Maximal subgroups of free idempotent generated semigroups Dandan - PowerPoint PPT Presentation

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 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

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 ) 1 a = a = a 1 . Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 2 / 17

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 ) 1 a = a = a 1 . 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

Idempotent generated semigroups Example 1 Let X be a set. Then T X = { 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

Idempotent generated semigroups Example 1 Let X be a set. Then T X = { 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 T X by T n , where n ∈ N is finite. Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 3 / 17

Idempotent generated semigroups Example 1 Let X be a set. Then T X = { 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 T X by T n , 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

Idempotent generated semigroups Example 1 Let X be a set. Then T X = { 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 T X by T n , 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 ∼ = M n ( D ). Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 3 / 17

Idempotent generated semigroups An element e ∈ S is called an idempotent if e 2 = e . Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

Idempotent generated semigroups An element e ∈ S is called an idempotent if e 2 = e . Let E = E ( S ) = { e 2 = e : e ∈ S } . Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

Idempotent generated semigroups An element e ∈ S is called an idempotent if e 2 = e . Let E = E ( S ) = { e 2 = 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

Idempotent generated semigroups An element e ∈ S is called an idempotent if e 2 = e . Let E = E ( S ) = { e 2 = 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

Idempotent generated semigroups An element e ∈ S is called an idempotent if e 2 = e . Let E = E ( S ) = { e 2 = 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 ( T n ) = { α ∈ T n : rank α < n } = � E \ { I }� Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

Idempotent generated semigroups An element e ∈ S is called an idempotent if e 2 = e . Let E = E ( S ) = { e 2 = 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 ( T n ) = { α ∈ T n : rank α < n } = � E \ { I }� Example 4 Erdos (1967), Laffey (1973) S ( M n ( D )) = { A ∈ M n ( D ) : rank A < n } = � E \ { I }� . Dandan Yang (University of York) Free idempotent generated semigroups June 18, 2014 4 / 17

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

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

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

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

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

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

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

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

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

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

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 e ¯ IG( 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