R-cross-sections of the semigroup of order-preserving transformations of a finite chain

Transcription

1 R-cross-sections of the semigroup of order-preserving transformations of a finite chain Eugenija A. Bondar Ural Federal University AAA9+NSAC 07 Novi Sad, Serbia, June -8 Bondar Е.А. R-cross-sections of O n / 6

2 ρ-cross-section S ρ is an equivalence on S transversal If a transversal of ρ is a semigroup then it is called a ρ-cross-section. Bondar Е.А. R-cross-sections of O n / 6

3 Cross-sections of Green s relations in classical transformation semigroups [n] = {,,...,n}, T n transformation semigroup (written on the left) Green s relation K K - crosssections H R J = D L exist only for n =,, unique exist, unique up to isomorphism exist, no description is known exist, not unique, even up isomorphism Classical Finite Transformation Semigroups: An Introduction. (GanyushkinO., Mazorchuk V., 009 ) Bondar E. [0, 06] to Bondar Е.А. R-cross-sections of O n / 6

4 Semigroup of order-preserving transformations Semigroup O n of order-preserving transformations: α T n : for all x,y [n] x y implies xα yα. Green s relations of O n are just the restrictions of the corresponding Green s relations on T n : α,β O n a) α Rβ if and only if ker(α) = ker(β); b) α L β if and only if im(α) = im(β). L -cross-sections of O n The description of L -cross-sections of O n follows from the description of L -cross-sections for T n. Bondar Е.А. R-cross-sections of O n / 6

5 Semigroup of order-preserving transformations Semigroup O n of order-preserving transformations: α T n : for all x,y [n] x y implies xα yα. Green s relations of O n are just the restrictions of the corresponding Green s relations on T n : α,β O n a) α Rβ if and only if ker(α) = ker(β); b) α L β if and only if im(α) = im(β). L -cross-sections of O n The description of L -cross-sections of O n follows from the description of L -cross-sections for T n. Bondar Е.А. R-cross-sections of O n / 6

6 Higgins embedding O n can be embedded in dual O n+ (P. Higgins, 99) K = {k,k,...,k t } is the set, written in ascending order, of the maximum members of its kernel classes im(α) = {r,r,...,r t }, k i α = r i for all i t. { xα if x r, = k i + if r i < x < r i+, i t Bondar Е.А. R-cross-sections of O n / 6

7 Higgins embedding O n can be embedded in dual O n+ (P. Higgins, 99) n+... K = {k,k,...,k t } is the set, written in ascending order, of the maximum members of its kernel classes im(α) = {r,r,...,r t }, k i α = r i for all i t. { xα if x r, = k i + if r i < x < r i+, i t Bondar Е.А. R-cross-sections of O n / 6

8 Higgins embedding O n can be embedded in dual O n+ (P. Higgins, 99)... K = {k,k,...,k t } is the set, written in ascending order, of the maximum members of its kernel classes im(α) = {r,r,...,r t }, k i α = r i for all i t. { xα if x r, = k i + if r i < x < r i+, i t n+ n+ Bondar Е.А. R-cross-sections of O n / 6

9 Higgins embedding O n can be embedded in dual O n+ (P. Higgins, 99)... K = {k,k,...,k t } is the set, written in ascending order, of the maximum members of its kernel classes im(α) = {r,r,...,r t }, k i α = r i for all i t. { xα if x r, = k i + if r i < x < r i+, i t n+ n+ Bondar Е.А. R-cross-sections of O n / 6

10 Higgins embedding O n can be embedded in dual O n+ (P. Higgins, 99)... K = {k,k,...,k t } is the set, written in ascending order, of the maximum members of its kernel classes im(α) = {r,r,...,r t }, k i α = r i for all i t. { xα if x r, = k i + if r i < x < r i+, i t n+ n+ Bondar Е.А. R-cross-sections of O n / 6

11 Higgins embedding O n can be embedded in dual O n+ (P. Higgins, 99)... K = {k,k,...,k t } is the set, written in ascending order, of the maximum members of its kernel classes im(α) = {r,r,...,r t }, k i α = r i for all i t. { xα if x r, = k i + if r i < x < r i+, i t n+ n+ Bondar Е.А. R-cross-sections of O n / 6

12 Higgins embedding O n can be embedded in dual O n+ (P. Higgins, 99) K = {k,k,...,k t } is the set, written in ascending order, of the maximum members of its kernel classes im(α) = {r,r,...,r t }, k i α = r i for all i t. { xα if x r, = k i + if r i < x < r i+, i t n+ n n+ n Bondar Е.А. R-cross-sections of O n / 6

13 L -cross-section of O and its dual Bondar Е.А. R-cross-sections of O n / 6

14 L -cross-section of O and its dual Bondar Е.А. R-cross-sections of O n / 6

15 Respectful trees A homomorphism Γ Γ : sends the root to the root, preserves the parent child relation and the genders. Γ subordinates Γ if there exists a - homomorphism Γ Γ. Bondar Е.А. R-cross-sections of O n / 6

16 Respectful trees A homomorphism Γ Γ : sends the root to the root, preserves the parent child relation and the genders. Γ subordinates Γ if there exists a - homomorphism Γ Γ. A respectful binary tree is a full binary tree such that conditions: (S) if a male vertex has a nephew, the nephew subordinates his uncle; (S) if a female vertex has a niece, the niece subordinates her aunt. 7 Bondar Е.А. R-cross-sections of O n / 6

17 A full binary tree which is not respectful 6 Bondar Е.А. R-cross-sections of O n / 6

18 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 the following conditions hold true: ) if the root has the son or the daughter then s(r) < r and r < d(r) n respectively. ) if v T(n) is a vertex and for p(v) and some x,y [n] the condition 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 O n / 6

19 Order-preserving trees T() T () T () Bondar Е.А. R-cross-sections of O n / 6

20 Diagram presentation of ([n], ) Bondar Е.А. R-cross-sections of O n / 6

21 Diagram presentation of ([n], ) Bondar Е.А. R-cross-sections of O n / 6

22 Diagram presentation of ([n], ) Bondar Е.А. R-cross-sections of O n / 6

23 Diagram presentation of ([n], ) Bondar Е.А. R-cross-sections of O n / 6

24 «Inner» trees Bondar Е.А. R-cross-sections of O n / 6

25 «Inner» trees Γ l () [] [] [] [] [][] [] Bondar Е.А. R-cross-sections of O n / 6

26 «Inner» trees Γ l () [] [] [] [] [][] [] [] [] Γ r () [] [] [] [] [] Bondar Е.А. R-cross-sections of O n / 6

27 Sketch of an order-preserving tree for an R-cross-section of O n r Γ a Γ b Γ a Γ b Γ ak Γ ak Γ bt Γ ai (Γ bj ) respectful trees on a i - (b j -)element set respectively, each tree subordinates the tree above a +a +...+a k = r, a k a k... a, b +b +...+b t = n r, b b... b t Bondar Е.А. R-cross-sections of O n / 6

28 Description of R-cross-sections of O n ([n], ) an order-preserving tree K m a partition of [n] into m convex intervals Order-preserving tree T( K m ) of intervals K m ϕ K m a - homomorphism between the tree of partitions and([n], ). Φ a set ofϕ K m, where K m goes through all possible convex partitions of [n] Bondar Е.А. R-cross-sections of O n / 6

29 Description of R-cross-sections of O n ([n], ) an order-preserving tree K m a partition of [n] into m convex intervals Order-preserving tree T( K m ) of intervals K m ϕ K m a - homomorphism between the tree of partitions and([n], ). Φ a set ofϕ K m, where K m goes through all possible convex partitions of [n] Bondar Е.А. R-cross-sections of O n / 6

30 Description of R-cross-sections of O n ([n], ) an order-preserving tree K m a partition of [n] into m convex intervals Order-preserving tree T( K m ) of intervals K m ϕ K m a - homomorphism between the tree of partitions and([n], ). Φ a set ofϕ K m, where K m goes through all possible convex partitions of [n] Bondar Е.А. R-cross-sections of O n / 6

31 Description of R-cross-sections of O n ([n], ) an order-preserving tree K m a partition of [n] into m convex intervals Order-preserving tree T( K m ) of intervals K m ϕ K m a - homomorphism between the tree of partitions and([n], ). Φ a set ofϕ K m, where K m 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 O n. Conversely, every R-cross-section of O n is isomorphic to Φ for an order-preserving binary tree ([n], ). Bondar Е.А. R-cross-sections of O n / 6

32 Sketch of an order-preserving tree for an R-cross-section of O n r Γ a Γ b Γ a Γ b Γ ak Γ ak Γ bt Γ ai (Γ bj ) respectful trees on a i - (b j -)element set respectively, each tree subordinates the tree above a +a +...+a k = r, a k a k... a, b +b +...+b t = n r, b b... b t Bondar Е.А. R-cross-sections of O n / 6

33 Classification of R-cross-sections of O n Similar respectful trees (Γ Γ ) Γ Γ Bondar Е.А. R-cross-sections of O n / 6

34 Classification of R-cross-sections of O n Theorem Let R, R be two R-cross-sections of O n. R = R iff one of the following conditions holds () the diagram of R is a mirror reflection of the diagram of R ; () Γ ai Γ a i for some i k, or Γ bj Γ b j for some j t, while other components are the same. Bondar Е.А. R-cross-sections of O n / 6