Motzkin monoids. Micky East. York Semigroup University of York, 5 Aug, 2016

Transkript

1 Micky East York Semigroup University of York, 5 Aug, 206

2 Joint work with Igor Dolinka and Bob Gray 2

5 Any questions? Given a semigroup S: What is rank(s) = min { A : A S, S = A }? Is S idempotent-generated? If so, what is idrank(s) = min { A : A E(S), S = A }? Does idrank(s) = rank(s)? If not, what is E(S) = E(S)? What is rank(e(s))? idrank(e(s))? Are these equal? Can we enumerate minimal (idempotent) generating sets? Ditto for the ideals of S? 5

6 Transformation semigroups Full transformation semigroup Let T n be the set of all transformations of n = {,..., n}. T n is the full transformation semigroup of degree n. T n contains the symmetric group S n. Any finite group G embeds in S G. Any finite semigroup S embeds in T S +. For f T n and x n, write xf instead of f (x). For f, g T n, write fg = f g (do f first). 6

7 Transformation semigroups Anatomy of a transformation For f T n, define im(f ) = {xf : x n} the image of f, ker(f ) = {(x, y) : xf = yf } the kernel of f, rank(f ) = im(f ) = n/ ker(f ) the rank of f. For f = T 8 : im(f ) = {2, 4, 6, 7, 8}, ker(f ) = (, 2, 3 4, ), rank(f ) = 5. 7

8 Transformation semigroups Same box equal rank. Same column equal image. Same row equal kernel. Same cell equal image and kernel. A cell with a rank r idempotent (f = f 2 ) is a subgroup isomorphic to S r. f T n is idempotent f im(f ) = id im(f ) f = T 8 is idempotent T 4 E(T n ) = {f T n : f = f 2 }. E(T n ) = n r= ( n r) r n r. 8

9 Transformation semigroups D r = D r (T n ) = {f T n : rank(f ) = r}. I r = I r (T n ) = {f T n : rank(f ) r}. D n = S n. I n = T n. I n = T n \ S n. Theorem (Howie and others, 966 ) T n \ S n is idempotent-generated. rank(t n \ S n ) = ( ) n 2 = n(n ) 2 = idrank(t n \ S n ). I I 2 I n are the ideals of T n. I r = D r = E(D r ) for r < n. rank(i r ) = S(n, r) = idrank(i r ) for < r < n. T 4 9

10 Transformation semigroups Theorem (Howie and McFadden, 978) T n \ S n = E(D n ). E(D n ) = {e ij, e ji : i < j n}. e 47 = e 74 = {minimal idempotent generating sets of T n \ S n } {strongly connected tournaments on n} Theorem (Wright, 970) T 5 \ S 5 = 0 e2,e 3,e 4,e 5,e 32, e 24,e 25,e 43,e 53,e 45

11 Full linear monoids Theorem (Gray, 2008) M n (F q ) = semigroup of all n n matrices over F q. D r = D r (M n (F q )) = {A M n (F q ) : rank(a) = r}. I r = I r (M n (F q )) = {A M n (F q ) : rank(a) r}. I r = D r = E(D r ) for 0 r < n. rank(i r ) = idrank(i r ) = [ n r ] q for 0 r < n. rank(m n (F q ) \ G n (F q )) = idrank(m n (F q ) \ G n (F q )) = qn q Erdos, 967; Dawlings, 982.

12 Green s relations For a semigroup S: x R y xs = ys x L y S x = S y x J y S xs = S ys H = R L D = R L = J if S finite Eggbox diagram: D-related elements in same box R-related elements in same row L -related elements in same column H -related elements in same cell 2

13 Green s relations T n 4 f R g ker(f ) = ker(g) f R= b g a f = ga g = fb T 4 3

14 Green s relations T n f R g ker(f ) = ker(g) f L g im(f ) = im(g) f D g rank(f ) = rank(g) group H -classes are isomorphic to S r D-classes are D r = {f T n : rank(f ) = r} D r contains ( n r) L -classes D r contains S(n, r) R-classes T 4 D r = ( n r) S(n, r)r! T n = n r= D r n n = n r= ( n r) S(n, r)r! 3

15 Green s relations M n (F q ) A R B Col(A) = Col(B) A L B Row(A) = Row(B) A D B rank(a) = rank(b) group H -classes are isomorphic to GL(r, F q ) D-classes are D r = {A M n (F) : rank(a) = r} 0 M 3 (F 2 ) D r contains [ n r ] q L -classes (and R-classes) q n2 = n r=0 [ n r ] 2 q q(r 2) (q ) r [r] q! Note For T n and M n (F q ), rank(i r ) = idrank(i r ) is equal to the maximum of the number of R- or L -classes in D r. 4

16 Partition monoids Let n = {,..., n} and n = {,..., n }. The partition monoid on n is P n = { set partitions of n n } { (equiv. classes of) graphs on vertex set n n }. Eg: α = { {, 3, 4 }{, 3, 4 }, {2, 4}{2, 4}, {5, 6,, 6 }{5, 6,, 6 }, {2, 3 }{2, 3 P } n } n

17 Partition monoids product in P n Let α, β P n. To calculate αβ: () connect bottom of α to top of β, (2) remove middle vertices and floating components, (3) smooth out resulting graph to obtain αβ. { α { β 2 3 } αβ The operation is associative, so P n is a semigroup (monoid, etc). 6

18 Partition monoids submonoids of P n B n = {α P n : blocks of α have size 2} Brauer monoid TL n = {α B n : α is planar} Temperley-Lieb monoid PB n = {α P n : blocks of α have size 2} partial Brauer monoid M n = {α PB n : α is planar} Motzkin monoid B 6 PB 6 TL 6 M 6 7

19 Green s relations diagram monoids P 4 PB 4 M 4 B 4 TL 4 8

20 Green s relations diagram monoids For α P n : dom(α) = {i n : the α-block of i intersects n } ker(α) = {(i, j) : i and j belong to the same α-block} codom(α) = {i n : the α-block of i intersects n} coker(α) = {(i, j) : i and j belong to the same α-block} rank(α) = number of transversal α-blocks For α = P 8 : dom(α) = {, 2, 3, 5} ker(α) = (, 2, 3 4, 7 ) rank(α) = 2 codom(α) = {3, 4, 7} coker(α) = (, 2 3, 4 6, 8 ) 9

21 Green s relations P n α D β rank(α) = rank(β) { dom(α) = dom(β) and α R β ker(α) = ker(β) { codom(α) = codom(β) α L β coker(α) = coker(β) and Group H -classes are isomorphic to S r P 6 P 4 20

22 Green s relations B n α D β rank(α) = rank(β) α R β ker(α) = ker(β) α L β coker(α) = coker(β) Group H -classes are isomorphic to S r Ranks are restricted to n, n 2, n 4,... B 6 B 4 2

23 Green s relations TL n α D β rank(α) = rank(β) α R β ker(α) = ker(β) α L β coker(α) = coker(β) Group H -classes are trivial Ranks are restricted to n, n 2, n 4,... TL 6 TL 4 22

24 Green s relations PB n α D β rank(α) = rank(β) { dom(α) = dom(β) and α R β ker(α) = ker(β) { codom(α) = codom(β) α L β coker(α) = coker(β) and Group H -classes are isomorphic to S r PB 6 PB 4 23

25 Green s relations M n α D β rank(α) = rank(β) { dom(α) = dom(β) and α R β ker(α) = ker(β) { codom(α) = codom(β) α L β coker(α) = coker(β) and Group H -classes are trivial M 6 M 4 24

26 Green s relations TL8 TL 25

27 Green s relations TL 5 TL 7 Thanks to Attila Egri-Nagy for these... 26

28 Green s relations PB 5 27

29 Green s relations PB 5 28

30 Green s relations M 5 29

31 Idempotent generators P n Theorem (E, 20) P n \ S n is idempotent generated. P n \ S n = t r, t ij : r n, i < j n. t r = r n t ij = i j n rank(p n \ S n ) = idrank(p n \ S n ) = ( ) n+ 2 = n(n+) 2. Defining relations also given. 30

32 Idempotent generators P n \ S n Minimal (idempotent) generating sets (E and Gray, 204) P 5 \ S 5 = t 2, t 3, t 5, t 24, t 34, t 45, t 4, e 4, f 4, e 23, f 23, e 35, f 35, e 52, f 52. 3

33 Idempotent generators B n Theorem (Maltcev and Mazorchuk, 2007) B n \ S n is idempotent generated. B n \ S n = u ij : i < j n. u ij = i j n rank(b n \ S n ) = idrank(b n \ S n ) = ( ) n 2 = n(n ) 2. Defining relations also given. 32

34 Idempotent generators B n \ S n Minimal (idempotent) generating sets (E and Gray, 204)

35 Idempotent generators TL n Theorem (Borisavljević, Došen, Petrić, 2002, etc) TL n is idempotent generated. TL n = u,..., u n. u i = i n rank(tl n ) = idrank(tl n ) = n

36 Idempotent generators E(PB n ) = E(PB n ) PB 5 PB 5 \ S 5!E(PB 5 ) PB 5 \ S 5! 35

37 Idempotent generators E(PB n ) = E(PB n ) PB 5 E(PB 5 ) PB 5 \ S 5! 36

38 Idempotent generators E(PB n ) = E(PB n ) PB 5 E(PB 5 ) PB 5 \ S 5! 37

39 Idempotent generators E(PB n ) = E(PB n ) E(PB 7 ) 38

40 Idempotent generators E(PB n ) = E(PB n ) Theorem (Dolinka, E, Gray, 205) E(PB n ) = {} {t r : r n} I r 2 (PB n ) E(PB n ) = t r, u ij : r n, i < j n. t r = r n u ij = i j n rank(e(pb n )) = idrank(e(pb n )) = ( ) n+ 2 = n(n+) 2. 39

41 Idempotent generators E(M n ) = E(M n ) M 5 E(M 5 ) 40

43 Idempotent generators E(M n ) = E(M n ) A n is cosparse if: i n \ A i + A. Theorem (Dolinka, E, Gray, 205) E(M n ) = {} {id A : A n cosparse} {α M n : dom(α) and codom(α) non-cosparse} E(M n ) = t,..., t n, u,..., u n, v,..., v n 2. t i = i n u j = j n v k = k n rank(e(m n )) = idrank(e(m n )) = 3n 3. 42

45 Ideals P n Theorem (E and Gray, 204) The ideals of P n are I r = I r (P n ) = {α P n : rank(α) r} for 0 r n. I r = D r = E(D r ) for 0 r < n, where D r = {α P n : rank(α) = r}. rank(i r ) = idrank(i r ) = n j=r S(n, j)( j r). P 4 44

46 Ideals B n Theorem (E and Gray, 204) The ideals of B n are I r = I r (B n ) = {α B n : rank(α) r} for 0 r = n 2k n. I r = D r = E(D r ) for 0 r < n, where D r = {α B n : rank(α) = r}. rank(i r ) = idrank(i r ) = n! 2 k k!r!. B 4 45

47 Ideals TL n Theorem (E and Gray, 204) The ideals of TL n are I r = I r (TL n ) = {α TL n : rank(α) r} for 0 r = n 2k n. I r = D r = E(D r ) for 0 r < n, where D r = {α TL n : rank(α) = r}. ). rank(i r ) = idrank(i r ) = r+ n+ ( n+ k TL 4 46

48 Ideals PB n Theorem (Dolinka, E, Gray, 205) The ideals of PB n are I r = I r (PB n ) = {α PB n : rank(α) r} for 0 r n. I r = D r D r for 0 r < n = D r iff r = 0 or r n (mod 2) rank(i r ) = ( n r ) (n r)!! + ( n r) a(n r), where a(k) = number of involutions of k. I r is idempotent-generated iff r n 2. PB 4 idrank(i r ) = rank(i r ) where appropriate. 47

49 Ideals M n Theorem (Dolinka, E, Gray, 205) The ideals of M n are I r = I r (M n ) = {α M n : rank(α) r} for 0 r n. I r = D r D r for 0 r < n. rank(i r ) = m(n, r) + m (n, r), Motzkin and Riordan numbers. I r is idempotent-generated iff r < n 2. rank(e(i r )) = idrank(e(i r )) = rank(i r ) for 0 r n 2. M 4 48

50 Motzkin triangle A06489 m(n, r) = number of Motzkin paths from (0, 0) to (n, r) stay in st quadrant, use steps U(, ), D(, ), F (, 0) m(4, 0) = 9: m(0, 0) =, m(n, r) = 0 if (n, r) out-of-bounds m(n +, r) = m(n, r ) + m(n, r) + m(n, r + ) 49

51 Motzkin triangle A m(0, 0) =, m(n, r) = 0 if (n, r) out-of-bounds m(n +, r) = m(n, r ) + m(n, r) + m(n, r + ) 50

52 Riordan triangle A m (n, r) = number of Motzkin paths from (0, 0) to (n, 0) with r flats at level 0. m (0, 0) =, m (n, r) = 0 if (n, r) out-of-bounds m (n +, r) = m (n, r ) + m (n, r + ) + + m (n, n)

53 Riordan triangle A m (n, r) = number of Motzkin paths from (0, 0) to (n, 0) with r flats at level 0. m (0, 0) =, m (n, r) = 0 if (n, r) out-of-bounds m (n +, r) = m (n, r ) + m (n, r + ) + + m (n, n) T H A N K 0 Y O U 4 0!! 6 5 5

54 Special thanks to the York Maths Dept! 52