Universal locally finite maximally homogeneous semigroups Robert D. Gray 1 (joint work with I. Dolinka) Leeds, September 2017 1 This work was supported by the EPSRC grant EP/N033353/1 ‘Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem’, and by the LMS Research in Pairs grant ‘Universal locally finite partially homogeneous semigroups and inverse semigroups’ (Ref: 41530).
Hall’s group In 1959 Philip Hall constructed a countably infinite group U with the following properties: ▸ Universal: contains every finite group as a subgroup ▸ Locally finite: every finitely generated subgroup is finite ▸ Homogeneous: every isomorphism φ ∶ A → B between finite subgroups A , B of U extends to an automorphism of U . In fact, any two isomorphic subgroups of U are conjugate in U . U is the unique countable group satisfying these properties.
Hall’s group In 1959 Philip Hall constructed a countably infinite group U with the following properties: ▸ Universal: contains every finite group as a subgroup ▸ Locally finite: every finitely generated subgroup is finite ▸ Homogeneous: every isomorphism φ ∶ A → B between finite subgroups A , B of U extends to an automorphism of U . In fact, any two isomorphic subgroups of U are conjugate in U . U is the unique countable group satisfying these properties. AAA83, Novi Sad, 2012, Manfred Droste asked: “Is there a countable universal locally finite homogeneous semigroup?”
Constructing Hall’s group Example: Let G = S 4 , the symmetric group, and K = {() , ( 1 2 )} , L = {() , ( 1 2 )( 3 4 )} . Then K , L ≤ G , with K ≅ L but they are not conjugate in G .
Constructing Hall’s group Example: Let G = S 4 , the symmetric group, and K = {() , ( 1 2 )} , L = {() , ( 1 2 )( 3 4 )} . Then K , L ≤ G , with K ≅ L but they are not conjugate in G . Now embed φ ∶ S 4 = G → S G = S S 4 using Cayley’s Theorem g ↦ ρ g , x ρ g = xg for x ∈ G . Now φ ( K ) and φ ( L ) are conjugate in S G = S S 4 .
Constructing Hall’s group Example: Let G = S 4 , the symmetric group, and K = {() , ( 1 2 )} , L = {() , ( 1 2 )( 3 4 )} . Then K , L ≤ G , with K ≅ L but they are not conjugate in G . Now embed φ ∶ S 4 = G → S G = S S 4 using Cayley’s Theorem g ↦ ρ g , x ρ g = xg for x ∈ G . Now φ ( K ) and φ ( L ) are conjugate in S G = S S 4 . Construct U by iterating this process Set G 0 = S 4 , G 1 = S S 4 , G 2 = S S S 4 , . . . and let φ ∶ G i → G i + 1 be given by the right regular representation g ↦ ρ g , giving φ 0 φ 1 φ 2 � → G 1 � → G 2 � → ... G 0 Then U = ⋃ i ≥ 0 G i is the direct limit of this chain of symmetric groups.
Amalgamation
Amalgamation
Amalgamation and Fraïssé’s Theorem Definition (Amalgamation property for a class C ) If S , A , B ∈ C and f 1 ∶ S → A and f 2 ∶ S → B are embeddings then ∃ C ∈ C and embeddings g 1 ∶ A → C and g 2 ∶ B → C such that f 1 g 1 = f 2 g 2 . ▸ The class of finite groups has the amalgamation property. It is an amalgamation class and its Fraïssé limit is U . ▸ Fraïssé’s Theorem implies that a countable homogeneous structure is uniquely determined by its finitely generated substructures (called its age ). Conclusion: Hall’s group U is the unique countable homogeneous locally finite group.
Locally finite structures with maximal symmetry Groups Inverse semigroups Semigroups Permutations Partial bijections Transformations ( 1 3 ) ( 1 −) ( 1 2 ) 2 3 4 2 3 4 2 3 4 − 4 1 2 4 2 2 3 3 S n -limit I n -limit T n -limit S n 1 ≤ S n 2 ≤ ... I n 1 ≤ I n 2 ≤ ... T n 1 ≤ T n 2 ≤ ... U (Hall’s group) I T General philosophy Even though neither T nor I is homogeneous, they still display a high degree of symmetry in their combinatorial and algebraic structure.
Amalgamation bases for finite semigroups Kimura (1957): The class of finite semigroups does not have the amalgamation property. Therefore, there is no countable universal locally finite homogeneous semigroup.
Amalgamation bases for finite semigroups Kimura (1957): The class of finite semigroups does not have the amalgamation property. Therefore, there is no countable universal locally finite homogeneous semigroup. “How homogeneous can a countable universal locally finite semigroup be?”
Amalgamation bases for finite semigroups Kimura (1957): The class of finite semigroups does not have the amalgamation property. Therefore, there is no countable universal locally finite homogeneous semigroup. “How homogeneous can a countable universal locally finite semigroup be?” Definition. A finite semigroup S is an amalgamation base for all finite semigroups if in the class of finite semigroups every can be completed to some The class B of all such semigroups contains all finite: groups, inverse semigroups whose principal ideals form a chain, full transformation semigroups T n (K. Shoji (2016))
Maximal homogeneity B = { S ∶ S is an amalgamation base for all finite semigroups } T – a countable universal locally finite semigroup, S – a finite semigroup. Definition We say Aut ( T ) acts homogeneously on copies of S in T if for all U 1 , U 2 ≤ T with U 1 ≅ S ≅ U 2 , every isomorphism φ ∶ U 1 → U 2 extends to an automorphism of T . Proposition Aut ( T ) acts homogeneously on copies of S in T � ⇒ S ∈ B Definition We say T is maximally homogeneous if, for all S ∈ B , Aut ( T ) acts homogeneously on copies of S in T .
The maximally homogeneous semigroup T T n = the full transformation semigroup of all maps from [ n ] = { 1 , 2 ,... n } to itself under composition. Definition If we have a chain M 0 → M 1 → M 2 → ... of embeddings of semigroups, where each M i ≅ T n i , then the limit T = ⋃ i ≥ 0 M i is a full transformation limit semigroup. Fact: Every infinite full transformation limit semigroup is universal and locally finite.
The maximally homogeneous semigroup T T n = the full transformation semigroup of all maps from [ n ] = { 1 , 2 ,... n } to itself under composition. Definition If we have a chain M 0 → M 1 → M 2 → ... of embeddings of semigroups, where each M i ≅ T n i , then the limit T = ⋃ i ≥ 0 M i is a full transformation limit semigroup. Fact: Every infinite full transformation limit semigroup is universal and locally finite. Theorem (Dolinka & RDG (2017)) There is a unique maximally homogeneous full transformation limit semigroup T .
Existence and uniqueness of T Theorem (Dolinka & RDG (2017)) There is a unique maximally homogeneous full transformation limit semigroup T . ▸ Since T is not homogeneous it cannot be constructed using Fraïssé’s Theorem. ▸ We instead make use of a well-known generalisation, sometimes called the Hrushovski construction. ▸ See D. Evans’s Lecture notes from his talks at the Hausdorff Institute for Mathematics, Bonn, September 2013. ▸ T is not obtainable by iterating Cayley’s theorem for semigroups T n → T T n → T T Tn → ...
Structure of T n * S 4 123 124 134 234 * * ⇔ 12|3|4 α J β α & β generate the same ideal * * 13|2|4 ⇔ ∣ im α ∣ = ∣ im β ∣ . * * 14|2|3 * * 23|1|4 Set J r = { α ∈ T n ∶ ∣ im α ∣ = r } . * * 24|1|3 * * 34|1|2 Each idempotent ǫ in J r is contained in a 12 13 14 23 24 34 maximal subgroup H ǫ of S r . * * * 123|4 * * * 124|3 Example * * * 134|2 ǫ = ( 1 3 ) ∈ T 4 2 3 4 * * * 234|1 1 2 3 * * * * 12|34 * * * * 13|24 H ǫ = {( 1 k ) ∶ { i , j , k } = { 1 , 2 , 3 }} 2 3 4 * * * * 14|23 i j k 1 2 3 4 * * * * 1234
Structure of the maximally homogeneous semigroup T Theorem (Dolinka & RDG (2017)) 1. T is countable universal and locally finite. 2. T / J is a chain isomorphic to ( Q , ≤ ) . 3. Every maximal subgroup is isomorphic to Hall’s group U . 4. Aut (T ) acts transitively on the set of J -classes of T (so all principal factors J ∗ are isomorphic to each other).
Graham–Houghton graphs – local structure α = ( 1 3 ) , 2 3 4 5 6 5 3 3 5 2 ker α = 1 4 ∣ 2 3 6 ∣ 5 ⇔ α R β α & β generate same right ideal ⇔ ker α = ker β. ⇔ α L β α & β generate same left ideal ⇔ im α = im β. = R ∩ L H I - r -element set, P - partition with r parts H P , I is a group ⇔ H P , I contains an idempotent ⇔ I a transversal of P Γ ( J 2 )
Graham–Houghton graphs in T Definition (The countable random bipartite graph) It is the unique countable universal homogeneous bipartite graph. It is characterised as the countably infinite bipartite graph satisfying: ( ∗ ) for any two finite disjoint sets U, V from one part of the bipartition, there is a vertex w in the other part with w ∼ U but w / ∼ V .
Graham–Houghton graphs in T Definition (The countable random bipartite graph) It is the unique countable universal homogeneous bipartite graph. It is characterised as the countably infinite bipartite graph satisfying: ( ∗ ) for any two finite disjoint sets U, V from one part of the bipartition, there is a vertex w in the other part with w ∼ U but w / ∼ V . Theorem (Dolinka & RDG (2017)) Every Graham–Houghton graph of T is isomorphic to the countable random bipartite graph.
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