`39`42`'613a``45`47`'603ar-cross-sections of the …2017/09/01 · r-cross-sections of...

R-cross-sections of the semigroup of

order-preserving transformations of a finite chain

Eugenija A. Bondar

Ural Federal University

AAA94+NSAC 2017Novi Sad, Serbia, June 15-18

Bondar Е.А. R-cross-sections of On 15.06.2017 1 / 16

ρ-cross-section

S ρ is an equivalence on S

transversal

If a transversal of ρ is a semigroup then it iscalled a ρ-cross-section.


Cross-sections of Green’s relations in classical transformation

semigroups

[n] = {1, 2, . . . , n},T n transformation semigroup (written on the left)

Green’srelationK

H R J = D L

K -cross-sections

existonly forn = 1, 2,unique

exist,unique up toisomorphism

exist,nodescriptionis known

exist,notunique, evenup toisomorphism

• Classical Finite Transformation Semigroups: An Introduction.(GanyushkinO., Mazorchuk V., 2009 )• Bondar E. [2014, 2016]


Semigroup of order-preserving transformations

Semigroup On of order-preserving transformations:

α ∈ Tn : for all x , y ∈ [n] x ≤ y implies xα ≤ yα.

Green’s relations of On are just the restrictions of the correspondingGreen’s relations on Tn:∀α, β ∈ On

a) αR β if and only if ker (α) = ker (β);

b) αL β if and only if im (α) = im (β).

L -cross-sections of On

The description of L -cross-sections of On follows from thedescription of L -cross-sections for Tn.


Higgins’ embedding

On can be embedded in dual O∗

n+1 (P. Higgins, 1995)

K = {k1, k2, . . . , kt} is the set, written inascending order, of the maximum membersof its kernel classes

im (α) = {r1, r2, . . . , rt},

kiα = ri for all 1 ≤ i ≤ t.

xα∗ =

{1 if x ≤ r1,

ki + 1 if ri < x < ri+1, 1 ≤ i ≤ t





1

3

5

. . .

2n+ 1


im (α) = {r1, r2, . . . , rt},


xα∗ =

{1 if x ≤ r1,

ki + 1 if ri < x < ri+1, 1 ≤ i ≤ t





1

3

5

. . .

2n+ 1

1

3

5

2n+ 1


im (α) = {r1, r2, . . . , rt},


xα∗ =

{1 if x ≤ r1,

ki + 1 if ri < x < ri+1, 1 ≤ i ≤ t





1

3

5

. . .

2n+ 1

1

3

5

2n+ 1

1

5

. . .

2n + 1


im (α) = {r1, r2, . . . , rt},


xα∗ =

{1 if x ≤ r1,

ki + 1 if ri < x < ri+1, 1 ≤ i ≤ t





1

3

5

. . .

2n+ 1

1

3

5

2n+ 1

1

5

. . .

2n + 1

1

3

2n + 1


im (α) = {r1, r2, . . . , rt},


xα∗ =

{1 if x ≤ r1,

ki + 1 if ri < x < ri+1, 1 ≤ i ≤ t





1

3

5

. . .

2n+ 1

1

3

5

2n+ 1

0

2

4

6

2n

0

2

4

6

2n


im (α) = {r1, r2, . . . , rt},


xα∗ =

{1 if x ≤ r1,

ki + 1 if ri < x < ri+1, 1 ≤ i ≤ t


L -cross-section of O3 and its dual

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3


L -cross-section of O3 and its dual

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4


Respectful trees

A homomorphism Γ1 → Γ2: sends the root to the root, preserves theparent–child relation and the genders.Γ1 subordinates Γ2 if there exists a 1-1 homomorphism Γ1 → Γ2.


Respectful trees

A homomorphism Γ1 → Γ2: sends the root to the root, preserves theparent–child relation and the genders.Γ1 subordinates Γ2 if there exists a 1-1 homomorphism Γ1 → Γ2.A respectful binary tree is a full binary tree such that conditions:(S1) if a male vertex has a nephew, the nephew subordinates hisuncle;(S2) if a female vertex has a niece, the niece subordinates her aunt.

1

1 1

2

3

1 1

2

5 2

7

1 1


A full binary tree which is not respectful

1

1 1

2

3

1 1

2

5 1

6


Order-preserving trees

r denotes the root,s(v) the son of a vertex v ,d(v) the daughter of a vertex v ,p(v) denotes the parent of v

We say a binary tree T (n) is order-preserving for ([n],≤), if thefollowing conditions hold true:1) if the root has the son or the daughter then

1 ≤ s(r) < r and r < d(r) ≤ n respectively.

2) if v ∈ T (n) is a vertex and for p(v) and some x , y ∈ [n] thecondition x ≤ p(v) ≤ y holds, then

{x ≤ v < p(v), if v is the sonp(v),p(v) < v ≤ y , if v is the daughter p(v).


Order-preserving trees

T (4)

1

2

4

3

T1(5)

2

1 4

3 5

T2(5)

5

1

4

3

2


Diagram presentation of ([n],≺)

1

5

9

4

3

2

8

7

6


«Inner» trees

1

5

9

4

3

2

8

7

6


«Inner» trees

Γl(5) [4]

[2] [2]

[1] [1] [1] [1]

1

5

9

4

3

2

8

7

6


«Inner» trees

Γl(5) [4]

[2] [2]

[1] [1] [1] [1]

Γr (5)[4]

[3]

[1][2]

[1]

[1] [1]

1

5

9

4

3

2

8

7

6


Sketch of an order-preserving tree

for an R-cross-section of On

. . .. . .

r

Γa1Γa2

Γak−1

Γak

Γb1Γb2

Γbt

Γai (Γbj ) respectful trees on ai - (bj -)element set respectively,each tree subordinates the tree abovea1 + a2 + . . .+ ak = r−1, ak ≤ ak−1 ≥ . . . ≥ a1,b1 + b2 + . . . + bt = n − r, b1 ≤ b2 ≤ . . . ≤ bt


Description of R-cross-sections of On

([n],≺) an order-preserving tree

K̃m a partition of [n] into m convex intervals

Order-preserving tree T (K̃m) of intervals K̃m

ϕK̃m≺ a 1-1 homomorphism between the tree of partitions and ([n],≺).

Φ≺ a set of ϕK̃m≺

, where K̃m goes through all possible convex partitionsof [n]


Description of R-cross-sections of On

([n],≺) an order-preserving tree

K̃m a partition of [n] into m convex intervals

Order-preserving tree T (K̃m) of intervals K̃m

ϕK̃m≺

a 1-1 homomorphism between the tree of partitions and ([n],≺).

Φ≺ a set of ϕK̃m≺ , where K̃m goes through all possible convex partitions

of [n]

Theorem

Given an order-preserving binary tree ([n],≺) the set Φ≺ constitutes

an R-cross-section of On. Conversely, every R-cross-section of On

is isomorphic to Φ≺ for an order-preserving binary tree ([n],≺).


Sketch of an order-preserving tree

for an R-cross-section of On

. . .. . .

r

Γa1Γa2

Γak−1

Γak

Γb1Γb2

Γbt

Γai (Γbj ) respectful trees on ai - (bj -)element set respectively,each tree subordinates the tree abovea1 + a2 + . . .+ ak = r−1, ak ≤ ak−1 ≥ . . . ≥ a1,b1 + b2 + . . . + bt = n − r, b1 ≤ b2 ≤ . . . ≤ bt


Classification of R-cross-sections of On

Similar respectful trees (Γ1 ∼ Γ2)

Γ14

1 3

1 2

1 1

Γ24

3 1

2 1

1 1


Classification of R-cross-sections of On

Theorem

Let R1, R2 be two R-cross-sections of On.

R1∼= R2 iff one of the following conditions holds

(1) the diagram of R1 is a mirror reflection of the diagram of R2;

(2) Γai ∼ Γ′ai for some 1 ≤ i ≤ k , or Γbj ∼ Γ′bj for some 1 ≤ j ≤ t,

while other components are the same.


`39`42`'613a``45`47`'603ar-cross-sections of the …2017/09/01 · r-cross-sections of...

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