International workshopon graphs, semigroups,and semigroup acts

celebrating the 75th birthdayof Ulrich Knauer

October 10 - October 13, 2017Institute of MathematicsTechnical University BerlinStrasse des 17. Juni 136, Berlin

On ranks of the planarity of

semigroup varietiesOmSPU

D. V. Solomatin

11 October, 2017

International Workshop on Graphs, Semigroups, and Semigroup Acts

Institute of Mathematics of the Technical University Berlin,October 10 - October 13, 2017

Cayley graphs for semigroups

Example:

Cayley graphs for semigroups

Example: 0,1,, mraaaC rmrmr

Cayley graphs for semigroups

Example:

2aa 1ra ra

1ra

1mra

a a a

a

a

a

0,1,, mraaaC rmrmr

Simple Cayley graphs for semigroups

Example:

2aa 1ra ra

1ra

1mra

0,1,, mraaaC rmrmr

)( ,mrCSCay

Cayley graphs for semigroups

Example: 1,,...,,,,,|,...,, 2121 naaatzyxztxyaaaZ nnn

1a 2a na

1a

2ana

1a

2a

na

1a

2a

na

na

2a

1a

0

Simple Cayley graphs for semigroups

Example:

1a 2a na

0

1,,...,,,,,|,...,, 2121 naaatzyxztxyaaaZ nnn

)( nZSCay

Planar graphs

Planar graphs

Planar graphs

• Knauer K., Knauer U. On planar right groups // Semigroup Forum, 2015. vol.92.

5K 3,3K

• C.Droms, Infinite-ended groups with planar Cayley graphs // Department of Mathematics & Statistics James Madison University, Harrisonburg, 2006

5K 3,3K

• C.Droms, Infinite-ended groups with planar Cayley graphs // Department of Mathematics & Statistics James Madison University, Harrisonburg, 2006

• Xia Zhang, Clifford semigroups with genus zero // Proceedings of the International Conference on Semigroups, acts and categories with applications to graphs : to celebrate the 65th birthdays of M.Kilp and U.Knauer, University of Tartu, June 27-30, 2007

5K 3,3K

• C.Droms, Infinite-ended groups with planar Cayley graphs // Department of Mathematics & Statistics James Madison University, Harrisonburg, 2006

• Xia Zhang, Clifford semigroups with genus zero // Proceedings of the International Conference on Semigroups, acts and categories with applications to graphs : to celebrate the 65th birthdays of M.Kilp and U.Knauer, University of Tartu, June 27-30, 2007

• Behnam Khosravi, Bahman Khosravi, A Characterization of Cayley Graphs of Brandt Semigroups // Bull. Malays. Math. Sci. Soc. (2) 35(2) (2012), 399–410

5K 3,3K

• C.Droms, Infinite-ended groups with planar Cayley graphs // Department of Mathematics & Statistics James Madison University, Harrisonburg, 2006

• Xia Zhang, Clifford semigroups with genus zero // Proceedings of the International Conference on Semigroups, acts and categories with applications to graphs : to celebrate the 65th birthdays of M.Kilp and U.Knauer, University of Tartu, June 27-30, 2007

• Behnam Khosravi, Bahman Khosravi, A Characterization of Cayley Graphs of Brandt Semigroups // Bull. Malays. Math. Sci. Soc. (2) 35(2) (2012), 399–410

• M.Ruangnai1, S.Panma, S.Arworn, On Cayley isomorphisms of left and right groups // International Journal of Pure and Applied Mathematics, Volume 80 No. 4 2012, 561-571

5K 3,3K

• C.Droms, Infinite-ended groups with planar Cayley graphs // Department of Mathematics & Statistics James Madison University, Harrisonburg, 2006

• Xia Zhang, Clifford semigroups with genus zero // Proceedings of the International Conference on Semigroups, acts and categories with applications to graphs : to celebrate the 65th birthdays of M.Kilp and U.Knauer, University of Tartu, June 27-30, 2007

• Behnam Khosravi, Bahman Khosravi, A Characterization of Cayley Graphs of Brandt Semigroups // Bull. Malays. Math. Sci. Soc. (2) 35(2) (2012), 399–410

• M.Ruangnai1, S.Panma, S.Arworn, On Cayley isomorphisms of left and right groups // International Journal of Pure and Applied Mathematics, Volume 80 No. 4 2012, 561-571

• A.Georgakopoulos, The Planar Cubic Cayley Graphs // Habilitationsschrift, Hamburg 2012

5K 3,3K

• C.Droms, Infinite-ended groups with planar Cayley graphs // Department of Mathematics & Statistics James Madison University, Harrisonburg, 2006

• Xia Zhang, Clifford semigroups with genus zero // Proceedings of the International Conference on Semigroups, acts and categories with applications to graphs : to celebrate the 65th birthdays of M.Kilp and U.Knauer, University of Tartu, June 27-30, 2007

• Behnam Khosravi, Bahman Khosravi, A Characterization of Cayley Graphs of Brandt Semigroups // Bull. Malays. Math. Sci. Soc. (2) 35(2) (2012), 399–410

• M.Ruangnai1, S.Panma, S.Arworn, On Cayley isomorphisms of left and right groups // International Journal of Pure and Applied Mathematics, Volume 80 No. 4 2012, 561-571

• A.Georgakopoulos, The Planar Cubic Cayley Graphs // Habilitationsschrift, Hamburg 2012

• Q.Meng, B.Zhang, Generalized Cayley graphs of a class ofsemigroups // South Asian Journal of Mathematics 2013 , Vol. 3 ( 4 ) : 272-278

5K 3,3K

• C.Droms, Infinite-ended groups with planar Cayley graphs // Department of Mathematics & Statistics James Madison University, Harrisonburg, 2006

• Xia Zhang, Clifford semigroups with genus zero // Proceedings of the International Conference on Semigroups, acts and categories with applications to graphs : to celebrate the 65th birthdays of M.Kilp and U.Knauer, University of Tartu, June 27-30, 2007

• Behnam Khosravi, Bahman Khosravi, A Characterization of Cayley Graphs of Brandt Semigroups // Bull. Malays. Math. Sci. Soc. (2) 35(2) (2012), 399–410

• M.Ruangnai1, S.Panma, S.Arworn, On Cayley isomorphisms of left and right groups // International Journal of Pure and Applied Mathematics, Volume 80 No. 4 2012, 561-571

• A.Georgakopoulos, The Planar Cubic Cayley Graphs // Habilitationsschrift, Hamburg 2012

• Q.Meng, B.Zhang, Generalized Cayley graphs of a class ofsemigroups // South Asian Journal of Mathematics 2013 , Vol. 3 ( 4 ) : 272-278

• A.Georgakopoulos, M.Hamann, The planar Cayley graphs are effectively enumerable// Supported by EPSRC grant EP/L002787/1, Hamburg, June 10, 2015

Trees

5K 3,3K

Trees

5K 3,3K

Trees

5K 3,3K

• A.L.Makariev, The ordinal sums of semigroup with acyclic Cayley graphs // Herald of Omsk University – Omsk: OmSU n.a. F.M. Dostoevsky, 4 (2008), pp. 12–17. (in Russian).

3K 5K 3,3K

Outerplanar graphs

5K 3,3K3K

Outerplanar graphs

5K 3,3K3K

• D.V.Solomatin, Some semigroups with outerplanar Cayley graphs // Siberian Electronic Mathematical Reports, 8 (2011), pp. 191–212. (in Russian).

3K 4K 3,2K5K 3,3K

Generalized outerplanar graphs

4K 3,2K5K 3,3K

3K

Generalized outerplanar graphs

4K 3,2K5K 3,3K

3K

• Sedláček J. On a generalization of outerplanar graphs (in Czech) // Časopis Pěst. Mat, 1988. –Vol. 113, No. 2. – P. 213–218.

Generalized outerplanar graphs

4K 3,2K5K 3,3K

3K

• Sedláček J. On a generalization of outerplanar graphs (in Czech) // Časopis Pěst. Mat, 1988. –Vol. 113, No. 2. – P. 213–218.

• D.V. Solomatin, P.O. Martynov. Finite free commutative semigroups and semigroups with zero, admits a generalized outerplanar Cayley graphs.

4K 3,2K5K 3,3K

121 GG 3K

Linkless embedding

4K 3,2K5K 3,3K

121 GG 3K

Linkless embedding

4K 3,2K5K 3,3K

121 GG

6K

3K

Linkless embedding

• Sachs, 1983; Robertson, Seymour, Thomas, 1995.

4K 3,2K5K 3,3K

121 GG

6K

3K

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

11 October, 2017

How to systematize isolated examples of semigroups?

Question:

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

11 October, 2017

Answer:

SEE TO SEMIGROUP VARIETIES

How to systematize isolated examples of semigroups?

Question:

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

11 October, 2017

We study the concept of the planarity rank suggested by L.M.Martynov for semigroup varieties [*].

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

11 October, 2017

We study the concept of the planarity rank suggested by L.M.Martynov for semigroup varieties [*].

New Problems of Algebra and Logic, Omsk Algebraic Seminar. Available at:http://www.mathnet.ru/php/seminars.phtml?presentid=12900 (in Russian)

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

11 October, 2017

Let V be a variety of semigroups.

DEFINITION

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

11 October, 2017

Let V be a variety of semigroups.

If there is a natural number r ≥ 1 that all V-free semigroups of ranks

≤ r allow planar Cayley graphs

DEFINITION

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

11 October, 2017

Let V be a variety of semigroups.

If there is a natural number r ≥ 1 that all V-free semigroups of ranks

≤ r allow planar Cayley graphs and the V-free semigroup of a rank

r+1 doesn’t allow planar Cayley graph,

DEFINITION

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

11 October, 2017

Let V be a variety of semigroups.

If there is a natural number r ≥ 1 that all V-free semigroups of ranks

≤ r allow planar Cayley graphs and the V-free semigroup of a rank

r+1 doesn’t allow planar Cayley graph, then this number r is called

the planarity rank for variety V.

DEFINITION

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

11 October, 2017

Let V be a variety of semigroups.

If there is a natural number r ≥ 1 that all V-free semigroups of ranks

≤ r allow planar Cayley graphs and the V-free semigroup of a rank

r+1 doesn’t allow planar Cayley graph, then this number r is called

the planarity rank for variety V. If such a number r doesn’t exist, then

we say that the variety V has the infinite planarity rank.

DEFINITION

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

11 October, 2017

Let V be a variety of semigroups.

If there is a natural number r ≥ 1 that all V-free semigroups of ranks

≤ r allow planar Cayley graphs and the V-free semigroup of a rank

r+1 doesn’t allow planar Cayley graph, then this number r is called

the planarity rank for variety V. If such a number r doesn’t exist, then

we say that the variety V has the infinite planarity rank.

DEFINITION

(L. M. Martynov)

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

11 October, 2017

a a2 an1

1, bbaS

Theorem 1 (in the class of commutative monoids):

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

11 October, 2017

a a2 an1

1, bbaS

bt

b

b2

bt-1

a

ab

ab2

abt-1

ar

arb

arb2

arbt-1

ar+1

ar+1b

ar+1b2

ar+1bt-1

ar+m-1

ar+m-1b

ar+m-1b2

ar+m-1bt-1

1,, trmr baabaS

Theorem 1 (in the class of commutative monoids):

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

11 October, 2017

a a2 an1

1, bbaS

bt

b

b2

bt-1

a

ab

ab2

abt-1

ar

arb

arb2

arbt-1

ar+1

ar+1b

ar+1b2

ar+1bt-1

ar+m-1

ar+m-1b

ar+m-1b2

ar+m-1bt-1

1,, trmr baabaS

c2

c

bc

abc

arbc

ac

ar+m-1bc

arc

ar+m-1c

babarbar+m-1b

ar+m-1 ar a

1,,,,,,, 22 cbbaacbbccaacbaabcbaS rmr

Theorem 1 (in the class of commutative monoids):

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

11 October, 2017

Theorem 1 (in the class of commutative monoids):

Am = var{xm = 1} }var{1,

ipipi xx

S M – variety of all commutative monoids

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

11 October, 2017

Theorem 1 (in the class of commutative monoids):

Am = var{xm = 1} }var{1,

ipipi xx

S

1) r(V) = 1 V = Am or V = , ,1,S pi 2m )21( pi

2) r(V) = 2 V = M or V = or V = , 11,1

Si1

1,2Si

)12( 21 ii

3) r(V) = 3 V = A2 or 11,1S

M – variety of all commutative monoids

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

11 October, 2017

Theorem 1 (in the class of commutative monoids):

D. V. Solomatin, Planarity ranks of varieties of commutativemonoids, Herald of Omsk University – Omsk: OmSU n.a. F.M.Dostoevsky, 4 (2012), pp. 41–45. (in Russian).

Am = var{xm = 1} }var{1,

ipipi xx

S

1) r(V) = 1 V = Am or V = , ,1,S pi 2m )21( pi

2) r(V) = 2 V = M or V = or V = , 11,1

Si1

1,2Si

)12( 21 ii

3) r(V) = 3 V = A2 or 11,1S

M – variety of all commutative monoids

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

11 October, 2017

Theorem 2 (in the class of commutative semigroups):

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

11 October, 2017

Theorem 2 (in the class of commutative semigroups):

var{x1+m = x1} , m > 1

Illustration

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

11 October, 2017

Theorem 2 (in the class of commutative semigroups):

var{x1+m = x1} , m > 1

Illustration dcbayxyxxydcbaCay ,,,,|,,,

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

11 October, 2017

Theorem 2 (in the class of commutative semigroups):

var{x1+m = x1} , m > 1

Illustration

not planar

dcbayxyxxydcbaCay ,,,,|,,,

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

11 October, 2017

Theorem 2 (in the class of commutative semigroups):

Non-trivial variety of commutative semigroups either has infinite rank of planarity and at the same time coincides with the variety of semigroups with zero multiplication or has a rank of planarity 1, 2 or 3.

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

11 October, 2017

Theorem 2 (in the class of commutative semigroups):

D. V. Solomatin, The ranks of planarity for varieties of commutative semigroups, Prikladnaya DiskretnayaMatematika – Tomsk: TSU, 4 (2016), pp. 50–64. (in Russian).

Non-trivial variety of commutative semigroups either has infinite rank of planarity and at the same time coincides with the variety of semigroups with zero multiplication or has a rank of planarity 1, 2 or 3.

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

11 October, 2017

Theorem 3 (in the general case):

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

11 October, 2017

Theorem 3 (in the general case):

The rank of the planarity of the variety of idempotent semigroups is equal to 3;

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

11 October, 2017

Theorem 3 (in the general case):

The rank of the planarity of the variety of idempotent semigroups is equal to 3; the rank of the planarity of the variety var{xw = w; wx = w} of nilsemigroups for any word w that does not contain the variable x is equal to infinity;

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

11 October, 2017

Theorem 3 (in the general case):

The rank of the planarity of the variety of idempotent semigroups is equal to 3; the rank of the planarity of the variety var{xw = w; wx = w} of nilsemigroups for any word w that does not contain the variable x is equal to infinity; the rank of planarity of the permutation variety of semigroups is equal to 1 or 2.

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

11 October, 2017

Theorem 3 (in the general case):

D. V. Solomatin, On ranks of the planarity of varieties of all idempotent semigroups, nilsemigroups, and semigroups with the permutation identity, Herald of Omsk University – Omsk: OmSU n.a. F.M. Dostoevsky, (2017), 10 p, to appear. (in Russian)

The rank of the planarity of the variety of idempotent semigroups is equal to 3; the rank of the planarity of the variety var{xw = w; wx = w} of nilsemigroups for any word w that does not contain the variable x is equal to infinity; the rank of planarity of the permutation variety of semigroups is equal to 1 or 2.

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

11 October, 2017

Theorem 3 (in the general case):

},var{ wwxwxww Ν }var{ 2121 nnn xxxxxx P

}var{ xxx I

Illustration

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

11 October, 2017

Theorem 3 (in the general case):

1|)( 111 axxxxaF I

6,|,)( 21212 aaxxxxaaF I

159,,|,,)( 3213213 aaaxxxxaaaF I

380332,,,|,,,)( 432143214 aaaaxxxxaaaaF I

7655148847512,,,,|,,,,)( 54321543215 aaaaaxxxxaaaaaF I

}var{ xxx I

},var{ wwxwxww Ν }var{ 2121 nnn xxxxxx P

VV nn aaaF ,,,)( 21

Illustration

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

11 October, 2017

Theorem 3 (in the general case):

))(( 3 IFSCay

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

11 October, 2017

Theorem 3 (in the general case):

))(( 3 IFSCay

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

11 October, 2017

Theorem 3 (in the general case):

))(( 3 IFSCay

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

11 October, 2017

Theorem 3 (in the general case):

planar))(( 3 IFSCay

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

11 October, 2017

Theorem 3 (in the general case):

))(( 4 IFSCay

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

11 October, 2017

Theorem 3 (in the general case):

abcd abcda

abcdabc abcdab

abcdb

abcdba

abcdbac

abcdbacd

abcdbacda

abcdac

abcdacb

abcdacbd

abcdacbda

abcdacbdab

abcdbab

abcdbabd

abcdbabda

abcdbabdac

))(( 4 IFSCay

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

11 October, 2017

Theorem 3 (in the general case):

abcd abcda

abcdabc abcdab

abcdb

abcdba

abcdbac

abcdbacd

abcdbacda

abcdac

abcdacb

abcdacbd

abcdacbda

abcdacbdab

abcdbab

abcdbabd

abcdbabda

abcdbabdacnot

planar

))(( 4 IFSCay

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

11 October, 2017

Theorem 3 (in the general case):

3)( IrplanarnotFSCay ))(( 4 I

planarFSCay

FSCay

FSCay

))((

)),((

)),((

3

2

1

I

I

I

Illustration

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

11 October, 2017

SUBDEFINITION

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

11 October, 2017

The variety, each semigroup of which admits a planar Cayley graph, called planar variety.

SUBDEFINITION

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

11 October, 2017

Theorem 4 (planar variety of commutative semigroups):

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

11 October, 2017

Theorem 4 (planar variety of commutative semigroups):

1a 2a na

0

1,,...,,,,,|,...,, 2121 naaatzyxztxyaaaZ nnn

)( nZSCay

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

11 October, 2017

Theorem 4 (planar variety of commutative semigroups):

The variety of semigroups with zero multiplication is only one nontrivial planar variety of commutative semigroups.

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

11 October, 2017

Theorem 4 (planar variety of commutative semigroups):

D. V. Solomatin, Planar varieties of commutative semigroups,Herald of Omsk University – Omsk: OmSU n.a. F.M. Dostoevsky,2(2015), pp.17–22. (in Russian)

The variety of semigroups with zero multiplication is only one nontrivial planar variety of commutative semigroups.

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

11 October, 2017

Theorem 5 (planar varieties of semigroups):

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

11 October, 2017

Theorem 5 (planar varieties of semigroups):

The variety var{xy = zt} of semigroups with zero multiplication, the variety var{xy = x} of left-zero semigroups, the variety var{xy = xz} and only they are non-trivial planar varieties of semigroups

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

11 October, 2017

Theorem 5 (planar varieties of semigroups):

D. V. Solomatin, Planar varieties of semigroups, Sib. Electr.Math. Reports, 12 (2015), pp. 232–247. (in Russian)

The variety var{xy = zt} of semigroups with zero multiplication, the variety var{xy = x} of left-zero semigroups, the variety var{xy = xz} and only they are non-trivial planar varieties of semigroups

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

11 October, 2017

Theorem 5 (planar varieties of semigroups):

ztxy var

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

11 October, 2017

Theorem 5 (planar varieties of semigroups):

ztxy var

xxy var

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

11 October, 2017

Theorem 5 (planar varieties of semigroups):

ztxy var

xxy var

xzxy var

Thank you for attention! Solomatin_DV

@omgpu.ru