Linear sandwich semigroups

Transcription

1 University of Western Sydney Permutation groups and transformation semigroups Durham University, 21 July, 215

2 Don t mention the cri%$et 1

3 Joint work with Igor Dolinka 2

4 Sandwiches? (Lyapin, 196; cf Brown 1955) Let M mn be the set of all m n matrices over a field F. Fix A M nm. For X, Y M mn, define X Y = XAY. M A mn = (M mn, ) is a linear sandwich semigroup. Example If m = n and A = I, then M A mn = M n is the full linear monoid. 3

5 Egg-box diagrams for M n (F = Z 2 )

6 Egg sandwiches 6

7 Egg sandwiches

8 Plan (non-linear) I Green s relations I regular elements I ideals I idempotent generation I small (idempotent) generating sets I bigger sandwiches? 8

9 Easy lemmas Lemma If A, B M nm and rank(a) = rank(b), then M A mn = M B mn. So we study M J mn, where J = J r = [ I r O O O ] Mnm ( r min(m, n) is fixed). Write elements of M mn in 2 2 block form: Lemma [ A B C D ] [ E F G H [ A B ] C D, where A Mrr, B M r,n r, etc. ] [ = AE AF ] CE CF. 9

10 Green s relations Let X, Y M mn. Write X RY X M n = Y M n, X L Y M m X = M m Y, H = R L, D = R L. X J Y M m X M n = M m Y M n, For X M mn, write R X = {Y M mn : X RY }, etc. Proposition R X = {Y M mn : Col(X ) = Col(Y )}, L X = {Y M mn : Row(X ) = Row(Y )}, J X = D X = {Y M mn : rank(x ) = rank(y )} 1

11 Green s relations Let X, Y M mn. Write X R J Y X M mn = Y M mn, i.e., XJM mn = YJM mn, X L J Y M mn X = M mn Y, X J J Y M mn X M mn = M mn Y M mn, H J = R J L J, D J = R J L J. These are the usual Green s relations on M J mn. For X M mn, write R J X = {Y M mn : X R J Y }, etc. 11

12 Green s relations Let X = [ A B C D XJ = [ A O C O Define sets ] Mmn. Easy to check: ], JX = [ A B O O ] [, JXJ = A O ] O O. P 1 = {X M mn : Col(XJ) = Col(X )}, P 2 = {X M mn : Row(JX ) = Row(X )}, P = P 1 P 2 = {X M mn : rank(jxj) = rank(x )} = Reg(M J mn) M J mn. 12

13 Green s relations Proposition For X M mn, R J X = { R X P 1 if X P 1 {X } if X M mn \ P 1, L J X = { L X P 2 if X P 2 {X } if X M mn \ P 2, { HX J = H X if X P {X } if X M mn \ P, 13

14 Green s relations Proposition (continued) For X M mn, D X P if X P DX J = L X P 2 if X P 2 \ P 1 R X P 1 if X P 1 \ P 2 {X } if X M mn \ (P 1 P 2 ). 14

15 High energy semigroup theory P 1 P 1 P 2 P 2 15

16 Small generating sets Theorem Suppose r < min(m, n). The D J -maximal elements generate M J mn. Any generating set must contain all D J -maximal elements. rank(m J mn) = min(m,n) s=r+1 [ m s ] q [ n s ] q q (s 2) (q 1) s [s] q!. 16

17 Small generating sets Theorem Suppose r = min(m, n). M J mn has a unique maximal D J -class a rectangular group. [ ] max(m, n) rank(m J mn) = min(m, n). q

18 Regular elements unscrambling the egg M J

19 Regular elements unscrambling the egg M J

20 Look familiar? 2 1 Reg(M J 1 32 ) M 1 Reg(M J 2 33 ) M 2 2

21 Look familiar? 2 Reg(M J 2 33 ) Reg(MJ 2 34 ) Reg(MJ 2 43 ) M 2 21

22 Look familiar? Reg(M J 3 43 ) Reg(MJ 3 34 ) M 3 22

23 Big bang Theorem P = Reg(M Jr mn) M Jr mn is an inflated M r. M r has a chain of D-classes: D < D 1 < < D r. P has a chain of D J -classes: D J < DJ 1 < < DJ r. D s (M r ) maps onto [ r s ] q R-classes of M r. Each of these expands into q s(m r) R J -classes in D J s (P). Group H J -classes in D s (M r ) and D J s (P) are = G s. P = r s= qs(m+n 2r) q (s 2) (q 1) s [s] q! [ r s ] 2 q. rank(p) = q r(max(m,n) r)

24 Idempotent generators Theorem E(M r ) = r s= qs(r s) [ s r ] q. E(M J mn) = r s= qs(m+n r s) [ s r ] q. E r = E(M r ) = {I r } (M r \ G r ) Erdos. rank(e r ) = idrank(e r ) = 1 + (q r 1)/(q 1) Dawlings. E J mn = E(M J mn) = E(D) (P \ D), where D = D r (P). rank(e J mn) = idrank(e J mn) = q r(max(m,n) r) + (q r 1)/(q 1). 24

25 Ideals Theorem The ideals of M r form a chain: {O r } = I I 1 I r = M r. rank(i s ) = idrank(i s ) = [ s r ] q for s < r Gray. The ideals of P = Reg(M J mn) form a chain: {O mn } = I J I J 1 I J r = P. rank(i J s ) = idrank(i J s ) = q s(max(m,n) r) [ r s ] q for s < r. 25

26 More sandwiches... Let C = (Ob, Hom) be a small category. S = Hom has the structure of a partial semigroup. f (g h) = (f g) h if the products are defined, etc. Partialise notions like Green s relations, regularity, stability. If θ Hom(Y, X ) is fixed, Hom(X, Y ) is a semigroup under: f g = f θ g. M A mn arises when C is a linear category. Work is under way for diagram categories and more... 26

27 Thank you! Variants of finite full transformation semigroups Dolinka and East Semigroups of rectangular matrices under a sandwich operation Dolinka and East 27