UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 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 - 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
- 3-2. 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
- 4-3. 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
- 5-5. (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
Q1 (advanced topic) MTHE6011A and MTHE7011A Semigroup Theory Feedback on the 2016 2017 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.
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).