is the direct product of two free semigroups (and hence is residually finite by (ii) together with results from the module on direct products). Q1 Students preformed well on this question. Most marks were lost in the second part of (iii): proving that S has a single D-class. Since D = L R one can fix an element and compute its D-class by first computing its R-class, call it R, and then taking the union of the L-classes of the elements from R. Examples of computing D-classes like this did appear in the exercise sheets for the module, so it was surprising that so many students were unable to do this. Q2 Students performed well in this question. Marks were lost in part (i) for not explaining how every word can be transformed into a word of the forms a i b j by repeatedly applying the relation ba = a 4 b. Marks were lost in part (iii) because some students either failed to work out how to write down the H-classes, and some students did not remember that an H-class is a group iff it contains an idempotent. So all that was needed was to check that each H-class contains an idempotent. Q3 Students did well in parts (i) and (ii), but found part (iii) more challenging. For (iii)(a) the semigroup in question is isomorphic to a right zero semigroup, and one of the exercises in the module was to show that any equivalence relation on a right zero semigroup is a congruence. Almost every student failed to get (iii)(b) correct. This question was designed to test whether you know (a) what an equivalence relation is and (b) whether you know what a congruence is. If I = {1, 2} and Λ = {1, 2} then one example of an equivalence relation on T which is not a congruence would be the equivalence relation with equivalence classes {(1, 1)}, {(1, 2)}, and {(2, 1), (2, 2)}. To show it is not a congruence it suffices to note that (2, 1) (2, 2) but when we left-multiply by (1, 1) we get a pair that is not related. Q4 Students did quite well on this question. A lot of marks were lost in part (iii). Part (iii) could be proved either as (a) an application of parts (i) and (ii) or (b) by using the Miller Clifford Theorem.

7 Q5 Students found this question difficult. Part (i) was bookwork and most students did well on it. Part (ii) was new and students struggled with it. Most students made a decent attempt at (ii)(a). For (ii)(b) some students assumed that Green s relations in D 1 and D 2 were given by restricting the corresponding relations from B. This is true, but one needs to show that D 1 and D 2 are regular before being able to deduce this (the most natural thing is to prove D 1 = D2 = B). Nobody managed to prove that T is simple. Many did see that J D follows easily from the previous parts of the question. Q6 Very few students attempted this question. Most that did found it hard. For those that did attempt is, most marks were lost in part (iii). To show that a f.g. Clifford semigroup has finitely many idempotents one needs to prove (a) H is a congruence (b) S/H is a semilattice and that (c) any finitely generated semilattice is finite. Then apply part (ii).

Attempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator.

Attempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 SEMIGROUP THEORY MTHE6011A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are not permitted

More information

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 CRYPTOGRAPHY MTHD6025A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt

More information

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 SETS, NUMBERS AND PROBABILITY MTHA4001Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. penalised

More information

Pseudo-finite monoids and semigroups

Pseudo-finite monoids and semigroups University of York NBSAN, 07-08 January 2018 Based on joint work with Victoria Gould and Dandan Yang Contents Definitions: what does it mean for a monoid to be pseudo-finite, or pseudo-generated by a finite

More information

Finiteness conditions and index in semigroup theory

Finiteness conditions and index in semigroup theory Finiteness conditions and index in semigroup theory Robert Gray University of Leeds Leeds, January 2007 Robert Gray (University of Leeds) 1 / 39 Outline 1 Motivation and background Finiteness conditions

More information

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 CALCULUS AND MULTIVARIABLE CALCULUS MTHA4005Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.

More information

Attempt THREE questions. You will not be penalised if you attempt additional questions.

Attempt THREE questions. You will not be penalised if you attempt additional questions. UNIVERITY OF EAT ANGLIA chool of Mathematics Main eries UG Examination 07 8 MATHEMATIC FOR CIENTIT C MTHB5007B Time allowed: Hours Attempt THREE questions. You will not be penalised if you attempt additional

More information

Semigroup presentations via boundaries in Cayley graphs 1

Semigroup presentations via boundaries in Cayley graphs 1 Semigroup presentations via boundaries in Cayley graphs 1 Robert Gray University of Leeds BMC, Newcastle 2006 1 (Research conducted while I was a research student at the University of St Andrews, under

More information

Axioms of Kleene Algebra

Axioms of Kleene Algebra Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.

More information

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE612B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.

More information

FLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y

FLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 FLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.

More information

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 FERMAT S LAST THEOREM MTHD6024B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. penalised

More information

SEMIGROUP PRESENTATIONS FOR CONGRUENCES ON GROUPS

SEMIGROUP PRESENTATIONS FOR CONGRUENCES ON GROUPS Bull. Korean Math. Soc. 50 (2013), No. 2, pp. 445 449 http://dx.doi.org/10.4134/bkms.2013.50.2.445 SEMIGROUP PRESENTATIONS FOR CONGRUENCES ON GROUPS Gonca Ayık and Basri Çalışkan Abstract. We consider

More information

Binary Operations. Chapter Groupoids, Semigroups, Monoids

Binary Operations. Chapter Groupoids, Semigroups, Monoids 36 Chapter 5 Binary Operations In the last lecture, we introduced the residue classes Z n together with their addition and multiplication. We have also shown some properties that these two operations have.

More information

Completely regular semigroups and (Completely) (E, H E )-abundant semigroups (a.k.a. U-superabundant semigroups): Similarities and Contrasts

Completely regular semigroups and (Completely) (E, H E )-abundant semigroups (a.k.a. U-superabundant semigroups): Similarities and Contrasts Completely regular semigroups and (Completely) (E, H E )-abundant semigroups (a.k.a. U-superabundant semigroups): Similarities and Contrasts Xavier MARY Université Paris-Ouest Nanterre-La Défense, Laboratoire

More information

Name: Student ID: Instructions:

Name: Student ID: Instructions: Instructions: Name: CSE 322 Autumn 2001: Midterm Exam (closed book, closed notes except for 1-page summary) Total: 100 points, 5 questions, 20 points each. Time: 50 minutes 1. Write your name and student

More information

Equational Logic. Chapter Syntax Terms and Term Algebras

Equational Logic. Chapter Syntax Terms and Term Algebras Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from

More information

CHAPTER 4. βs as a semigroup

CHAPTER 4. βs as a semigroup CHAPTER 4 βs as a semigroup In this chapter, we assume that (S, ) is an arbitrary semigroup, equipped with the discrete topology. As explained in Chapter 3, we will consider S as a (dense ) subset of its

More information

arxiv: v1 [math.ra] 25 May 2013

arxiv: v1 [math.ra] 25 May 2013 Quasigroups and Related Systems 20 (2012), 203 209 Congruences on completely inverse AG -groupoids Wieslaw A. Dudek and Roman S. Gigoń arxiv:1305.6858v1 [math.ra] 25 May 2013 Abstract. By a completely

More information

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE712B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.

More information

MATHEMATICAL MODELLING, MECHANICS AND MOD- ELLING MTHA4004Y

MATHEMATICAL MODELLING, MECHANICS AND MOD- ELLING MTHA4004Y UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 MATHEMATICAL MODELLING, MECHANICS AND MOD- ELLING MTHA4004Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and ONE other

More information

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2012 2013 MECHANICS AND MODELLING MTH-1C32 Time allowed: 2 Hours Attempt QUESTIONS 1 AND 2 and THREE other questions. Notes are

More information

Adequate and Ehresmann semigroups

Adequate and Ehresmann semigroups Adequate and Ehresmann semigroups NSAC2013: June 8th, 2013, Novi Sad Victoria Gould University of York What is this talk about? Classes of semigroups with semilattices of idempotents inverse ample adequate

More information

Approaching cosets using Green s relations and Schützenberger groups

Approaching cosets using Green s relations and Schützenberger groups Approaching cosets using Green s relations and Schützenberger groups Robert Gray British Mathematical Colloquium March 2008 1 / 13 General question How are the properties of a semigroup related to those

More information

Permutation Groups and Transformation Semigroups Lecture 2: Semigroups

Permutation Groups and Transformation Semigroups Lecture 2: Semigroups Permutation Groups and Transformation Semigroups Lecture 2: Semigroups Peter J. Cameron Permutation Groups summer school, Marienheide 18 22 September 2017 I am assuming that you know what a group is, but

More information

5 Group theory. 5.1 Binary operations

5 Group theory. 5.1 Binary operations 5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1

More information

INTRODUCTION TO THE GROUP THEORY

INTRODUCTION TO THE GROUP THEORY Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher

More information

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 SETS, NUMBERS AND PROBABILITY MTHA4001Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Janacek

More information

1 + 1 = 2: applications to direct products of semigroups

1 + 1 = 2: applications to direct products of semigroups 1 + 1 = 2: applications to direct products of semigroups Nik Ruškuc nik@mcs.st-and.ac.uk School of Mathematics and Statistics, University of St Andrews Lisbon, 16 December 2010 Preview: 1 + 1 = 2... for

More information

Congruences on Inverse Semigroups using Kernel Normal System

Congruences on Inverse Semigroups using Kernel Normal System (GLM) 1 (1) (2016) 11-22 (GLM) Website: http:///general-letters-in-mathematics/ Science Reflection Congruences on Inverse Semigroups using Kernel Normal System Laila M.Tunsi University of Tripoli, Department

More information

MATHEMATICAL PROBLEM SOLVING, MECHANICS AND MODELLING MTHA4004Y

MATHEMATICAL PROBLEM SOLVING, MECHANICS AND MODELLING MTHA4004Y UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2014 2015 MATHEMATICAL PROBLEM SOLVING, MECHANICS AND MODELLING MTHA4004Y Time allowed: 2 Hours Attempt QUESTIONS 1 AND 2 and

More information

MATH 2200 Final Review

MATH 2200 Final Review MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics

More information

UNIVERSITY OF EAST ANGLIA. School of Mathematics UG End of Year Examination MATHEMATICAL LOGIC WITH ADVANCED TOPICS MTH-4D23

UNIVERSITY OF EAST ANGLIA. School of Mathematics UG End of Year Examination MATHEMATICAL LOGIC WITH ADVANCED TOPICS MTH-4D23 UNIVERSITY OF EAST ANGLIA School of Mathematics UG End of Year Examination 2003-2004 MATHEMATICAL LOGIC WITH ADVANCED TOPICS Time allowed: 3 hours Attempt Question ONE and FOUR other questions. Candidates

More information

GENERALIZED GREEN S EQUIVALENCES ON THE SUBSEMIGROUPS OF THE BICYCLIC MONOID

GENERALIZED GREEN S EQUIVALENCES ON THE SUBSEMIGROUPS OF THE BICYCLIC MONOID GENERALIZED GREEN S EQUIVALENCES ON THE SUBSEMIGROUPS OF THE BICYCLIC MONOID L. Descalço and P.M. Higgins Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United

More information

Candidates must show on each answer book the type of calculator used. Only calculators permitted under UEA Regulations may be used.

Candidates must show on each answer book the type of calculator used. Only calculators permitted under UEA Regulations may be used. UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2010 2011 CRYPTOGRAPHY Time allowed: 2 hours Attempt THREE questions. Candidates must show on each answer book the type of calculator

More information

Research Statement. MUHAMMAD INAM 1 of 5

Research Statement. MUHAMMAD INAM 1 of 5 MUHAMMAD INAM 1 of 5 Research Statement Preliminaries My primary research interests are in geometric inverse semigroup theory and its connections with other fields of mathematics. A semigroup M is called

More information

0.2 Vector spaces. J.A.Beachy 1

0.2 Vector spaces. J.A.Beachy 1 J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a

More information

Monoids. Definition: A binary operation on a set M is a function : M M M. Examples:

Monoids. Definition: A binary operation on a set M is a function : M M M. Examples: Monoids Definition: A binary operation on a set M is a function : M M M. If : M M M, we say that is well defined on M or equivalently, that M is closed under the operation. Examples: Definition: A monoid

More information

Approaches to tiling semigroups

Approaches to tiling semigroups Nick Gilbert Heriot-Watt University, Edinburgh (joint work with Erzsi Dombi) Periodic tilings Non-periodic tilings Ulrich Kortenkamp: Paving the Alexanderplatz Tilings of R n A tile in R n is a connected

More information

Module Contact: Dr Xiaoming Zhai, ENV Copyright of the University of East Anglia Version 2

Module Contact: Dr Xiaoming Zhai, ENV Copyright of the University of East Anglia Version 2 UNIVERSITY OF EAST ANGLIA School of Environmental Sciences Main Series UG Examination 2017-2018 OCEAN CIRCULATION ENV-5016A Time allowed: 2 hours Answer THREE questions Write each answer in a SEPARATE

More information

Strongly Regular Congruences on E-inversive Semigroups

Strongly Regular Congruences on E-inversive Semigroups International Mathematical Forum, Vol. 10, 2015, no. 1, 47-56 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2015.411188 Strongly Regular Congruences on E-inversive Semigroups Hengwu Zheng

More information

Approaching cosets using Green s relations and Schützenberger groups

Approaching cosets using Green s relations and Schützenberger groups Approaching cosets using Green s relations and Schützenberger groups Robert Gray Mini workshop in Algebra, CAUL April 2008 1 / 14 General question How are the properties of a semigroup related to those

More information

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3

More information

On the Complete Join of Permutative Combinatorial Rees-Sushkevich Varieties

On the Complete Join of Permutative Combinatorial Rees-Sushkevich Varieties International Journal of Algebra, Vol. 1, 2007, no. 1, 1 9 On the Complete Join of Permutative Combinatorial Rees-Sushkevich Varieties Edmond W. H. Lee 1 Department of Mathematics, Simon Fraser University

More information

arxiv: v4 [math.gr] 22 Nov 2017

arxiv: v4 [math.gr] 22 Nov 2017 UNIVERSAL LOCALLY FINITE MAXIMALLY HOMOGENEOUS SEMIGROUPS AND INVERSE SEMIGROUPS IGOR DOLINKA AND ROBERT D. GRAY arxiv:1704.00641v4 [math.gr] 22 Nov 2017 Abstract. In 1959, P. Hall introduced the locally

More information

On the lattice of congruences on a fruitful semigroup

On the lattice of congruences on a fruitful semigroup On the lattice of congruences on a fruitful semigroup Department of Mathematics University of Bielsko-Biala POLAND email: rgigon@ath.bielsko.pl or romekgigon@tlen.pl The 54th Summer School on General Algebra

More information

INVERSE SEMIQUANDLES. Michael Kinyon. 4th Mile High, 2 August University of Lisbon. Department of Mathematics 1 / 21

INVERSE SEMIQUANDLES. Michael Kinyon. 4th Mile High, 2 August University of Lisbon. Department of Mathematics 1 / 21 INVERSE SEMIQUANDLES Michael Kinyon Department of Mathematics University of Lisbon 4th Mile High, 2 August 2017 1 / 21 João This is joint work with João Araújo (Universidade Aberta). 2 / 21 Conjugation

More information

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2013 14 CALCULUS AND MULTIVARIABLE CALCULUS MTHA4005Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.

More information

DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY

DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY HANDOUT ABSTRACT ALGEBRA MUSTHOFA DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY 2012 BINARY OPERATION We are all familiar with addition and multiplication of two numbers. Both

More information

A graph theoretic approach to combinatorial problems in semigroup theory

A graph theoretic approach to combinatorial problems in semigroup theory A graph theoretic approach to combinatorial problems in semigroup theory Robert Gray School of Mathematics and Statistics University of St Andrews Thesis submitted for the Degree of Doctor of Philosophy

More information

SOLVABLE GROUPS OF EXPONENTIAL GROWTH AND HNN EXTENSIONS. Roger C. Alperin

SOLVABLE GROUPS OF EXPONENTIAL GROWTH AND HNN EXTENSIONS. Roger C. Alperin SOLVABLE GROUPS OF EXPONENTIAL GROWTH AND HNN EXTENSIONS Roger C. Alperin An extraordinary theorem of Gromov, [Gv], characterizes the finitely generated groups of polynomial growth; a group has polynomial

More information

Math 116 First Midterm October 17, 2014

Math 116 First Midterm October 17, 2014 Math 116 First Midterm October 17, 2014 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover AND IS DOUBLE SIDED. There are 9 problems.

More information

1 Last time: determinants

1 Last time: determinants 1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)

More information

Monoids of languages, monoids of reflexive. relations and ordered monoids. Ganna Kudryavtseva. June 22, 2010

Monoids of languages, monoids of reflexive. relations and ordered monoids. Ganna Kudryavtseva. June 22, 2010 June 22, 2010 J -trivial A monoid S is called J -trivial if the Green s relation J on it is the trivial relation, that is aj b implies a = b for any a, b S, or, equivalently all J -classes of S are one-element.

More information

MATH20101 Real Analysis, Exam Solutions and Feedback. 2013\14

MATH20101 Real Analysis, Exam Solutions and Feedback. 2013\14 MATH200 Real Analysis, Exam Solutions and Feedback. 203\4 A. i. Prove by verifying the appropriate definition that ( 2x 3 + x 2 + 5 ) = 7. x 2 ii. By using the Rules for its evaluate a) b) x 2 x + x 2

More information

2. Prime and Maximal Ideals

2. Prime and Maximal Ideals 18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let

More information

On a topological simple Warne extension of a semigroup

On a topological simple Warne extension of a semigroup ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 16, Number 2, 2012 Available online at www.math.ut.ee/acta/ On a topological simple Warne extension of a semigroup Iryna Fihel, Oleg

More information

The Membership Problem for a, b : bab 2 = ab

The Membership Problem for a, b : bab 2 = ab Semigroup Forum OF1 OF8 c 2000 Springer-Verlag New York Inc. DOI: 10.1007/s002330010009 RESEARCH ARTICLE The Membership Problem for a, b : bab 2 = ab D. A. Jackson Communicated by Gerard J. Lallement Throughout,

More information

A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING. David F. Anderson and Elizabeth F. Lewis

A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING. David F. Anderson and Elizabeth F. Lewis International Electronic Journal of Algebra Volume 20 (2016) 111-135 A GENERAL HEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUAIVE RING David F. Anderson and Elizabeth F. Lewis Received: 28 April 2016 Communicated

More information

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 SETS, NUMBERS AND PROBABILITY MTHA4001Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. penalised

More information

Attempt ALL QUESTIONS IN SECTION A, ONE QUESTION FROM SECTION B and ONE QUESTION FROM SECTION C Linear graph paper will be provided.

Attempt ALL QUESTIONS IN SECTION A, ONE QUESTION FROM SECTION B and ONE QUESTION FROM SECTION C Linear graph paper will be provided. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2015-2016 ENERGY ENGINEERING PRINCIPLES ENG-5001Y Time allowed: 3 Hours Attempt ALL QUESTIONS IN SECTION A, ONE QUESTION FROM

More information

Definition: A binary relation R from a set A to a set B is a subset R A B. Example:

Definition: A binary relation R from a set A to a set B is a subset R A B. Example: Chapter 9 1 Binary Relations Definition: A binary relation R from a set A to a set B is a subset R A B. Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B.

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

More information

Groups. 3.1 Definition of a Group. Introduction. Definition 3.1 Group

Groups. 3.1 Definition of a Group. Introduction. Definition 3.1 Group C H A P T E R t h r e E Groups Introduction Some of the standard topics in elementary group theory are treated in this chapter: subgroups, cyclic groups, isomorphisms, and homomorphisms. In the development

More information

MA554 Assessment 1 Cosets and Lagrange s theorem

MA554 Assessment 1 Cosets and Lagrange s theorem MA554 Assessment 1 Cosets and Lagrange s theorem These are notes on cosets and Lagrange s theorem; they go over some material from the lectures again, and they have some new material it is all examinable,

More information

Algebraic structures I

Algebraic structures I MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one

More information

1.1 Definition. A monoid is a set M together with a map. 1.3 Definition. A monoid is commutative if x y = y x for all x, y M.

1.1 Definition. A monoid is a set M together with a map. 1.3 Definition. A monoid is commutative if x y = y x for all x, y M. 1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M M M, (x, y) x y such that (i) (x y) z = x (y z) x, y, z M (associativity); (ii) e M such that x e = e x = x for all x M (e

More information

Module Contact: Dr Steven Hayward, CMP Copyright of the University of East Anglia Version 1

Module Contact: Dr Steven Hayward, CMP Copyright of the University of East Anglia Version 1 UNIVERSITY OF EAST ANGLIA School of Computing Sciences Main Series UG Examination 2016-17 MATHEMATICS FOR COMPUTING B CMP-4005Y Time allowed: 2 hours Answer ANY SIX questions out of SEVEN. Notes are not

More information

Combinatorial Methods in Study of Structure of Inverse Semigroups

Combinatorial Methods in Study of Structure of Inverse Semigroups Combinatorial Methods in Study of Structure of Inverse Semigroups Tatiana Jajcayová Comenius University Bratislava, Slovakia Graphs, Semigroups, and Semigroup Acts 2017, Berlin October 12, 2017 Presentations

More information

WORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}

WORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...} WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not

More information

MAT 3271: Selected solutions to problem set 7

MAT 3271: Selected solutions to problem set 7 MT 3271: Selected solutions to problem set 7 Chapter 3, Exercises: 16. Consider the Real ffine Plane (that is what the text means by the usual Euclidean model ), which is a model of incidence geometry.

More information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

More information

Extensions and I-semidirect products. inverse semigroups

Extensions and I-semidirect products. inverse semigroups of inverse semigroups Bolyai Institute University of Szeged NBSAN 2018 York, 7 8 January, 2018 Introduction: groups and semigroups extension of groups semidirect / standard wreath product of groups Theorem

More information

The kernel of a monoid morphism. Part I Kernels and extensions. Outline. Basic definitions. The kernel of a group morphism

The kernel of a monoid morphism. Part I Kernels and extensions. Outline. Basic definitions. The kernel of a group morphism Outline The kernel of a monoid morphism Jean-Éric Pin1 (1) Kernels and extensions (2) The synthesis theoem (3) The finite case (4) Group radical and effective characterization (5) The topological approach

More information

Math 31 Lesson Plan. Day 22: Tying Up Loose Ends. Elizabeth Gillaspy. October 31, Supplies needed: Colored chalk.

Math 31 Lesson Plan. Day 22: Tying Up Loose Ends. Elizabeth Gillaspy. October 31, Supplies needed: Colored chalk. Math 31 Lesson Plan Day 22: Tying Up Loose Ends Elizabeth Gillaspy October 31, 2011 Supplies needed: Colored chalk Other topics V 4 via (P ({1, 2}), ) and Cayley table. D n for general n; what s the center?

More information

A four element semigroup that is inherently nonfinitely based?

A four element semigroup that is inherently nonfinitely based? A four element semigroup that is inherently nonfinitely based? Peter R. Jones Marquette University NBSAN Workshop, York, November 21, 2012 Finite basis properties A (finite) algebra A is finitely based

More information

Primitive Ideals of Semigroup Graded Rings

Primitive Ideals of Semigroup Graded Rings Sacred Heart University DigitalCommons@SHU Mathematics Faculty Publications Mathematics Department 2004 Primitive Ideals of Semigroup Graded Rings Hema Gopalakrishnan Sacred Heart University, gopalakrishnanh@sacredheart.edu

More information

GOOD LUCK! 2. a b c d e 12. a b c d e. 3. a b c d e 13. a b c d e. 4. a b c d e 14. a b c d e. 5. a b c d e 15. a b c d e. 6. a b c d e 16.

GOOD LUCK! 2. a b c d e 12. a b c d e. 3. a b c d e 13. a b c d e. 4. a b c d e 14. a b c d e. 5. a b c d e 15. a b c d e. 6. a b c d e 16. MA109 College Algebra Spring 017 Exam1 017-0-08 Name: Sec.: Do not remove this answer page you will turn in the entire exam. You have two hours to do this exam. No books or notes may be used. You may use

More information

Linear equations are equations involving only polynomials of degree one.

Linear equations are equations involving only polynomials of degree one. Chapter 2A Solving Equations Solving Linear Equations Linear equations are equations involving only polynomials of degree one. Examples include 2t +1 = 7 and 25x +16 = 9x 4 A solution is a value or a set

More information

Idempotent and Regular Elements of the Complete Semigroups of Binary Relations of the Class ( )

Idempotent and Regular Elements of the Complete Semigroups of Binary Relations of the Class ( ) Applied Mathematics 205 6 32-38 Published Online February 205 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/0.4236/am.205.62029 Idempotent and Regular Elements of the Complete Semigroups

More information

Monoids and Their Cayley Graphs

Monoids and Their Cayley Graphs Monoids and Their Cayley Graphs Nik Ruskuc nik@mcs.st-and.ac.uk School of Mathematics and Statistics, University of St Andrews NBGGT, Leeds, 30 April, 2008 Instead of an Apology Then, as for the geometrical

More information

Lecture 6c: Green s Relations

Lecture 6c: Green s Relations Lecture 6c: Green s Relations We now discuss a very useful tool in the study of monoids/semigroups called Green s relations Our presentation draws from [1, 2] As a first step we define three relations

More information

2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a.

2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a. Chapter 2 Groups Groups are the central objects of algebra. In later chapters we will define rings and modules and see that they are special cases of groups. Also ring homomorphisms and module homomorphisms

More information

Abel-Grassmann s bands. 1. Introduction

Abel-Grassmann s bands. 1. Introduction Quasigroups and Related Systems 11 (2004), 95 101 Abel-Grassmann s bands Petar V. Protić and Nebojša Stevanović Abstract Abel-Grassmann s groupoids or shortly AG-groupoids have been considered in a number

More information

Aperiodic languages and generalizations

Aperiodic languages and generalizations Aperiodic languages and generalizations Lila Kari and Gabriel Thierrin Department of Mathematics University of Western Ontario London, Ontario, N6A 5B7 Canada June 18, 2010 Abstract For every integer k

More information

Assigned homework problems S. L. Kleiman, fall 2008

Assigned homework problems S. L. Kleiman, fall 2008 18.705 Assigned homework problems S. L. Kleiman, fall 2008 Problem Set 1. Due 9/11 Problem R 1.5 Let ϕ: A B be a ring homomorphism. Prove that ϕ 1 takes prime ideals P of B to prime ideals of A. Prove

More information

Introduction to abstract algebra: definitions, examples, and exercises

Introduction to abstract algebra: definitions, examples, and exercises Introduction to abstract algebra: definitions, examples, and exercises Travis Schedler January 21, 2015 1 Definitions and some exercises Definition 1. A binary operation on a set X is a map X X X, (x,

More information

arxiv: v3 [math.gr] 18 Aug 2011

arxiv: v3 [math.gr] 18 Aug 2011 ON A PROBLEM OF M. KAMBITES REGARDING ABUNDANT SEMIGROUPS JOÃO ARAÚJO AND MICHAEL KINYON arxiv:1006.3677v3 [math.gr] 18 Aug 2011 Abstract. A semigroup is regular if it contains at least one idempotent

More information

Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

Identities in upper triangular tropical matrix semigroups and the bicyclic monoid Identities in upper triangular tropical matrix semigroups and the bicyclic monoid Marianne Johnson (Joint work with Laure Daviaud and Mark Kambites) York Semigroup Seminar, May 2017 Semigroup identities

More information

Linear Algebra. Chapter 5

Linear Algebra. Chapter 5 Chapter 5 Linear Algebra The guiding theme in linear algebra is the interplay between algebraic manipulations and geometric interpretations. This dual representation is what makes linear algebra a fruitful

More information

Solution Sheet (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = (i) gcd (97, 157) = 1 = ,

Solution Sheet (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = (i) gcd (97, 157) = 1 = , Solution Sheet 2 1. (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = 3. 2. (i) gcd (97, 157) = 1 = 34 97 21 157, (ii) gcd (527, 697) = 17 = 4 527 3 697, (iii) gcd (2323, 1679) =

More information

SYNTACTIC SEMIGROUP PROBLEM FOR THE SEMIGROUP REDUCTS OF AFFINE NEAR-SEMIRINGS OVER BRANDT SEMIGROUPS

SYNTACTIC SEMIGROUP PROBLEM FOR THE SEMIGROUP REDUCTS OF AFFINE NEAR-SEMIRINGS OVER BRANDT SEMIGROUPS SYNTACTIC SEMIGROUP PROBLEM FOR THE SEMIGROUP REDUCTS OF AFFINE NEAR-SEMIRINGS OVER BRANDT SEMIGROUPS JITENDER KUMAR AND K. V. KRISHNA Abstract. The syntactic semigroup problem is to decide whether a given

More information

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that:

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: Equivalence MA Relations 274 and Partitions Scott Taylor 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is

More information

CHAPTER 1. Relations. 1. Relations and Their Properties. Discussion

CHAPTER 1. Relations. 1. Relations and Their Properties. Discussion CHAPTER 1 Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition 1.1.1. A binary relation from a set A to a set B is a subset R A B. If (a, b) R we say a is Related to b

More information

Sets. We discuss an informal (naive) set theory as needed in Computer Science. It was introduced by G. Cantor in the second half of the nineteenth

Sets. We discuss an informal (naive) set theory as needed in Computer Science. It was introduced by G. Cantor in the second half of the nineteenth Sets We discuss an informal (naive) set theory as needed in Computer Science. It was introduced by G. Cantor in the second half of the nineteenth century. Most students have seen sets before. This is intended

More information

Complete Semigroups of Binary Relations Defined by Semilattices of the Class Σ X,10

Complete Semigroups of Binary Relations Defined by Semilattices of the Class Σ X,10 Applied Mathematics 05 6 74-4 Published Online February 05 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/0.46/am.05.606 Complete Semigroups of Binary Relations efined by Semilattices of

More information

Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations

Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.

More information

ON SOME SEMIGROUPS GENERATED FROM CAYLEY FUNCTIONS LEJO J. MANAVALAN, P.G. ROMEO

ON SOME SEMIGROUPS GENERATED FROM CAYLEY FUNCTIONS LEJO J. MANAVALAN, P.G. ROMEO Available online at http://scikorg J Semigroup Theory Appl 2018, 2018:5 https://doiorg/1028919/jsta/3562 ISSN: 2051-2937 ON SOME SEMIGROUPS GENERATED FROM CAYLEY FUNCTIONS LEJO J MANAVALAN, PG ROMEO Department

More information

* 8 Groups, with Appendix containing Rings and Fields.

* 8 Groups, with Appendix containing Rings and Fields. * 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that

More information

Lectures - XXIII and XXIV Coproducts and Pushouts

Lectures - XXIII and XXIV Coproducts and Pushouts Lectures - XXIII and XXIV Coproducts and Pushouts We now discuss further categorical constructions that are essential for the formulation of the Seifert Van Kampen theorem. We first discuss the notion

More information