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Maximal subgroups of free idempotent generated semigroups Dandan Yang University of York, UK Based on my ongoing work with Victoria Gould Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 1 / 17 Idempotent

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Maximal subgroups of free idempotent generated semigroups

Dandan Yang University of York, UK Based on my ongoing work with Victoria Gould

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 1 / 17

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Idempotent generated semigroups

A semigroup is a non-empty set S equipped with a binary operation such that associativity holds: (∀a, b, c ∈ S) a(bc) = (ab)c.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 2 / 17

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Idempotent generated semigroups

A semigroup is a non-empty set S equipped with a binary operation such that associativity holds: (∀a, b, c ∈ S) a(bc) = (ab)c. Further, S is called a monoid if (∃1 ∈ S)(∀a ∈ S) 1a = a = a1.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 2 / 17

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Idempotent generated semigroups

A semigroup is a non-empty set S equipped with a binary operation such that associativity holds: (∀a, b, c ∈ S) a(bc) = (ab)c. Further, S is called a monoid if (∃1 ∈ S)(∀a ∈ S) 1a = a = a1. A semigroup S is (Von-Neumann) regular if (∀a ∈ S)(∃x ∈ S) axa = a.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 2 / 17

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Idempotent generated semigroups

Example 1 Let X be a set. Then TX = {α| α : X − → X is a map} forms a regular monoid with multiplication being composition of maps (left to right), called the full transformation monoid over X.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 3 / 17

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Idempotent generated semigroups

Example 1 Let X be a set. Then TX = {α| α : X − → X is a map} forms a regular monoid with multiplication being composition of maps (left to right), called the full transformation monoid over X. If |X| = n, denote TX by Tn, where n ∈ N is finite.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 3 / 17

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Idempotent generated semigroups

Example 1 Let X be a set. Then TX = {α| α : X − → X is a map} forms a regular monoid with multiplication being composition of maps (left to right), called the full transformation monoid over X. If |X| = n, denote TX by Tn, where n ∈ N is finite. Example 2 Let V be a vector space over a division ring D. Then End V = {α| α : V − → V is a linear map} is a regular monoid with multiplication being composition of maps (left to right), called the full linear monoid of V.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 3 / 17

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Idempotent generated semigroups

Example 1 Let X be a set. Then TX = {α| α : X − → X is a map} forms a regular monoid with multiplication being composition of maps (left to right), called the full transformation monoid over X. If |X| = n, denote TX by Tn, where n ∈ N is finite. Example 2 Let V be a vector space over a division ring D. Then End V = {α| α : V − → V is a linear map} is a regular monoid with multiplication being composition of maps (left to right), called the full linear monoid of V. If V is n-dimensional, where n ∈ N is finite, then End V ∼ = Mn(D).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 3 / 17

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Idempotent generated semigroups

An element e ∈ S is called an idempotent if e2 = e.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

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Idempotent generated semigroups

An element e ∈ S is called an idempotent if e2 = e. Let E = E(S) = {e2 = e : e ∈ S}.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

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Idempotent generated semigroups

An element e ∈ S is called an idempotent if e2 = e. Let E = E(S) = {e2 = e : e ∈ S}. Surprisingly, E(S) may determine the whole structure of S!!!

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

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Idempotent generated semigroups

An element e ∈ S is called an idempotent if e2 = e. Let E = E(S) = {e2 = e : e ∈ S}. Surprisingly, E(S) may determine the whole structure of S!!! A semigroup S is idempotent generated if S = E.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

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Idempotent generated semigroups

An element e ∈ S is called an idempotent if e2 = e. Let E = E(S) = {e2 = e : e ∈ S}. Surprisingly, E(S) may determine the whole structure of S!!! A semigroup S is idempotent generated if S = E. Example 3 Howie (1966) S(Tn) = {α ∈ Tn : rank α < n} = E \ {I}

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

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Idempotent generated semigroups

An element e ∈ S is called an idempotent if e2 = e. Let E = E(S) = {e2 = e : e ∈ S}. Surprisingly, E(S) may determine the whole structure of S!!! A semigroup S is idempotent generated if S = E. Example 3 Howie (1966) S(Tn) = {α ∈ Tn : rank α < n} = E \ {I} Example 4 Erdos (1967), Laffey (1973) S(Mn(D)) = {A ∈ Mn(D) : rank A < n} = E \ {I}.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

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Biordered sets

Let S be a semigroup and E = E(S). For any e, f ∈ E, define e ≤R f ⇔ fe = e and e ≤L f ⇔ ef = e.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 5 / 17

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Biordered sets

Let S be a semigroup and E = E(S). For any e, f ∈ E, define e ≤R f ⇔ fe = e and e ≤L f ⇔ ef = e. Note e ≤R f (e ≤L f ) implies both ef and fe are idempotents.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 5 / 17

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Biordered sets

Let S be a semigroup and E = E(S). For any e, f ∈ E, define e ≤R f ⇔ fe = e and e ≤L f ⇔ ef = e. Note e ≤R f (e ≤L f ) implies both ef and fe are idempotents. We say that (e, f ) is a basic pair if e ≤R f , f ≤R e, e ≤L f or f ≤L e, i.e. {e, f } ∩ {ef , fe} = ∅;

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 5 / 17

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Biordered sets

Let S be a semigroup and E = E(S). For any e, f ∈ E, define e ≤R f ⇔ fe = e and e ≤L f ⇔ ef = e. Note e ≤R f (e ≤L f ) implies both ef and fe are idempotents. We say that (e, f ) is a basic pair if e ≤R f , f ≤R e, e ≤L f or f ≤L e, i.e. {e, f } ∩ {ef , fe} = ∅; and ef , fe are called basic products.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 5 / 17

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Biordered sets

Let S be a semigroup and E = E(S). For any e, f ∈ E, define e ≤R f ⇔ fe = e and e ≤L f ⇔ ef = e. Note e ≤R f (e ≤L f ) implies both ef and fe are idempotents. We say that (e, f ) is a basic pair if e ≤R f , f ≤R e, e ≤L f or f ≤L e, i.e. {e, f } ∩ {ef , fe} = ∅; and ef , fe are called basic products. Consider E(S) as a set, we define a partial binary operation · by e · f = ef if (e, f ) is basic,

  • therwise, undefined.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 5 / 17

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Biordered sets

Facts

1 The partial algebra E satisfies a number of axioms; if S is regular, an

extra axiom holds.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 6 / 17

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Biordered sets

Facts

1 The partial algebra E satisfies a number of axioms; if S is regular, an

extra axiom holds.

2 A biordered set is a partial algebra satisfying these axioms; if the

extra one also holds it is a regular biordered set.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 6 / 17

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Biordered sets

Facts

1 The partial algebra E satisfies a number of axioms; if S is regular, an

extra axiom holds.

2 A biordered set is a partial algebra satisfying these axioms; if the

extra one also holds it is a regular biordered set.

3 A biordered set is regular if and only if E = E(S) for a regular

semigroup S Nambooripad (1979).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 6 / 17

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Biordered sets

Facts

1 The partial algebra E satisfies a number of axioms; if S is regular, an

extra axiom holds.

2 A biordered set is a partial algebra satisfying these axioms; if the

extra one also holds it is a regular biordered set.

3 A biordered set is regular if and only if E = E(S) for a regular

semigroup S Nambooripad (1979).

4 Any biordered set E is E(S) for some semigroup S Easdown (1985). Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 6 / 17

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Free idempotent generated semigroups

Let E be a biordered set (equivalently, a set of idempotents E of a semigroup S).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 7 / 17

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Free idempotent generated semigroups

Let E be a biordered set (equivalently, a set of idempotents E of a semigroup S). The free idempotent generated semigroup IG(E) is a free object in the category of semigroups that are generated by E, defined by IG(E) = E : ¯ e¯ f = ef , e, f ∈ E, {e, f } ∩ {ef , fe} = ∅. where E = {¯ e : e ∈ E}.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 7 / 17

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Free idempotent generated semigroups

Facts

1 IG(E) = E. Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 8 / 17

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Free idempotent generated semigroups

Facts

1 IG(E) = E. 2 The natural map φ : IG(E) → S, given by ¯

eφ = e, is a morphism onto S′ = E(S).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 8 / 17

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Free idempotent generated semigroups

Facts

1 IG(E) = E. 2 The natural map φ : IG(E) → S, given by ¯

eφ = e, is a morphism onto S′ = E(S).

3 The restriction of φ to the set of idempotents of IG(E) is a bijection,

so that E(IG(E)) ∼ = E as biordered sets.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 8 / 17

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Free idempotent generated semigroups

Facts

1 IG(E) = E. 2 The natural map φ : IG(E) → S, given by ¯

eφ = e, is a morphism onto S′ = E(S).

3 The restriction of φ to the set of idempotents of IG(E) is a bijection,

so that E(IG(E)) ∼ = E as biordered sets.

4 The morphism φ is an onto morphism from He to He. Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 8 / 17

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Free idempotent generated semigroups

Facts

1 IG(E) = E. 2 The natural map φ : IG(E) → S, given by ¯

eφ = e, is a morphism onto S′ = E(S).

3 The restriction of φ to the set of idempotents of IG(E) is a bijection,

so that E(IG(E)) ∼ = E as biordered sets.

4 The morphism φ is an onto morphism from He to He. 5 IG(E) has some pleasant properties, particularly with respect to

Green’s relations.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 8 / 17

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Maximal subgroups of IG(E)

Work of Pastijn (1977, 1980), Nambooripad and Pastijn (1980), McElwee (2002) led to a conjecture that all these groups must be free groups.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 9 / 17

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Maximal subgroups of IG(E)

Work of Pastijn (1977, 1980), Nambooripad and Pastijn (1980), McElwee (2002) led to a conjecture that all these groups must be free groups. Brittenham, Margolis and Meakin (2009) Z ⊕ Z can be a maximal subgroup of IG(E), for some E.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 9 / 17

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Maximal subgroups of IG(E)

Work of Pastijn (1977, 1980), Nambooripad and Pastijn (1980), McElwee (2002) led to a conjecture that all these groups must be free groups. Brittenham, Margolis and Meakin (2009) Z ⊕ Z can be a maximal subgroup of IG(E), for some E. Gray and Ruskuc (2012) Any group occurs as a maximal subgroup of some IG(E), a general presentation and a special choice of E are needed.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 9 / 17

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Maximal subgroups of IG(E)

Work of Pastijn (1977, 1980), Nambooripad and Pastijn (1980), McElwee (2002) led to a conjecture that all these groups must be free groups. Brittenham, Margolis and Meakin (2009) Z ⊕ Z can be a maximal subgroup of IG(E), for some E. Gray and Ruskuc (2012) Any group occurs as a maximal subgroup of some IG(E), a general presentation and a special choice of E are needed. Gould and Yang (2012) Any group occurs as a maximal subgroup of a natural IG(E), a simple approach suffices. Dolinka and Ruskuc (2013) Any group occurs as a maximal subgroup of IG(E) for some band E.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 9 / 17

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Maximal subgroups of IG(E)

Given a special biordered set E, which kind of groups can be the maximal subgroups of IG(E)?

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 10 / 17

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Maximal subgroups of IG(E)

Given a special biordered set E, which kind of groups can be the maximal subgroups of IG(E)? Let S be a semigroup with E = E(S). Let e ∈ E. Our aim is to find the relationship between the maximal subgroup He of IG(E) with identity e and the maximal subgroup He of S with identity e.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 10 / 17

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Maximal subgroups of IG(E)

Given a special biordered set E, which kind of groups can be the maximal subgroups of IG(E)? Let S be a semigroup with E = E(S). Let e ∈ E. Our aim is to find the relationship between the maximal subgroup He of IG(E) with identity e and the maximal subgroup He of S with identity e. There is an onto morphism from He to He.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 10 / 17

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Maximal subgroups of IG(E)

Given a special biordered set E, which kind of groups can be the maximal subgroups of IG(E)? Let S be a semigroup with E = E(S). Let e ∈ E. Our aim is to find the relationship between the maximal subgroup He of IG(E) with identity e and the maximal subgroup He of S with identity e. There is an onto morphism from He to He. Is He ∼ = He, for some E and some e ∈ E?

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 10 / 17

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Full transformation monoids and full linear monoids

Tn (PT n) - full (partial) transformation monoid, E - its biordered set. Gray and Ruskuc (2012); Dolinka (2013) rank e = r < n − 1, He ∼ = He ∼ = Sr.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 11 / 17

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Full transformation monoids and full linear monoids

Tn (PT n) - full (partial) transformation monoid, E - its biordered set. Gray and Ruskuc (2012); Dolinka (2013) rank e = r < n − 1, He ∼ = He ∼ = Sr. Brittenham, Margolis and Meakin (2010) Mn(D) - full linear monoid, E - its biordered set. rank e = 1 and n ≥ 3, He ∼ = He ∼ = D∗.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 11 / 17

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Full transformation monoids and full linear monoids

Tn (PT n) - full (partial) transformation monoid, E - its biordered set. Gray and Ruskuc (2012); Dolinka (2013) rank e = r < n − 1, He ∼ = He ∼ = Sr. Brittenham, Margolis and Meakin (2010) Mn(D) - full linear monoid, E - its biordered set. rank e = 1 and n ≥ 3, He ∼ = He ∼ = D∗. Dolinka and Gray (2012) rank e = r < n/3 and n ≥ 4, He ∼ = He ∼ = GLr(D).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 11 / 17

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Full transformation monoids and full linear monoids

Tn (PT n) - full (partial) transformation monoid, E - its biordered set. Gray and Ruskuc (2012); Dolinka (2013) rank e = r < n − 1, He ∼ = He ∼ = Sr. Brittenham, Margolis and Meakin (2010) Mn(D) - full linear monoid, E - its biordered set. rank e = 1 and n ≥ 3, He ∼ = He ∼ = D∗. Dolinka and Gray (2012) rank e = r < n/3 and n ≥ 4, He ∼ = He ∼ = GLr(D). Note rank e = n − 1, He is free; rank e = n, He is trivial.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 11 / 17

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Free G-acts

Let G be a group, n ∈ N, n ≥ 3. Let Fn(G) be a rank n free left G-act.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 12 / 17

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Free G-acts

Let G be a group, n ∈ N, n ≥ 3. Let Fn(G) be a rank n free left G-act. Recall that, as a set, Fn(G) = {gxi : g ∈ G, i ∈ [1, n]};

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 12 / 17

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Free G-acts

Let G be a group, n ∈ N, n ≥ 3. Let Fn(G) be a rank n free left G-act. Recall that, as a set, Fn(G) = {gxi : g ∈ G, i ∈ [1, n]}; identify xi with 1xi, where 1 is the identity of G;

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 12 / 17

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Free G-acts

Let G be a group, n ∈ N, n ≥ 3. Let Fn(G) be a rank n free left G-act. Recall that, as a set, Fn(G) = {gxi : g ∈ G, i ∈ [1, n]}; identify xi with 1xi, where 1 is the identity of G; gxi = hxj if and only if g = h and i = j;

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 12 / 17

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Free G-acts

Let G be a group, n ∈ N, n ≥ 3. Let Fn(G) be a rank n free left G-act. Recall that, as a set, Fn(G) = {gxi : g ∈ G, i ∈ [1, n]}; identify xi with 1xi, where 1 is the identity of G; gxi = hxj if and only if g = h and i = j; the action of G is given by g(hxi) = (gh)xi.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 12 / 17

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Free G-acts

Let G be a group, n ∈ N, n ≥ 3. Let Fn(G) be a rank n free left G-act. Recall that, as a set, Fn(G) = {gxi : g ∈ G, i ∈ [1, n]}; identify xi with 1xi, where 1 is the identity of G; gxi = hxj if and only if g = h and i = j; the action of G is given by g(hxi) = (gh)xi. A map α : Fn(G) − → Fn(G) is called an endomorphism if for all i ∈ [1, n] and g ∈ G, we have (gxi)α = g(xiα)

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 12 / 17

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Free G-acts

Let G be a group, n ∈ N, n ≥ 3. Let Fn(G) be a rank n free left G-act. Recall that, as a set, Fn(G) = {gxi : g ∈ G, i ∈ [1, n]}; identify xi with 1xi, where 1 is the identity of G; gxi = hxj if and only if g = h and i = j; the action of G is given by g(hxi) = (gh)xi. A map α : Fn(G) − → Fn(G) is called an endomorphism if for all i ∈ [1, n] and g ∈ G, we have (gxi)α = g(xiα) and the rank of α is the minimal number of (free) generators in its image.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 12 / 17

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Endomorphism monoids of free left G-acts

Let End Fn(G) be monoid of all endomorphisms of Fn(G) with E = E(End Fn(G)). Facts

1 End Fn(G) ∼

= G ≀ Tn.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 13 / 17

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Endomorphism monoids of free left G-acts

Let End Fn(G) be monoid of all endomorphisms of Fn(G) with E = E(End Fn(G)). Facts

1 End Fn(G) ∼

= G ≀ Tn.

2 Fountain and Lewin (1992) S(End Fn(G)) = E \ {I}. Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 13 / 17

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Endomorphism monoids of free left G-acts

Let End Fn(G) be monoid of all endomorphisms of Fn(G) with E = E(End Fn(G)). Facts

1 End Fn(G) ∼

= G ≀ Tn.

2 Fountain and Lewin (1992) S(End Fn(G)) = E \ {I}.

Dolinka, Gould and Yang (2014) rank e = r < n − 1, He ∼ = He ∼ = G ≀ Sr.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 13 / 17

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Endomorphism monoids of free left G-acts

Let End Fn(G) be monoid of all endomorphisms of Fn(G) with E = E(End Fn(G)). Facts

1 End Fn(G) ∼

= G ≀ Tn.

2 Fountain and Lewin (1992) S(End Fn(G)) = E \ {I}.

Dolinka, Gould and Yang (2014) rank e = r < n − 1, He ∼ = He ∼ = G ≀ Sr. Note If r = n, then He is trivial; if r = n − 1, then He is free.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 13 / 17

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Endomorphism monoids of free left G-acts

Let End Fn(G) be monoid of all endomorphisms of Fn(G) with E = E(End Fn(G)). Facts

1 End Fn(G) ∼

= G ≀ Tn.

2 Fountain and Lewin (1992) S(End Fn(G)) = E \ {I}.

Dolinka, Gould and Yang (2014) rank e = r < n − 1, He ∼ = He ∼ = G ≀ Sr. Note If r = n, then He is trivial; if r = n − 1, then He is free. If r = 1, then He = G and so that: Corollary Every group can be a maximal subgroup of a naturally occurring IG(E).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 13 / 17

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Endomorphism monoids of free left G-acts

Let End Fn(G) be monoid of all endomorphisms of Fn(G) with E = E(End Fn(G)). Facts

1 End Fn(G) ∼

= G ≀ Tn.

2 Fountain and Lewin (1992) S(End Fn(G)) = E \ {I}.

Dolinka, Gould and Yang (2014) rank e = r < n − 1, He ∼ = He ∼ = G ≀ Sr. Note If r = n, then He is trivial; if r = n − 1, then He is free. If r = 1, then He = G and so that: Corollary Every group can be a maximal subgroup of a naturally occurring IG(E). If G is trivial, then End Fn(G) is essentially Tn, so we deduce the result of Gray and Ruskuc (2012).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 13 / 17

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v ∗-algebras

‘What then do vector spaces and sets have in common which forces End V and TX to support a similar pleasing structure’? The above question was asked by Gould (1995).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 14 / 17

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v ∗-algebras

‘What then do vector spaces and sets have in common which forces End V and TX to support a similar pleasing structure’? The above question was asked by Gould (1995). To answer it, Gould investigated the endomorphism monoid of a class of universal algebra, called v∗-algebra, also known as independence algebra.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 14 / 17

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v ∗-algebras

‘What then do vector spaces and sets have in common which forces End V and TX to support a similar pleasing structure’? The above question was asked by Gould (1995). To answer it, Gould investigated the endomorphism monoid of a class of universal algebra, called v∗-algebra, also known as independence algebra. Let A be an (universal) algebra. A constant in A is the image of a basic nullary operation. An algebraic constant in A is the image of a nullary term operation.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 14 / 17

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v ∗-algebras

‘What then do vector spaces and sets have in common which forces End V and TX to support a similar pleasing structure’? The above question was asked by Gould (1995). To answer it, Gould investigated the endomorphism monoid of a class of universal algebra, called v∗-algebra, also known as independence algebra. Let A be an (universal) algebra. A constant in A is the image of a basic nullary operation. An algebraic constant in A is the image of a nullary term operation. Note ∅ = ∅ if and only if A has no algebraic constants; if and only if A has no constants.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 14 / 17

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SLIDE 60

v ∗-algebras

‘What then do vector spaces and sets have in common which forces End V and TX to support a similar pleasing structure’? The above question was asked by Gould (1995). To answer it, Gould investigated the endomorphism monoid of a class of universal algebra, called v∗-algebra, also known as independence algebra. Let A be an (universal) algebra. A constant in A is the image of a basic nullary operation. An algebraic constant in A is the image of a nullary term operation. Note ∅ = ∅ if and only if A has no algebraic constants; if and only if A has no constants.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 14 / 17

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SLIDE 61

v ∗-algebras

We say that A satisfies the exchange property (EP), if for every subset X of A and all elements x, y ∈ A if y ∈ X ∪ {x} and y ∈ X = ⇒ x ∈ X ∪ {y}.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 15 / 17

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SLIDE 62

v ∗-algebras

We say that A satisfies the exchange property (EP), if for every subset X of A and all elements x, y ∈ A if y ∈ X ∪ {x} and y ∈ X = ⇒ x ∈ X ∪ {y}. A subset X of A is called independent if for each x ∈ X we have x ∈ X\{x}.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 15 / 17

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SLIDE 63

v ∗-algebras

We say that A satisfies the exchange property (EP), if for every subset X of A and all elements x, y ∈ A if y ∈ X ∪ {x} and y ∈ X = ⇒ x ∈ X ∪ {y}. A subset X of A is called independent if for each x ∈ X we have x ∈ X\{x}. We say that a subset X of A is a basis of A if X generates A and is independent.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 15 / 17

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SLIDE 64

v ∗-algebras

We say that A satisfies the exchange property (EP), if for every subset X of A and all elements x, y ∈ A if y ∈ X ∪ {x} and y ∈ X = ⇒ x ∈ X ∪ {y}. A subset X of A is called independent if for each x ∈ X we have x ∈ X\{x}. We say that a subset X of A is a basis of A if X generates A and is independent. We have the following results:

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 15 / 17

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SLIDE 65

v ∗-algebras

We say that A satisfies the exchange property (EP), if for every subset X of A and all elements x, y ∈ A if y ∈ X ∪ {x} and y ∈ X = ⇒ x ∈ X ∪ {y}. A subset X of A is called independent if for each x ∈ X we have x ∈ X\{x}. We say that a subset X of A is a basis of A if X generates A and is independent. We have the following results: (i) Any algebra satisfying the exchange property (EP) has a basis.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 15 / 17

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SLIDE 66

v ∗-algebras

We say that A satisfies the exchange property (EP), if for every subset X of A and all elements x, y ∈ A if y ∈ X ∪ {x} and y ∈ X = ⇒ x ∈ X ∪ {y}. A subset X of A is called independent if for each x ∈ X we have x ∈ X\{x}. We say that a subset X of A is a basis of A if X generates A and is independent. We have the following results: (i) Any algebra satisfying the exchange property (EP) has a basis. (ii) A subset X is a basis if and only if X is a minimal generating set if and

  • nly if X is the maximal independent set.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 15 / 17

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SLIDE 67

v ∗-algebras

We say that A satisfies the exchange property (EP), if for every subset X of A and all elements x, y ∈ A if y ∈ X ∪ {x} and y ∈ X = ⇒ x ∈ X ∪ {y}. A subset X of A is called independent if for each x ∈ X we have x ∈ X\{x}. We say that a subset X of A is a basis of A if X generates A and is independent. We have the following results: (i) Any algebra satisfying the exchange property (EP) has a basis. (ii) A subset X is a basis if and only if X is a minimal generating set if and

  • nly if X is the maximal independent set.

(iii) All of bases of A has the same cardinality, called the rank of A.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 15 / 17

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SLIDE 68

v ∗-algebras

We say that a mapping θ from A into itself is an endomorphism if for any n-ary term operation t(x1, · · · , xn) we have t(x1, · · · , xn)θ = t(x1θ, · · · , xnθ).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 16 / 17

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SLIDE 69

v ∗-algebras

We say that a mapping θ from A into itself is an endomorphism if for any n-ary term operation t(x1, · · · , xn) we have t(x1, · · · , xn)θ = t(x1θ, · · · , xnθ). The rank of θ is defined as the cardinality of the basis of im θ.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 16 / 17

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SLIDE 70

v ∗-algebras

We say that a mapping θ from A into itself is an endomorphism if for any n-ary term operation t(x1, · · · , xn) we have t(x1, · · · , xn)θ = t(x1θ, · · · , xnθ). The rank of θ is defined as the cardinality of the basis of im θ. An algebra A satisfying the exchange property is called a v∗-algebra if it satisfies the free basis property, by which we mean that for any basis X

  • f A and a map α : X −

→ A, α can be extended to an endomorphism of A.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 16 / 17

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SLIDE 71

v ∗-algebras

We say that a mapping θ from A into itself is an endomorphism if for any n-ary term operation t(x1, · · · , xn) we have t(x1, · · · , xn)θ = t(x1θ, · · · , xnθ). The rank of θ is defined as the cardinality of the basis of im θ. An algebra A satisfying the exchange property is called a v∗-algebra if it satisfies the free basis property, by which we mean that for any basis X

  • f A and a map α : X −

→ A, α can be extended to an endomorphism of A. Note Sets, vector spaces over division rings and free G-acts all are examples of v∗-algebras.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 16 / 17

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SLIDE 72

v ∗-algebras

Question: Let E = E(End A). Which groups can be the maximal subgroups of IG(E).

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 17 / 17

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v ∗-algebras

Question: Let E = E(End A). Which groups can be the maximal subgroups of IG(E). Let e ∈ E with rank r, where 1 ≤ r ≤ n.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 17 / 17

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SLIDE 74

v ∗-algebras

Question: Let E = E(End A). Which groups can be the maximal subgroups of IG(E). Let e ∈ E with rank r, where 1 ≤ r ≤ n. If r = n, the maximal subgroups are trivial.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 17 / 17

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SLIDE 75

v ∗-algebras

Question: Let E = E(End A). Which groups can be the maximal subgroups of IG(E). Let e ∈ E with rank r, where 1 ≤ r ≤ n. If r = n, the maximal subgroups are trivial. If r = n − 1, the maximal subgroups are free.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 17 / 17

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SLIDE 76

v ∗-algebras

Question: Let E = E(End A). Which groups can be the maximal subgroups of IG(E). Let e ∈ E with rank r, where 1 ≤ r ≤ n. If r = n, the maximal subgroups are trivial. If r = n − 1, the maximal subgroups are free. If r = 1 and A has no constants. The set H of all unary term operations of A forms a group under the multiplication given as the composition of functions.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 17 / 17

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SLIDE 77

v ∗-algebras

Question: Let E = E(End A). Which groups can be the maximal subgroups of IG(E). Let e ∈ E with rank r, where 1 ≤ r ≤ n. If r = n, the maximal subgroups are trivial. If r = n − 1, the maximal subgroups are free. If r = 1 and A has no constants. The set H of all unary term operations of A forms a group under the multiplication given as the composition of functions. The maximal subgroup He ∼ = He ∼ = H.

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 17 / 17

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SLIDE 78

v ∗-algebras

Question: Let E = E(End A). Which groups can be the maximal subgroups of IG(E). Let e ∈ E with rank r, where 1 ≤ r ≤ n. If r = n, the maximal subgroups are trivial. If r = n − 1, the maximal subgroups are free. If r = 1 and A has no constants. The set H of all unary term operations of A forms a group under the multiplication given as the composition of functions. The maximal subgroup He ∼ = He ∼ = H. Recently, we are working on the case where r < n/3...

Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 17 / 17