Free idempotent generated semigroups Nik Ru skuc - PowerPoint PPT Presentation

Free idempotent generated semigroups Nik Ruˇ skuc nik@mcs.st-and.ac.uk School of Mathematics and Statistics, University of St Andrews Novi Sad, 6 June 2013

All of old. Nothing else ever. Ever tried. Ever failed. No matter. Try again. (S. Beckett, Worstword Ho) University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Free IG semigroups: idea ◮ To every semigroup S with idempotents E associate the free-est semigroup IG( E ) in which idempotents have the same structure as in S . ◮ To every regular semigroup S with idempotents E associate the free-est regular semigroup RIG( E ) in which idempotents have the same structure as in S . ◮ Structure = biorder. University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Free IG semigroups: definition E – the set of idempotents in a semigroup S . IG( E ) := � E | e 2 = e ( e ∈ E ) , e · f = ef ( { e , f } ∩ { ef , fe } � = ∅ ) � . Suppose now S is regular. S ( e , f ) = { h ∈ E : ehf = ef , fhe = h } � = ∅ (sandwich sets). RIG( E ) := � E | IG , e · h · f = e · f ( e , f ∈ E , h ∈ S ( e , f )) � . University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: V -semilattice e f Let S = z University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: V -semilattice e f Let S = z IG( S ) = � e , f , z | e 2 = e , f 2 = f , z 2 = z , ez = ze = fz = zf = z � : University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: V -semilattice e f Let S = z IG( S ) = � e , f , z | e 2 = e , f 2 = f , z 2 = z , ez = ze = fz = zf = z � : e f ( ef ) i e ( ef ) i ( fe ) i ( fe ) i f z University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: V -semilattice e f Let S = z IG( S ) = � e , f , z | e 2 = e , f 2 = f , z 2 = z , ez = ze = fz = zf = z � : e f ( ef ) i e ( ef ) i ( fe ) i ( fe ) i f z RIG = � e , f , z | IG , ef = fe = z � = S . University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: 2 × 2 rectangular band S = � e ij | e ij e kl = e il ( i , j , k , l ∈ { 1 , 2 } ) � : e 11 e 12 e 21 e 22 University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: 2 × 2 rectangular band S = � e ij | e ij e kl = e il ( i , j , k , l ∈ { 1 , 2 } ) � : e 11 e 12 e 21 e 22 IG( S ) = � e ij | e ij e kl = e il ( i , j , k , l ∈ { 1 , 2 } , i = k or j = l ) � : University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: 2 × 2 rectangular band S = � e ij | e ij e kl = e il ( i , j , k , l ∈ { 1 , 2 } ) � : e 11 e 12 e 21 e 22 IG( S ) = � e ij | e ij e kl = e il ( i , j , k , l ∈ { 1 , 2 } , i = k or j = l ) � : ( e 11 e 22 ) i e 11 ( e 12 e 21 ) i e 12 ( e 12 e 21 ) i ( e 11 e 22 ) i ( e 21 e 12 ) i e 21 ( e 22 e 11 ) i e 22 ( e 22 e 11 ) i ( e 21 e 12 ) i University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Example: 2 × 2 rectangular band S = � e ij | e ij e kl = e il ( i , j , k , l ∈ { 1 , 2 } ) � : e 11 e 12 e 21 e 22 IG( S ) = � e ij | e ij e kl = e il ( i , j , k , l ∈ { 1 , 2 } , i = k or j = l ) � : ( e 11 e 22 ) i e 11 ( e 12 e 21 ) i e 12 ( e 12 e 21 ) i ( e 11 e 22 ) i ( e 21 e 12 ) i e 21 ( e 22 e 11 ) i e 22 ( e 22 e 11 ) i ( e 21 e 12 ) i RIG( S ) = IG( S ). University of St Andrews Nik Ruˇ skuc: Free IG semigroups

S , IG( E ), RIG( E ) ◮ The sets of idempotents isomorphic (as biordered sets). ◮ The D -class of an idempotent e has the same dimensions in all three. ◮ The group H e in S is a homomorphic image of its counterparts in IG( E ), RIG( E ), which themselves are isomorphic. ◮ IG( E ) may contain other, non-regular D -classes. Question Describe maximal subgroups of IG( E ) and RIG( E ). University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: big picture S University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: big picture IG( E ) S University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: big picture IG( E ) S e University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: big picture IG( E ) S e e University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: big picture IG( E ) S e e University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: big picture IG( E ) S e e University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: big picture IG( E ) S e e G =??? University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Setting the problem: zoom in IG( E ) e (11) S e 12 e 13 G =??? e (11) e 12 e 13 e 22 e 24 e 22 e 24 e 31 e 32 e 33 e 34 e 31 e 32 e 33 e 34 D University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Generators Fact G is generated by a set in 1-1 correspondence with D ∩ E ( S ) . University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Generators Fact G is generated by a set in 1-1 correspondence with D ∩ E ( S ) . generators of G D e (11) e 12 e 13 f 11 f 12 f 13 e 22 e 24 f 22 f 24 e 31 e 32 e 33 e 34 f 31 f 32 f 33 f 34 University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Generators Fact G is generated by a set in 1-1 correspondence with D ∩ E ( S ) . generators of G D e (11) e 12 e 13 f 11 f 12 f 13 e 22 e 24 f 22 f 24 e 31 e 32 e 33 e 34 f 31 f 32 f 33 f 34 G = � f ij ( e ij ∈ D ∩ E ) | ??? � University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Typical relations: f − 1 ij f il = f − 1 kj f kl h = h 2 j 1 l e 11 1 − · h e ij e il i h · − − · h e kj e kl k h · − � e ij � e il ; relation: f − 1 f il = f − 1 Singular square kj f kl . e kj e kl ij University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Presentation Theorem (Nambooripad ’79; Gray, NR ’12) The maximal subgroup G of e ∈ E in IG ( E ) or RIG ( E ) is defined by the presentation: � f ij | f i ,π ( i ) = 1 ( i ∈ I ) , f ij = f il ( if r j e il is a Schreier rep ) , � e ij � e il f − 1 f il = f − 1 kj f kl ( sing. sq. ) � . ij e kj e kl University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (1) ◮ Proof: Reidemeister–Schreier followed by Tietze transformations. ◮ Two types of relations: ◮ Initial conditions: declaring some generators equal to 1 or each other; ◮ Main relations: one per singular square. ◮ All relations of length ≤ 4. ◮ If no singular squares, the group is free. ◮ They have been conjectured to always be free. ◮ Brittenham, Margolis, Meakin ’09 construct a 73-element semigroup such that IG( E ) and RIG( E ) have Z × Z as a maximal subgroup. University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij � 1 � b ⇒ ab = c . ◮ a c University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij � 1 � b ⇒ ab = c . ◮ a c ◮ But: Every semigroup can be defined by relations of the form ab = c . University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij � 1 � b ⇒ ab = c . ◮ a c ◮ But: Every semigroup can be defined by relations of the form ab = c . ◮ Even better: Every finitely presented semigroup can be defined by finitely many relations of the form ab = c . University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij � 1 � b ⇒ ab = c . ◮ a c ◮ But: Every semigroup can be defined by relations of the form ab = c . ◮ Even better: Every finitely presented semigroup can be defined by finitely many relations of the form ab = c . ◮ Some more special squares . . . University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij � 1 � b ⇒ ab = c . ◮ a c ◮ But: Every semigroup can be defined by relations of the form ab = c . ◮ Even better: Every finitely presented semigroup can be defined by finitely many relations of the form ab = c . ◮ Some more special squares . . . � a � a ⇒ ◮ b c University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij � 1 � b ⇒ ab = c . ◮ a c ◮ But: Every semigroup can be defined by relations of the form ab = c . ◮ Even better: Every finitely presented semigroup can be defined by finitely many relations of the form ab = c . ◮ Some more special squares . . . � a � a ⇒ b = c . ◮ b c University of St Andrews Nik Ruˇ skuc: Free IG semigroups

Remarks (2) ◮ What can be defined by relations f − 1 f il = f − 1 kj f kl ? ij � 1 � b ⇒ ab = c . ◮ a c ◮ But: Every semigroup can be defined by relations of the form ab = c . ◮ Even better: Every finitely presented semigroup can be defined by finitely many relations of the form ab = c . ◮ Some more special squares . . . � a � a ⇒ b = c . ◮ b c � 1 � 1 ⇒ ◮ 1 a University of St Andrews Nik Ruˇ skuc: Free IG semigroups