International workshop on graphs, semigroups, and semigroup acts

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

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

Cyley grphs for semigroups Exmple:

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

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

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

Cyley grphs for semigroups Exmple: Z,..., xy zt x, y, z, t,,...,, n 1 n 1, 2 n 1 2 n 1 2 n 1 1 2 n 2 n 2 n 1 1 2 0 n

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

Plnr grphs

Plnr grphs Knuer K., Knuer U. On plnr right groups // Semigroup Forum, 2015. vol.92.

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

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), 399 410 M.Rungni1, S.Pnm, S.Arworn, On Cyley isomorphisms of left nd right groups // Interntionl Journl of Pure nd Applied Mthemtics, Volume 80 No. 4 2012, 561-571 A.Georgkopoulos, The Plnr Cubic Cyley Grphs // Hbilittionsschrift, Hmburg 2012 K K 5 3, 3

Trees K K 5 3, 3

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. 12 17. (in Russin). K K 5 3, 3

K3 K K 5 3, 3

Outerplnr grphs K 3 K K 5 3, 3

Outerplnr grphs D.V.Solomtin, Some semigroups with outerplnr Cyley grphs // Siberin Electronic Mthemticl Reports, 8 (2011), pp. 191 212. (in Russin). K 3 K K 5 3, 3

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

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

Generlized outerplnr grphs Sedláček J. On generliztion of outerplnr grphs (in Czech) // Čsopis Pěst. Mt, 1988. Vol. 113, No. 2. P. 213 218. K 3 K K 4 2, 3 K K 5 3, 3

Generlized outerplnr grphs Sedláček J. On generliztion of outerplnr grphs (in Czech) // Čsopis Pěst. Mt, 1988. Vol. 113, No. 2. P. 213 218. 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.

K 3 K K 4 2, 3 G K K 1 G 12 5 3, 3

Linkless embedding K 3 K K 4 2, 3 G K K 1 G 12 5 3, 3

Linkless embedding K 6 K 3 K K 4 2, 3 G K K 1 G 12 5 3, 3

Linkless embedding K 6 K 3 K K 4 2, 3 G K K 1 G 12 5 3, 3 Schs, 1983; Robertson, Seymour, Thoms, 1995.

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?

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

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 [*].

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: http://www.mthnet.ru/php/seminrs.phtml? presentid=12900 (in Russin)

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

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

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.

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

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

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

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

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. 41 45. (in Russin).

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):

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

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

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

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. 50 64. (in Russin).

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):

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;

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)

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}

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 ) ( 1 1 1 x x xx F I 6,, ) ( 2 1 2 1 2 x x xx F I 159,,,, ) ( 3 2 1 3 2 1 3 x x xx F I 380 332,,,,,, ) ( 4 3 2 1 4 3 2 1 4 x x xx F I 765 2 751884 514,,,,,,,, ) ( 5 4 3 2 1 5 4 3 2 1 5 x x xx F I vr{ x} xx I }, vr{ w wx w xw w Ν } vr{ 2 1 2 1 n n n x x x x x x P V V n n F,,, ) ( 2 1 Illustrtion

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))

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

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))

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

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

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

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

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.

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):

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

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.17 22. (in Russin)

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):

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

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

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

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

Thnk you for ttention! Solomtin_DV @omgpu.ru