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TRANSCRIPT
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.
Bondar Е.А. R-cross-sections of On 15.06.2017 2 / 16
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]
Bondar Е.А. R-cross-sections of On 15.06.2017 3 / 16
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.
Bondar Е.А. R-cross-sections of On 15.06.2017 4 / 16
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.
Bondar Е.А. R-cross-sections of On 15.06.2017 4 / 16
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
Bondar Е.А. R-cross-sections of On 15.06.2017 5 / 16
Higgins’ embedding
On can be embedded in dual O∗
n+1 (P. Higgins, 1995)
1
3
5
. . .
2n+ 1
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
Bondar Е.А. R-cross-sections of On 15.06.2017 5 / 16
Higgins’ embedding
On can be embedded in dual O∗
n+1 (P. Higgins, 1995)
1
3
5
. . .
2n+ 1
1
3
5
2n+ 1
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
Bondar Е.А. R-cross-sections of On 15.06.2017 5 / 16
Higgins’ embedding
On can be embedded in dual O∗
n+1 (P. Higgins, 1995)
1
3
5
. . .
2n+ 1
1
3
5
2n+ 1
1
5
. . .
2n + 1
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
Bondar Е.А. R-cross-sections of On 15.06.2017 5 / 16
Higgins’ embedding
On can be embedded in dual O∗
n+1 (P. Higgins, 1995)
1
3
5
. . .
2n+ 1
1
3
5
2n+ 1
1
5
. . .
2n + 1
1
3
2n + 1
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
Bondar Е.А. R-cross-sections of On 15.06.2017 5 / 16
Higgins’ embedding
On can be embedded in dual O∗
n+1 (P. Higgins, 1995)
1
3
5
. . .
2n+ 1
1
3
5
2n+ 1
1
5
. . .
2n + 1
1
3
2n + 1
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
Bondar Е.А. R-cross-sections of On 15.06.2017 5 / 16
Higgins’ embedding
On can be embedded in dual O∗
n+1 (P. Higgins, 1995)
1
3
5
. . .
2n+ 1
1
3
5
2n+ 1
0
2
4
6
2n
0
2
4
6
2n
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
Bondar Е.А. R-cross-sections of On 15.06.2017 5 / 16
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
Bondar Е.А. R-cross-sections of On 15.06.2017 6 / 16
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
Bondar Е.А. R-cross-sections of On 15.06.2017 6 / 16
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.
Bondar Е.А. R-cross-sections of On 15.06.2017 7 / 16
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
Bondar Е.А. R-cross-sections of On 15.06.2017 7 / 16
A full binary tree which is not respectful
1
1 1
2
3
1 1
2
5 1
6
Bondar Е.А. R-cross-sections of On 15.06.2017 8 / 16
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).
Bondar Е.А. R-cross-sections of On 15.06.2017 9 / 16
Order-preserving trees
T (4)
1
2
4
3
T1(5)
2
1 4
3 5
T2(5)
5
1
4
3
2
Bondar Е.А. R-cross-sections of On 15.06.2017 10 / 16
Diagram presentation of ([n],≺)
1
5
9
4
3
2
8
7
6
Bondar Е.А. R-cross-sections of On 15.06.2017 11 / 16
Diagram presentation of ([n],≺)
1
5
9
4
3
2
8
7
6
Bondar Е.А. R-cross-sections of On 15.06.2017 11 / 16
Diagram presentation of ([n],≺)
1
5
9
4
3
2
8
7
6
Bondar Е.А. R-cross-sections of On 15.06.2017 11 / 16
Diagram presentation of ([n],≺)
1
5
9
4
3
2
8
7
6
Bondar Е.А. R-cross-sections of On 15.06.2017 11 / 16
«Inner» trees
1
5
9
4
3
2
8
7
6
Bondar Е.А. R-cross-sections of On 15.06.2017 12 / 16
«Inner» trees
Γl(5) [4]
[2] [2]
[1] [1] [1] [1]
1
5
9
4
3
2
8
7
6
Bondar Е.А. R-cross-sections of On 15.06.2017 12 / 16
«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
Bondar Е.А. R-cross-sections of On 15.06.2017 12 / 16
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
Bondar Е.А. R-cross-sections of On 15.06.2017 13 / 16
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]
Bondar Е.А. R-cross-sections of On 15.06.2017 14 / 16
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]
Bondar Е.А. R-cross-sections of On 15.06.2017 14 / 16
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]
Bondar Е.А. R-cross-sections of On 15.06.2017 14 / 16
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],≺).
Bondar Е.А. R-cross-sections of On 15.06.2017 14 / 16
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
Bondar Е.А. R-cross-sections of On 15.06.2017 15 / 16
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
Bondar Е.А. R-cross-sections of On 15.06.2017 16 / 16
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.
Bondar Е.А. R-cross-sections of On 15.06.2017 16 / 16