Endomorphism Monoids of Graphs and Partial Orders
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1 Endomorphism Monoids of Graphs and Partial Orders Nik Ruskuc School of Mathematics and Statistics, University of St Andrews Leeds, 3rd August 2007
2 S.J. Pride:
3 S.J. Pride: Semigroups : Groups = C : R
4 S.J. Pride: Semigroups : Groups C : R
5 S.J. Pride: Semigroups : Groups C : R
6 S.J. Pride: Semigroups : Groups C : R Groups : permutations Semigroups : mappings
7 S.J. Pride: Semigroups : Groups C : R Groups : automorphisms Semigroups : endomorphisms
8 Morphisms Definition Let X = (X ; R i (i I )) be a relational structure. An endomorphism is a mapping θ : X X which respects all the relations R i, i.e. (x 1,..., x k ) R i (x 1 θ,..., x k θ) R i. An automorphism is a bijective endomorphism θ for which θ 1 is also an endomorphism. Remark For posets and graphs the above becomes: x y xθ yθ, x y xθ yθ.
9 Morphisms End(X ) = the endomorphism monoid of X Aut(X ) = the automorphism group of X General Problem For a given X, how are End(X ) and Aut(X ), and their properties, related?
10 Trans(X ) and Sym(X ) Let E = E(X ) be a trivial relational structure on X. Trans(X ) = End(E) the full transformation monoid on X Sym(X ) = Aut(E) the symmetric group on X Facts Trans(n) = n n, Sym(n) = n! (X infinite) Trans(X ) = Sym(X ) = 2 X
11 Finite chains C n : 1 < 2 <... < n Facts Aut(C n ) = {id} End(C n ) = ( ) 2n 1 n 1
12 Rank Definition The rank of a semigroup S is the smallest number of elements needed to generate S; notation: rank(s). Facts rank(sym(n)) = 2 rank(trans(n)) = 3 rank(end(c n )) = n
13 Relative ranks Definition Let S be a semigroup, and let T be a subsemigroup of S. The relative rank of S modulo T (denoted rank(s : T )) is the smallest size of a set A such that S = T A. Example rank(trans(n) : Sym(n)) = 1
14 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2.
15 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Remark Semigroups : Groups = C : R
16 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Remark rank(trans(x ) : Sym(X )) = [C : R]:-)
17 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof.
18 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof. ( ) X X X X δ 1 1
19 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof. ( ) X X X X γ onto
20 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof. ( ) X X X X
21 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof. ( ) X X X X δ
22 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof. ( ) X X X X δ π permutation
23 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof. ( ) X X X X δ π permutation γ
24 Example: rank(trans(x ) : Sym(X )) Proposition (Higgins, Howie, NR 98) For X infinite, rank(trans(x ) : Sym(X )) = 2. Proof. ( ) Suppose Trans(X ) = Sym(X ), τ. If τ is not injective, then every injection in Sym(X ), τ is a bijection. If τ is not surjective, then every surjection in Sym(X ), τ is a bijection.
25 Some more relative ranks rank(trans(x ) : E(Trans(X )) ) = 2 rank(trans(x ) : Inj(X )) = 1 rank(trans(x ) : Surj(X )) = 1
26 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ).
27 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35]
28 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β
29 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β
30 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β
31 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] γ
32 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] γ
33 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] γ
34 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] γ like θ 1
35 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] γ like θ 2
36 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] γ like θ 3
37 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ
38 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ
39 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ
40 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ
41 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ
42 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ
43 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ
44 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Proof. [Banach 35] β γ β β β γ γ = θ 3
45 Sierpinski s Theorem Theorem (Sierpinski 35) Every countable subset of Trans(X ) (X infinite) is contained in a 2-generated subsemigroup of Trans(X ). Corollary The relative rank of any subsemigroup of Trans(X ) is 0, 1, 2 or uncountable. Proof. Suppose rank(trans(x ) : S) = ℵ 0, with Trans(X ) = S, τ 1, τ 2,.... By Sierpinski s Theorem, τ 1, τ 2,... α, β for some α, β. But then Trans(X ) = S, α, β.
46 Sierpinski/Galvin Theorem (Galvin 93) Every countable subset of Sym(X ) (X infinite) is contained in a 2-generator subgroup of Sym(X ). Corollary The relative rank of any subgroup G of Sym(X ) is 0, 1 or uncountable. Remark Where has 2 disappeared to?
47 Partial bijections SymInv(X ) the monoid of all partial bijections of X, i.e. bijections between subsets of X (the symmetric inverse monoid) Theorem (Mitchell 02) Every countable subset of SymInv(X ) (X infinite) is contained in a 2-generated inverse submonoid. Corollary The relative rank of any inverse subsemigroup S of SymInv(X ) is 0, 1, 2 or uncountable. Question Does there exist an inverse subsemigroup of SymInv(X ) with relative rank 2?
48 Lineraly ordered set N Theorem (Higgins, Howie, Mitchell, NR 03) rank(trans(n) : End(N)) = 1.
49 Lineraly ordered set N Theorem (Higgins, Howie, Mitchell, NR 03) rank(trans(n) : End(N)) = 1. Proof. Let δ : N N be any onto, infinite-to-one, mapping.
50 Lineraly ordered set N Theorem (Higgins, Howie, Mitchell, NR 03) rank(trans(n) : End(N)) = 1. Proof. Let δ : N N be any onto, infinite-to-one, mapping. Let γ Trans(N) be arbitrary.
51 Lineraly ordered set N Theorem (Higgins, Howie, Mitchell, NR 03) rank(trans(n) : End(N)) = 1. Proof. Let δ : N N be any onto, infinite-to-one, mapping. Let γ Trans(N) be arbitrary. Define ɛ End(N) inductively by:
52 Lineraly ordered set N Theorem (Higgins, Howie, Mitchell, NR 03) rank(trans(n) : End(N)) = 1. Proof. Let δ : N N be any onto, infinite-to-one, mapping. Let γ Trans(N) be arbitrary. Define ɛ End(N) inductively by: kɛ (kγ)δ 1 and kɛ > (k 1)ɛ.
53 Lineraly ordered set N Theorem (Higgins, Howie, Mitchell, NR 03) rank(trans(n) : End(N)) = 1. Proof. Let δ : N N be any onto, infinite-to-one, mapping. Let γ Trans(N) be arbitrary. Define ɛ End(N) inductively by: kɛ (kγ)δ 1 and kɛ > (k 1)ɛ. Then γ = ɛδ End(N), δ.
54 Linearly ordered sets Higgins, Mitchell, NR 03 Theorem If L is a countable linearly ordered set then rank(trans(l) : End(L)) = 1. Theorem If L is a well-ordered set then rank(trans(l) : End(L)) = 1. Example rank(trans(r) : End(R)) is uncountable. Question Does there exist an infinite linearly ordered set L such that rank(trans(l) : End(L)) = 2? Question Classify, in order-theoretic terms, all linearly ordered sets L with finite relative ranks in Trans(L).
55 Partially ordered sets Definition Let P be a partially ordered set. For x P write: x = {p P : x p}, x = {p P : p x}
56 Partially ordered sets Definition Let P be a partially ordered set. For x P write: x = {p P : x p}, x = {p P : p x} Theorem (Higgins, Mitchell, Morayne, NR 06) Let P be a countably infinite poset, and let c be the number of connected components of P.
57 Partially ordered sets Definition Let P be a partially ordered set. For x P write: x = {p P : x p}, x = {p P : p x} Theorem (Higgins, Mitchell, Morayne, NR 06) Let P be a countably infinite poset, and let c be the number of connected components of P. (I) Suppose c is finite. Then rank(trans(p), End(P)) 2 if and only if there exists x P such that at least one of x or x is infinite.
58 Partially ordered sets Definition Let P be a partially ordered set. For x P write: x = {p P : x p}, x = {p P : p x} Theorem (Higgins, Mitchell, Morayne, NR 06) Let P be a countably infinite poset, and let c be the number of connected components of P. (I) Suppose c is finite. Then rank(trans(p), End(P)) 2 if and only if there exists x P such that at least one of x or x is infinite. (II) If c is infinite and P is not an antichain, then rank(trans(p), End(P)) = 1.
59 Partially ordered sets Definition Let P be a partially ordered set. For x P write: x = {p P : x p}, x = {p P : p x} Theorem (Higgins, Mitchell, Morayne, NR 06) Let P be a countably infinite poset, and let c be the number of connected components of P. (I) Suppose c is finite. Then rank(trans(p), End(P)) 2 if and only if there exists x P such that at least one of x or x is infinite. (II) If c is infinite and P is not an antichain, then rank(trans(p), End(P)) = 1. (III) If P is an antichain, then rank(trans(p), End(P)) = 0:-)
60 Partially ordered sets Corollary rank(trans(p), End(P)) 2 provided any of the following hold: P has a smallest element; P has a largest element; P is a lattice.
61 A poset with relative rank 2 (I) Higgins, Mitchell, Morayne, NR 06 Lemma If P is a poset with no non-trivial mono- or epi-morphisms then rank(trans(p), End(P)) 2.
62 A poset with relative rank 2 (I) Higgins, Mitchell, Morayne, NR 06 Lemma If P is a poset with no non-trivial mono- or epi-morphisms then rank(trans(p), End(P)) 2. Proof. Suppose Trans(P) = End(P), µ.
63 A poset with relative rank 2 (I) Higgins, Mitchell, Morayne, NR 06 Lemma If P is a poset with no non-trivial mono- or epi-morphisms then rank(trans(p), End(P)) 2. Proof. Suppose Trans(P) = End(P), µ. Let π Sym(P); write π = γ 1 γ 2... γ n, γ i End(P) {µ}.
64 A poset with relative rank 2 (I) Higgins, Mitchell, Morayne, NR 06 Lemma If P is a poset with no non-trivial mono- or epi-morphisms then rank(trans(p), End(P)) 2. Proof. Suppose Trans(P) = End(P), µ. Let π Sym(P); write π = γ 1 γ 2... γ n, γ i End(P) {µ}. γ 1 is injective; γ n is surjective.
65 A poset with relative rank 2 (I) Higgins, Mitchell, Morayne, NR 06 Lemma If P is a poset with no non-trivial mono- or epi-morphisms then rank(trans(p), End(P)) 2. Proof. Suppose Trans(P) = End(P), µ. Let π Sym(P); write π = γ 1 γ 2... γ n, γ i End(P) {µ}. γ 1 is injective; γ n is surjective. Hence γ 1 = γ n = µ Sym(P).
66 A poset with relative rank 2 (I) Higgins, Mitchell, Morayne, NR 06 Lemma If P is a poset with no non-trivial mono- or epi-morphisms then rank(trans(p), End(P)) 2. Proof. Suppose Trans(P) = End(P), µ. Let π Sym(P); write π = γ 1 γ 2... γ n, γ i End(P) {µ}. γ 1 is injective; γ n is surjective. Hence γ 1 = γ n = µ Sym(P). By induction π = µ n, a contradiction.
67 Poset with no monomorphisms c 4 c 0 c 6 c 2 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
68 Poset with no monomorphisms c 0 c 4 c 2 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
69 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
70 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
71 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 3 c c 5 1 c 7 c 9 c 11
72 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
73 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
74 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
75 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
76 Poset with no monomorphisms c 0 c 2 c 4 c 6 c 8 c 10 c 1 c 3 c 5 c 7 c 9 c 11
77 Poset with no epimorphisms
78 Poset with no epimorphisms Let A = {a 1, a 2,...} be a countably infinite set.
79 Poset with no epimorphisms Let A = {a 1, a 2,...} be a countably infinite set. Let E be the set of finite subsets E of A satisfying: a n E 2 E n + 1.
80 Poset with no epimorphisms Let A = {a 1, a 2,...} be a countably infinite set. Let E be the set of finite subsets E of A satisfying: a n E 2 E n + 1. Enumerate E: E 1, E 2,...
81 Poset with no epimorphisms Let A = {a 1, a 2,...} be a countably infinite set. Let E be the set of finite subsets E of A satisfying: a n E 2 E n + 1. Enumerate E: E 1, E 2,... Let B = {b 1, b 2,...}.
82 Poset with no epimorphisms Let A = {a 1, a 2,...} be a countably infinite set. Let E be the set of finite subsets E of A satisfying: a n E 2 E n + 1. Enumerate E: E 1, E 2,... Let B = {b 1, b 2,...}. A partial order on A B: b m > a n a n E m. b m B E m A
83 A poset with relative rank 2 (II) c 0 c 2 c 4 B b m A E m c 1 c 3 c 5 c 2m+1 Proposition (Higgins, Mitchell, Morayne, NR 06) The above poset P satisfies the conditions from the part (I) of the Theorem, but has no non-trivial mono- or epi-morphisms; consequently rank(trans(p) : End(P)) = 2.
84 Random graph R the random graph Theorem (Truss 85) Every countable group embeds into Aut(R).
85 Random graph R the random graph Theorem (Truss 85) Every countable group embeds into Aut(R). Theorem (Bonato, Delic, Dolinka, to appear) Every countable monoid embeds into End(R).
86 Random graph Theorem (Truss 85) Aut(R) is simple.
87 Random graph Theorem (Truss 85) Aut(R) is simple. Theorem (Delic, Dolinka 04) End(R) has uncountably many ideals. Remark To every ideal I of a monoid M there corresponds the Rees congruence Φ I M on M; but, not every congruence is a Rees congruence.
88 Random graph Theorem (Truss 85) Aut(R) is simple. Theorem (Delic, Dolinka 04) End(R) has uncountably many ideals. Remark To every ideal I of a monoid M there corresponds the Rees congruence Φ I M on M; but, not every congruence is a Rees congruence. Question How many non-rees congruences does End(R) have?
89 Random graph Theorem (Truss 85) Aut(R) is simple. Theorem (Delic, Dolinka 04) End(R) has uncountably many ideals. Remark To every ideal I of a monoid M there corresponds the Rees congruence Φ I M on M; but, not every congruence is a Rees congruence. Question How many non-rees congruences does End(R) have? Question How many congruences does Trans(N) have?
90 Random graph Definition The semigroup S is said to have Sierpinski index n if every countable subset of S is contained in an n-generated subsemigroup of S.
91 Random graph Definition The semigroup S is said to have Sierpinski index n if every countable subset of S is contained in an n-generated subsemigroup of S. Question Does Aut(R) have a finite Sierpinski index?
92 Random graph Definition The semigroup S is said to have Sierpinski index n if every countable subset of S is contained in an n-generated subsemigroup of S. Question Does Aut(R) have a finite Sierpinski index? Question Does End(R) have a finite Sierpinski index?
93 Random graph Definition The semigroup S is said to have Sierpinski index n if every countable subset of S is contained in an n-generated subsemigroup of S. Question Does Aut(R) have a finite Sierpinski index? Question Does End(R) have a finite Sierpinski index? Facts rank(sym(r) : Aut(R)) = 1, rank(trans(r) : Aut(R)) = 2, rank(trans(r) : End(R)) = 1. Question rank(end(r) : Aut(R)) =?
94 Homogeneous structures Question Let H be a homogeneous relational structure. Find general/interesting conditions under which: rank(sym(h) : Aut(H)) is finite; rank(trans(h) : End(H)) is finite; rank(end(h) : Aut(H)) is finite; End(H) and/or Aut(H) have finite Sierpinski index.
95 Homomorphism homogeneous structures P.J. Cameron, J. Nesetril, Homomorphism-homogeneous relational structures, preprint P.J. Cameron, D. Lockett, Posets, homomorphisms and homogeneity, preprint Question Investigate algebraic and combinatorial properties of endomorphism monoids of these new types of homogeneous structures, and how they relate to the corresponding automorphism groups.
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