International workshop on graphs, semigroups, and semigroup acts

Transcription

1 Interntionl workshop on grphs, semigroups, nd semigroup cts celebrting the 75th birthdy of Ulrich Knuer October 10 - October 13, 2017 Institute of Mthemtics Technicl University Berlin Strsse des 17. Juni 136, Berlin

2 Interntionl Workshop on Grphs, Semigroups, nd Semigroup Acts Institute of Mthemtics of the Technicl University Berlin, October 10 - October 13, 2017 On rnks of the plnrity of semigroup vrieties OmSPU D. V. Solomtin

3 Cyley grphs for semigroups Exmple:

4 Cyley grphs for semigroups rm r Exmple: C, r 1, m 0 r, m

5 Cyley grphs for semigroups Exmple: C rm r r, m, r 1, m 0 rm1 2 r1 r r1

6 Simple Cyley grphs for semigroups Exmple: C rm r r, m, r 1, m 0 rm1 2 r1 r SCy( C,m ) r r1

7 Cyley grphs for semigroups Exmple: Z,..., xy zt x, y, z, t,,...,, n 1 n 1, 2 n 1 2 n 1 2 n n 2 n 2 n n

8 Simple Cyley grphs for semigroups Exmple: Z n xy zt x, y, z, t,,...,, n 1 1, 2,..., n 1 2 n 1 2 n SCy( Zn ) 0

9 Plnr grphs

10 Plnr grphs

11 Plnr grphs Knuer K., Knuer U. On plnr right groups // Semigroup Forum, vol.92.

12 K K 5 3, 3 C.Droms, Infinite-ended groups with plnr Cyley grphs // Deprtment of Mthemtics & Sttistics Jmes Mdison University, Hrrisonburg, 2006

13 C.Droms, Infinite-ended groups with plnr Cyley grphs // Deprtment of Mthemtics & Sttistics Jmes Mdison University, Hrrisonburg, 2006 Xi Zhng, Clifford semigroups with genus zero // Proceedings of the Interntionl Conference on Semigroups, cts nd ctegories with pplictions to grphs : to celebrte the 65th birthdys of M.Kilp nd U.Knuer, University of Trtu, June 27-30, 2007 K K 5 3, 3

14 C.Droms, Infinite-ended groups with plnr Cyley grphs // Deprtment of Mthemtics & Sttistics Jmes Mdison University, Hrrisonburg, 2006 Xi Zhng, Clifford semigroups with genus zero // Proceedings of the Interntionl Conference on Semigroups, cts nd ctegories with pplictions to grphs : to celebrte the 65th birthdys of M.Kilp nd U.Knuer, University of Trtu, June 27-30, 2007 Behnm Khosrvi, Bhmn Khosrvi, A Chrcteriztion of Cyley Grphs of Brndt Semigroups // Bull. Mlys. Mth. Sci. Soc. (2) 35(2) (2012), K K 5 3, 3

15 C.Droms, Infinite-ended groups with plnr Cyley grphs // Deprtment of Mthemtics & Sttistics Jmes Mdison University, Hrrisonburg, 2006 Xi Zhng, Clifford semigroups with genus zero // Proceedings of the Interntionl Conference on Semigroups, cts nd ctegories with pplictions to grphs : to celebrte the 65th birthdys of M.Kilp nd U.Knuer, University of Trtu, June 27-30, 2007 Behnm Khosrvi, Bhmn Khosrvi, A Chrcteriztion of Cyley Grphs of Brndt Semigroups // Bull. Mlys. Mth. Sci. Soc. (2) 35(2) (2012), M.Rungni1, S.Pnm, S.Arworn, On Cyley isomorphisms of left nd right groups // Interntionl Journl of Pure nd Applied Mthemtics, Volume 80 No , K K 5 3, 3

16 C.Droms, Infinite-ended groups with plnr Cyley grphs // Deprtment of Mthemtics & Sttistics Jmes Mdison University, Hrrisonburg, 2006 Xi Zhng, Clifford semigroups with genus zero // Proceedings of the Interntionl Conference on Semigroups, cts nd ctegories with pplictions to grphs : to celebrte the 65th birthdys of M.Kilp nd U.Knuer, University of Trtu, June 27-30, 2007 Behnm Khosrvi, Bhmn Khosrvi, A Chrcteriztion of Cyley Grphs of Brndt Semigroups // Bull. Mlys. Mth. Sci. Soc. (2) 35(2) (2012), M.Rungni1, S.Pnm, S.Arworn, On Cyley isomorphisms of left nd right groups // Interntionl Journl of Pure nd Applied Mthemtics, Volume 80 No , A.Georgkopoulos, The Plnr Cubic Cyley Grphs // Hbilittionsschrift, Hmburg 2012 K K 5 3, 3

17 C.Droms, Infinite-ended groups with plnr Cyley grphs // Deprtment of Mthemtics & Sttistics Jmes Mdison University, Hrrisonburg, 2006 Xi Zhng, Clifford semigroups with genus zero // Proceedings of the Interntionl Conference on Semigroups, cts nd ctegories with pplictions to grphs : to celebrte the 65th birthdys of M.Kilp nd U.Knuer, University of Trtu, June 27-30, 2007 Behnm Khosrvi, Bhmn Khosrvi, A Chrcteriztion of Cyley Grphs of Brndt Semigroups // Bull. Mlys. Mth. Sci. Soc. (2) 35(2) (2012), M.Rungni1, S.Pnm, S.Arworn, On Cyley isomorphisms of left nd right groups // Interntionl Journl of Pure nd Applied Mthemtics, Volume 80 No , A.Georgkopoulos, The Plnr Cubic Cyley Grphs // Hbilittionsschrift, Hmburg 2012 Q.Meng, B.Zhng, Generlized Cyley grphs of clss of semigroups // South Asin Journl of Mthemtics 2013, Vol. 3 ( 4 ) : K K 5 3, 3

18 C.Droms, Infinite-ended groups with plnr Cyley grphs // Deprtment of Mthemtics & Sttistics Jmes Mdison University, Hrrisonburg, 2006 Xi Zhng, Clifford semigroups with genus zero // Proceedings of the Interntionl Conference on Semigroups, cts nd ctegories with pplictions to grphs : to celebrte the 65th birthdys of M.Kilp nd U.Knuer, University of Trtu, June 27-30, 2007 Behnm Khosrvi, Bhmn Khosrvi, A Chrcteriztion of Cyley Grphs of Brndt Semigroups // Bull. Mlys. Mth. Sci. Soc. (2) 35(2) (2012), M.Rungni1, S.Pnm, S.Arworn, On Cyley isomorphisms of left nd right groups // Interntionl Journl of Pure nd Applied Mthemtics, Volume 80 No , A.Georgkopoulos, The Plnr Cubic Cyley Grphs // Hbilittionsschrift, Hmburg 2012 Q.Meng, B.Zhng, Generlized Cyley grphs of clss of semigroups // South Asin Journl of Mthemtics 2013, Vol. 3 ( 4 ) : A.Georgkopoulos, M.Hmnn, The plnr Cyley grphs re effectively enumerble// Supported by EPSRC grnt EP/L002787/1, Hmburg, June 10, 2015 K K 5 3, 3

19 Trees K K 5 3, 3

20 Trees K K 5 3, 3

21 Trees A.L.Mkriev, The ordinl sums of semigroup with cyclic Cyley grphs // Herld of Omsk University Omsk: OmSU n.. F.M. Dostoevsky, 4 (2008), pp (in Russin). K K 5 3, 3

22 K3 K K 5 3, 3

23 Outerplnr grphs K 3 K K 5 3, 3

24 Outerplnr grphs D.V.Solomtin, Some semigroups with outerplnr Cyley grphs // Siberin Electronic Mthemticl Reports, 8 (2011), pp (in Russin). K 3 K K 5 3, 3

25 K3 K K 4 2, 3 K K 5 3, 3

26 Generlized outerplnr grphs K 3 K K 4 2, 3 K K 5 3, 3

27 Generlized outerplnr grphs Sedláček J. On generliztion of outerplnr grphs (in Czech) // Čsopis Pěst. Mt, Vol. 113, No. 2. P K 3 K K 4 2, 3 K K 5 3, 3

28 Generlized outerplnr grphs Sedláček J. On generliztion of outerplnr grphs (in Czech) // Čsopis Pěst. Mt, Vol. 113, No. 2. P K 3 K K 4 2, 3 K K 5 3, 3 D.V. Solomtin, P.O. Mrtynov. Finite free commuttive semigroups nd semigroups with zero, dmits generlized outerplnr Cyley grphs.

29 K 3 K K 4 2, 3 G K K 1 G , 3

30 Linkless embedding K 3 K K 4 2, 3 G K K 1 G , 3

31 Linkless embedding K 6 K 3 K K 4 2, 3 G K K 1 G , 3

32 Linkless embedding K 6 K 3 K K 4 2, 3 G K K 1 G , 3 Schs, 1983; Robertson, Seymour, Thoms, 1995.

33 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Question: How to systemtize isolted exmples of semigroups?

34 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Question: How to systemtize isolted exmples of semigroups? Answer: SEE TO SEMIGROUP VARIETIES

35 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s We study the concept of the plnrity rnk suggested by L.M.Mrtynov for semigroup vrieties [*].

36 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s We study the concept of the plnrity rnk suggested by L.M.Mrtynov for semigroup vrieties [*]. New Problems of Algebr nd Logic, Omsk Algebric Seminr. Avilble t: presentid=12900 (in Russin)

37 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Let V be vriety of semigroups. DEFINITION

38 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s DEFINITION Let V be vriety of semigroups. If there is nturl number r 1 tht ll V-free semigroups of rnks r llow plnr Cyley grphs

39 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s DEFINITION Let V be vriety of semigroups. If there is nturl number r 1 tht ll V-free semigroups of rnks r llow plnr Cyley grphs nd the V-free semigroup of rnk r+1 doesn t llow plnr Cyley grph,

40 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s DEFINITION Let V be vriety of semigroups. If there is nturl number r 1 tht ll V-free semigroups of rnks r llow plnr Cyley grphs nd the V-free semigroup of rnk r+1 doesn t llow plnr Cyley grph, then this number r is clled the plnrity rnk for vriety V.

41 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s DEFINITION Let V be vriety of semigroups. If there is nturl number r 1 tht ll V-free semigroups of rnks r llow plnr Cyley grphs nd the V-free semigroup of rnk r+1 doesn t llow plnr Cyley grph, then this number r is clled the plnrity rnk for vriety V. If such number r doesn t exist, then we sy tht the vriety V hs the infinite plnrity rnk.

42 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s DEFINITION Let V be vriety of semigroups. If there is nturl number r 1 tht ll V-free semigroups of rnks r llow plnr Cyley grphs nd the V-free semigroup of rnk r+1 doesn t llow plnr Cyley grph, then this number r is clled the plnrity rnk for vriety V. If such number r doesn t exist, then we sy tht the vriety V hs the infinite plnrity rnk. (L. M. Mrtynov)

43 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 1 (in the clss of commuttive monoids): 1 2 n S, b b 1

44 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 1 (in the clss of commuttive monoids): 1 2 n r+m-1 r+1 r b t S, b b 1 r+m-1 b r+1 b r b b b r+m-1 b 2 r+1 b 2 r b 2 b 2 b 2 r+m-1 b t-1 r+1 b t-1 r b t-1 b t-1 b t-1 S rm, b, b 1 r t

45 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 1 (in the clss of commuttive monoids): 1 2 n r+m-1 r+1 r b t S, b b 1 r+m-1 b r+1 b r b b b r+m-1 r r+m-1 b 2 r+1 b 2 r b 2 b 2 b 2 r+m-1 c r+m-1 bc r+m-1 b r b b b r+m-1 b t-1 r+1 b t-1 r b t-1 b t-1 b t-1 r c r bc S rm, b, b 1 r t c 2 c bc c bc S, b, c b b, c c, bc cb, rm r, b 2 b, c 2 1

46 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 1 (in the clss of commuttive monoids): A m = vr{x m 1 i p i = 1} S vr{ x x } i, p M vriety of ll commuttive monoids

47 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 1 (in the clss of commuttive monoids): A m = vr{x m 1 i p i = 1} S vr{ x x } i, p M vriety of ll commuttive monoids 1 1) r(v) = 1 V = A m or V =, m, ( i 1 p 2) S i, p 2 1 2) r(v) = 2 V = M or V = or V = S i, i 2 i 1) 3) r(v) = 3 V = A 2 or 1 S 1,1 1 S i 1,1 2,1 ( 1 2

48 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 1 (in the clss of commuttive monoids): A m = vr{x m 1 i p i = 1} S vr{ x x } i, p M vriety of ll commuttive monoids 1 1) r(v) = 1 V = A m or V =, m, ( i 1 p 2) S i, p 2 1 2) r(v) = 2 V = M or V = or V = S i, i 2 i 1) 3) r(v) = 3 V = A 2 or 1 S 1,1 1 S i 1,1 2,1 ( 1 2 D. V. Solomtin, Plnrity rnks of vrieties of commuttive monoids, Herld of Omsk University Omsk: OmSU n.. F.M. Dostoevsky, 4 (2012), pp (in Russin).

49 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 2 (in the clss of commuttive semigroups):

50 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 2 (in the clss of commuttive semigroups): Illustrtion vr{x 1+m = x 1 }, m > 1

51 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 2 (in the clss of commuttive semigroups): Illustrtion Cy, b, c, d xy yx x, y, b, c, d vr{x 1+m = x 1 }, m > 1

52 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 2 (in the clss of commuttive semigroups): Illustrtion Cy, b, c, d xy yx x, y, b, c, d not plnr vr{x 1+m = x 1 }, m > 1

53 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 2 (in the clss of commuttive semigroups): Non-trivil vriety of commuttive semigroups either hs infinite rnk of plnrity nd t the sme time coincides with the vriety of semigroups with zero multipliction or hs rnk of plnrity 1, 2 or 3.

54 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 2 (in the clss of commuttive semigroups): Non-trivil vriety of commuttive semigroups either hs infinite rnk of plnrity nd t the sme time coincides with the vriety of semigroups with zero multipliction or hs rnk of plnrity 1, 2 or 3. D. V. Solomtin, The rnks of plnrity for vrieties of commuttive semigroups, Prikldny Diskretny Mtemtik Tomsk: TSU, 4 (2016), pp (in Russin).

55 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse):

56 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): The rnk of the plnrity of the vriety of idempotent semigroups is equl to 3;

57 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): The rnk of the plnrity of the vriety of idempotent semigroups is equl to 3; the rnk of the plnrity of the vriety vr{xw = w; wx = w} of nilsemigroups for ny word w tht does not contin the vrible x is equl to infinity;

58 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): The rnk of the plnrity of the vriety of idempotent semigroups is equl to 3; the rnk of the plnrity of the vriety vr{xw = w; wx = w} of nilsemigroups for ny word w tht does not contin the vrible x is equl to infinity; the rnk of plnrity of the permuttion vriety of semigroups is equl to 1 or 2.

59 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): The rnk of the plnrity of the vriety of idempotent semigroups is equl to 3; the rnk of the plnrity of the vriety vr{xw = w; wx = w} of nilsemigroups for ny word w tht does not contin the vrible x is equl to infinity; the rnk of plnrity of the permuttion vriety of semigroups is equl to 1 or 2. D. V. Solomtin, On rnks of the plnrity of vrieties of ll idempotent semigroups, nilsemigroups, nd semigroups with the permuttion identity, Herld of Omsk University Omsk: OmSU n.. F.M. Dostoevsky, (2017), 10 p, to pper. (in Russin)

60 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): Illustrtion Ν w vr{ xw w, wx w} P n vr{ x x2 xn x1 x2 x 1 n } I vr{ xx x}

61 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): 1 ) ( x x xx F I 6,, ) ( x x xx F I 159,,,, ) ( x x xx F I ,,,,,, ) ( x x xx F I ,,,,,,,, ) ( x x xx F I vr{ x} xx I }, vr{ w wx w xw w Ν } vr{ n n n x x x x x x P V V n n F,,, ) ( 2 1 Illustrtion

62 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): SCy( F 3 ( I))

63 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): SCy( F 3 ( I))

64 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): SCy( F 3 ( I))

65 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): SCy( F 3 ( I)) plnr

66 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): SCy( F 4 ( I))

67 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): SCy( F 4 ( I)) bcd bcdb bcdc bcd bcdbb bcdbbd bcdb bcdbbd bcdbbdc bcdcb bcdbc bcdcbd bcdcbd bcdbcd bcdcbdb bcdbcd bcdbc bcdb

68 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): SCy( F 4 ( I)) bcd bcdb bcdc bcd bcdb bcdbb bcdbbd bcdbbd bcdbbdc bcdcb not plnr bcdbc bcdcbd bcdcbd bcdbcd bcdcbdb bcdbcd bcdbc bcdb

69 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 3 (in the generl cse): Illustrtion SCy( F 1 SCy( F SCy( F 2 3 ( I)), ( I)), ( I)) SCy( F 4 ( I)) plnr not plnr r( I) 3

70 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s SUBDEFINITION

71 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s SUBDEFINITION The vriety, ech semigroup of which dmits plnr Cyley grph, clled plnr vriety.

72 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 4 (plnr vriety of commuttive semigroups):

73 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 4 (plnr vriety of commuttive semigroups): Z n xy zt x, y, z, t,,...,, n 1 1, 2,..., n 1 2 n 1 2 n SCy( Zn ) 0

74 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 4 (plnr vriety of commuttive semigroups): The vriety of semigroups with zero multipliction is only one nontrivil plnr vriety of commuttive semigroups.

75 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 4 (plnr vriety of commuttive semigroups): The vriety of semigroups with zero multipliction is only one nontrivil plnr vriety of commuttive semigroups. D. V. Solomtin, Plnr vrieties of commuttive semigroups, Herld of Omsk University Omsk: OmSU n.. F.M. Dostoevsky, 2(2015), pp (in Russin)

76 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 5 (plnr vrieties of semigroups):

77 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 5 (plnr vrieties of semigroups): The vriety vr{xy = zt} of semigroups with zero multipliction, the vriety vr{xy = x} of left-zero semigroups, the vriety vr{xy = xz} nd only they re non-trivil plnr vrieties of semigroups

78 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 5 (plnr vrieties of semigroups): The vriety vr{xy = zt} of semigroups with zero multipliction, the vriety vr{xy = x} of left-zero semigroups, the vriety vr{xy = xz} nd only they re non-trivil plnr vrieties of semigroups D. V. Solomtin, Plnr vrieties of semigroups, Sib. Electr. Mth. Reports, 12 (2015), pp (in Russin)

79 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 5 (plnr vrieties of semigroups): vr xy zt

80 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 5 (plnr vrieties of semigroups): vr xy zt vr xy x

81 O n r n k s o f t h e p l n r i t y o f s e m i g r o u p v r i e t i e s Theorem 5 (plnr vrieties of semigroups): vr xy zt vr xy x vr xy xz

82 Thnk you for ttention!