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Quotients of the magmatic operad: lattice structures and convergent rewrite systems Cyrille Chenavier 1 Christophe Cordero 2 Samuele Giraudo 2 1 INRIA Lille - Nord Europe, quipe GAIA 2 Universit Paris-Est Marne-la-Valle, LIGM December 20,

  1. Motivations Nonsymmetric operads ◮ A (nonsymmetric linear) operad is a positively graded ( K -)vector space � O = O ( n ) , n ∈ N together with ⊲ a distinguished element 1 ∈ O (1); ⊲ partial compositions ◦ i : O ( n ) ⊗ O ( m ) → O ( n + m − 1), ∀ 1 ≤ i ≤ n ; satisfying axioms (next slide). ◮ Example: the operad End V of (multi-)linear mappings on the vector space V ; ⊲ End V ( n ) := Hom � V ⊗ n , V � ∋ x : ( v 1 , · · · , v n ) �→ x ( v 1 , · · · , v n ) ; ⊲ End V (1) ∋ 1 = id V : v �→ v ; ⊲ ∀ x ∈ End V ( n ) , y ∈ End V ( m ) , 1 ≤ i ≤ n , x ◦ i y : ( v 1 , · · · , v n + m − 1 ) �→ x ( v 1 , · · · , v i − 1 , y ( v i , · · · , v i + m − 1 ) , v i + m , · · · , v m + n − 1 ) . ◮ How to construct operads? ⊲ Using presentations by generators and relations � X | R � . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 5 / 19

  2. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  3. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  4. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } · · · · · · · · · · · · z y z · · · y · · · x x C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  5. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } , 1 : the thread; · · · · · · · · · · · · z y z · · · y · · · x x C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  6. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } , 1 : the thread; · · · y · · · x ⊲ x ◦ i y : obtained by grafting the root of y on the i -th leaf of x . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  7. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } , 1 : the thread; · · · · · · · · · · · · z y z · · · y · · · x x ⊲ x ◦ i y : obtained by grafting the root of y on the i -th leaf of x . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  8. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } , 1 : the thread; · · · · · · · · · · · · z y z · · · y · · · x x ⊲ x ◦ i y : obtained by grafting the root of y on the i -th leaf of x . ◮ The compositions satisfy axioms: C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  9. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } , 1 : the thread; · · · · · · · · · · · · z y z · · · y · · · x x ⊲ x ◦ i y : obtained by grafting the root of y on the i -th leaf of x . ◮ The compositions satisfy axioms: ⊲ neutrality of 1 for each ◦ i : 1 ◦ 1 x = x = x ◦ i 1 ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  10. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } , 1 : the thread; · · · · · · · · · · · · z y z · · · y · · · x x ⊲ x ◦ i y : obtained by grafting the root of y on the i -th leaf of x . ◮ The compositions satisfy axioms: ⊲ neutrality of 1 for each ◦ i : 1 ◦ 1 x = x = x ◦ i 1 ; ⊲ associativity of sequential compositions : x ◦ i ( y ◦ j z ) = ( x ◦ i y ) ◦ i + j − 1 z ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  11. Motivations Free operads ◮ The free operad F ( X ) over a graded set X is constructed as follows: ⊲ x ∈ X ( n ) is represented by a labelled node with n leaves: · · · x ⊲ F ( X ) := { linear combinations of syntactic trees } , 1 : the thread; · · · · · · · · · · · · z y z · · · y · · · x x ⊲ x ◦ i y : obtained by grafting the root of y on the i -th leaf of x . ◮ The compositions satisfy axioms: ⊲ neutrality of 1 for each ◦ i : 1 ◦ 1 x = x = x ◦ i 1 ; ⊲ associativity of sequential compositions: x ◦ i ( y ◦ j z ) = ( x ◦ i y ) ◦ i + j − 1 z ; ⊲ commutativity of parallel compositions : ( x ◦ i y ) ◦ j + m − 1 z = ( x ◦ j z ) ◦ i y , where i < j and m is the arity of y . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 6 / 19

  12. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  13. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  14. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  15. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  16. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  17. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by ⊲ one 0-ary generator ( � the unit) • C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  18. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by ⊲ one 0-ary generator ( � the unit) and one binary generator ( � the multiplication); • C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  19. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by ⊲ one 0-ary generator ( � the unit) and one binary generator ( � the multiplication); • ⊲ the neutrality relations • •≡ ≡ C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  20. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by ⊲ one 0-ary generator ( � the unit) and one binary generator ( � the multiplication); • ⊲ the neutrality relations and the associativity relation; • •≡ ≡ ≡ C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  21. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by ⊲ one 0-ary generator ( � the unit) and one binary generator ( � the multiplication); • ⊲ the neutrality relations and the associativity relation; • •≡ ≡ ≡ ◮ 2 nd example: the differential associative operad is presented by C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  22. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by ⊲ one 0-ary generator ( � the unit) and one binary generator ( � the multiplication); • ⊲ the neutrality relations and the associativity relation; • •≡ ≡ ≡ ◮ 2 nd example: the differential associative operad is presented by ⊲ one 0-ary generator, one binary generator and one unary generator ( � the differential); d C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  23. Motivations Operadic ideals/congruences ◮ Given R ⊆ F ( X ), the operad presented by � X | R � is constructed as follows: ⊲ ≡ R : the operadic congruence generated by R , that is x ≡ R 0 for every x ∈ R ; ⊲ I ( R ) := { x ∈ F ( X ) | x ≡ R 0 } : the operadic ideal generated by R ; ⊲ O � X | R � := F ( X ) / I ( R ). ◮ 1 st example: the unital associative operad is presented by ⊲ one 0-ary generator ( � the unit) and one binary generator ( � the multiplication); • ⊲ the neutrality relations and the associativity relation; • •≡ ≡ ≡ ◮ 2 nd example: the differential associative operad is presented by ⊲ one 0-ary generator, one binary generator and one unary generator ( � the differential); d ⊲ the neutrality and associativity relations and the Leibniz’s identity; ≡ d + d d C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 7 / 19

  24. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  25. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  26. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  27. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  28. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  29. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  30. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  31. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  32. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  33. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  34. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance ⊲ some combinatorial consequences: right comb trees form a linear bases, the coefficients of the Hilbert series are equal to 1; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  35. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance ⊲ some combinatorial consequences: right comb trees form a linear bases, the coefficients of the Hilbert series are equal to 1; ⊲ a homological consequence: the nonunital associative operad is a Koszul operad. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  36. Motivations Gröbner bases for operads ◮ Gröbner bases for operads: ⊲ convergent (i.e. terminating and confluent) rewrite systems on F ( X ); ⊲ a confluence criterion: the Diamond’s Lemma [Dotsenko-Khoroshkin 2010]. ◮ Case of the untial associative operad: ⊲ a Gröbner basis is induced by the rewrite rules: • • ⊲ indeed, all critical pairs are confluent; for instance ⊲ some combinatorial consequences: right comb trees form a linear bases, the coefficients of the Hilbert series are equal to 1; ⊲ a homological consequence: the nonunital associative operad is a Koszul operad. ◮ Gröbner bases are computed by the Buchberger/Knuth-Bendix’s completion procedure. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 8 / 19

  37. Motivations Objectives of the talk ◮ We study magmatic quotients C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 9 / 19

  38. Motivations Objectives of the talk ◮ We study magmatic quotients; ⊲ the operad CAs (3) belongs to a set of operads CAs := � CAs ( γ ) | γ ≥ 1 � ; ⊲ CAs is included in the set of magmatic quotients. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 9 / 19

  39. Motivations Objectives of the talk ◮ We study magmatic quotients; ⊲ the operad CAs (3) belongs to a set of operads CAs := � CAs ( γ ) | γ ≥ 1 � ; ⊲ CAs is included in the set of magmatic quotients. ◮ We introduce a lattice structure on magmatic quotients: ⊲ we define this structure in terms of morphisms between magmatic quotients; ⊲ we present a Grassmann formula analog for this lattice. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 9 / 19

  40. Motivations Objectives of the talk ◮ We study magmatic quotients; ⊲ the operad CAs (3) belongs to a set of operads CAs := � CAs ( γ ) | γ ≥ 1 � ; ⊲ CAs is included in the set of magmatic quotients. ◮ We introduce a lattice structure on magmatic quotients: ⊲ we define this structure in terms of morphisms between magmatic quotients; ⊲ we present a Grassmann formula analog for this lattice. ◮ We study the induced poset on CAs : ⊲ we present new lattice operations on this poset; ⊲ we study the existence of finite Gröbner bases for CAs ( γ ) operads; ⊲ we deduce the complete expression of the Hilbert series of CAs (3) . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 9 / 19

  41. Lattice of magmatic quotients Plan II. Lattice of magmatic quotients C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 10 / 19

  42. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  43. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  44. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . ⊲ Alternatively: it is an operad over one binary generator [ ⋆ ] I . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  45. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . ⊲ Alternatively: it is an operad over one binary generator [ ⋆ ] I . ⊲ Q ( K Mag ) := { magmatic quotients } . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  46. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . ⊲ Alternatively: it is an operad over one binary generator [ ⋆ ] I . ⊲ Q ( K Mag ) := { magmatic quotients } . Lemma. Given O 1 = K Mag / I 1 and O 2 = K Mag I 2 , we have dim (Hom ( O 1 , O 2 )) ≤ 1. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  47. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . ⊲ Alternatively: it is an operad over one binary generator [ ⋆ ] I . ⊲ Q ( K Mag ) := { magmatic quotients } . Lemma. Given O 1 = K Mag / I 1 and O 2 = K Mag I 2 , we have dim (Hom ( O 1 , O 2 )) ≤ 1. Sketch of proof. Let ϕ ∈ Hom ( O 1 , O 2 ), ⊲ taking arities into account: ∃ λ ∈ K , s.t. ϕ ([ ⋆ ] I 1 ) = λ [ ⋆ ] I 2 ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  48. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . ⊲ Alternatively: it is an operad over one binary generator [ ⋆ ] I . ⊲ Q ( K Mag ) := { magmatic quotients } . Lemma. Given O 1 = K Mag / I 1 and O 2 = K Mag I 2 , we have dim (Hom ( O 1 , O 2 )) ≤ 1. Sketch of proof. Let ϕ ∈ Hom ( O 1 , O 2 ), ⊲ taking arities into account: ∃ λ ∈ K , s.t. ϕ ([ ⋆ ] I 1 ) = λ [ ⋆ ] I 2 ; ⊲ by the universal property of the quotient: if λ � = 0, then I 1 ⊆ I 2 ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  49. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . ⊲ Alternatively: it is an operad over one binary generator [ ⋆ ] I . ⊲ Q ( K Mag ) := { magmatic quotients } . Lemma. Given O 1 = K Mag / I 1 and O 2 = K Mag I 2 , we have dim (Hom ( O 1 , O 2 )) ≤ 1. Sketch of proof. Let ϕ ∈ Hom ( O 1 , O 2 ), ⊲ taking arities into account: ∃ λ ∈ K , s.t. ϕ ([ ⋆ ] I 1 ) = λ [ ⋆ ] I 2 ; ⊲ by the universal property of the quotient: if λ � = 0, then I 1 ⊆ I 2 ; ⊲ if I 1 ⊆ I 2 , then ∀ µ ∈ K \ { 0 } , ∃ ψ ∈ Hom ( O 1 , O 2 ) s.t. ψ ([ ⋆ ] I 1 ) = µ [ ⋆ ] I 2 . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  50. Lattice of magmatic quotients The category of magmatic quotients ◮ K : a fixed field s.t. char ( K ) � = 2. ◮ The magmatic operad K Mag is the free operad over one binary generator. ◮ A magmatic quotient is a quotient operad O = K Mag / I . ⊲ Alternatively: it is an operad over one binary generator [ ⋆ ] I . ⊲ Q ( K Mag ) := { magmatic quotients } . Lemma. Given O 1 = K Mag / I 1 and O 2 = K Mag I 2 , we have dim (Hom ( O 1 , O 2 )) ≤ 1. Sketch of proof. Let ϕ ∈ Hom ( O 1 , O 2 ), ⊲ taking arities into account: ∃ λ ∈ K , s.t. ϕ ([ ⋆ ] I 1 ) = λ [ ⋆ ] I 2 ; ⊲ by the universal property of the quotient: if λ � = 0, then I 1 ⊆ I 2 ; ⊲ if I 1 ⊆ I 2 , then ∀ µ ∈ K \ { 0 } , ∃ ψ ∈ Hom ( O 1 , O 2 ) s.t. ψ ([ ⋆ ] I 1 ) = µ [ ⋆ ] I 2 . Remark. A nonzero operad morphism between magmatic quotients is surjective. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 11 / 19

  51. Lattice of magmatic quotients Lattice structure of magmatic quotients ◮ Let O 1 = K Mag / I 1 and O 2 = K Mag / I 2 ; ⊲ we have dim (Hom ( O 1 , O 2 )) ≤ 1; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff I 1 ⊆ I 2 ; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff ∃ ϕ : O 1 → O 2 surjective. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 12 / 19

  52. Lattice of magmatic quotients Lattice structure of magmatic quotients ◮ Let O 1 = K Mag / I 1 and O 2 = K Mag / I 2 ; ⊲ we have dim (Hom ( O 1 , O 2 )) ≤ 1; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff I 1 ⊆ I 2 ; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff ∃ ϕ : O 1 → O 2 surjective. ◮ Let � i � Q ( K Mag ) × Q ( K Mag ) defined by ⊲ O 2 � i O 1 iff dim (Hom ( O 1 , O 2 )) = 1; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 12 / 19

  53. Lattice of magmatic quotients Lattice structure of magmatic quotients ◮ Let O 1 = K Mag / I 1 and O 2 = K Mag / I 2 ; ⊲ we have dim (Hom ( O 1 , O 2 )) ≤ 1; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff I 1 ⊆ I 2 ; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff ∃ ϕ : O 1 → O 2 surjective. ◮ Let � i � Q ( K Mag ) × Q ( K Mag ) defined by ⊲ O 2 � i O 1 iff dim (Hom ( O 1 , O 2 )) = 1; ◮ Let ∧ i , ∨ i : Q ( K Mag ) × Q ( K Mag ) → Q ( K Mag ) defined by ⊲ O 1 ∧ i O 2 = K Mag / I 1 + I 2 ; ⊲ O 1 ∨ i O 2 = K Mag / I 1 ∩ I 2 . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 12 / 19

  54. Lattice of magmatic quotients Lattice structure of magmatic quotients ◮ Let O 1 = K Mag / I 1 and O 2 = K Mag / I 2 ; ⊲ we have dim (Hom ( O 1 , O 2 )) ≤ 1; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff I 1 ⊆ I 2 ; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff ∃ ϕ : O 1 → O 2 surjective. ◮ Let � i � Q ( K Mag ) × Q ( K Mag ) defined by ⊲ O 2 � i O 1 iff dim (Hom ( O 1 , O 2 )) = 1; ◮ Let ∧ i , ∨ i : Q ( K Mag ) × Q ( K Mag ) → Q ( K Mag ) defined by ⊲ O 1 ∧ i O 2 = K Mag / I 1 + I 2 ; ⊲ O 1 ∨ i O 2 = K Mag / I 1 ∩ I 2 . Theorem [C.-Cordero-Giraudo, 2018]. Consider the notations introduced above . i. The tuple ( Q ( K Mag ) , � i , ∧ i , ∨ i ) is a lattice. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 12 / 19

  55. Lattice of magmatic quotients Lattice structure of magmatic quotients ◮ Let O 1 = K Mag / I 1 and O 2 = K Mag / I 2 ; ⊲ we have dim (Hom ( O 1 , O 2 )) ≤ 1; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff I 1 ⊆ I 2 ; ⊲ dim (Hom ( O 1 , O 2 )) = 1 iff ∃ ϕ : O 1 → O 2 surjective. ◮ Let � i � Q ( K Mag ) × Q ( K Mag ) defined by ⊲ O 2 � i O 1 iff dim (Hom ( O 1 , O 2 )) = 1; ◮ Let ∧ i , ∨ i : Q ( K Mag ) × Q ( K Mag ) → Q ( K Mag ) defined by ⊲ O 1 ∧ i O 2 = K Mag / I 1 + I 2 ; ⊲ O 1 ∨ i O 2 = K Mag / I 1 ∩ I 2 . Theorem [C.-Cordero-Giraudo, 2018]. Consider the notations introduced above . i. The tuple ( Q ( K Mag ) , � i , ∧ i , ∨ i ) is a lattice. ii. We have the following Grassmann formula analog: H O 1 ∨ i O 2 ( t ) + H O 1 ∧ i O 2 ( t ) = H O 1 ( t ) + H O 2 ( t ) . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 12 / 19

  56. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  57. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  58. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  59. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil ≡ I 2 Nil − C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  60. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil ≡ I 2 Nil − ⊲ so that I 2 Nil is generated by and C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  61. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil ≡ I 2 Nil − ⊲ so that I 2 Nil is generated by and C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  62. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil ≡ I 2 Nil − ⊲ so that I 2 Nil is generated by and C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  63. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil ≡ I 2 Nil − ⊲ so that I 2 Nil is generated by and ◮ Letting I K RC (3) := { x − y | x and y are trees of arity 4 } , we have K RC (3) = As ∨ i AAs ; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  64. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil ≡ I 2 Nil − ⊲ so that I 2 Nil is generated by and ◮ Letting I K RC (3) := { x − y | x and y are trees of arity 4 } , we have K RC (3) = As ∨ i AAs ; ⊲ one shows that I K RC (3) ⊆ I As ∩ I AAs , so that ∃ π : K RC (3) → As ∨ i AAs surjective; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  65. Lattice of magmatic quotients Example ◮ Let As := K Mag / I As and AAs := K Mag / I AAs , where I As and I AAs are generated by − + and ◮ Let 2 Nil := As ∧ i AAs , that is I 2 Nil = I As + I AAs ; ⊲ we have ≡ I 2 Nil ≡ I 2 Nil − ⊲ so that I 2 Nil is generated by and ◮ Letting I K RC (3) := { x − y | x and y are trees of arity 4 } , we have K RC (3) = As ∨ i AAs ; ⊲ one shows that I K RC (3) ⊆ I As ∩ I AAs , so that ∃ π : K RC (3) → As ∨ i AAs surjective; ⊲ using the Grassmann formula, one shows that π is an isomorphism. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 13 / 19

  66. Comb associative operads Plan III. Comb associative operads C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 14 / 19

  67. Comb associative operads Definition of CAs operads ◮ γ ≥ 1: a positive integer; ⊲ I CAs ( γ ) : the ideal generated by • • γ nodes γ nodes • − • • • C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 15 / 19

  68. Comb associative operads Definition of CAs operads ◮ γ ≥ 1: a positive integer; ⊲ I CAs ( γ ) : the ideal generated by • • γ nodes γ nodes • − • • • ⊲ CAs ( γ ) := Mag / I CAs ( γ ) is called the γ -comb associative operad . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 15 / 19

  69. Comb associative operads Definition of CAs operads ◮ γ ≥ 1: a positive integer; ⊲ I CAs ( γ ) : the ideal generated by • • γ nodes γ nodes • − • • • ⊲ CAs ( γ ) := Mag / I CAs ( γ ) is called the γ -comb associative operad. ◮ For instance, ⊲ CAs (1) = K Mag , CAs (2) = As C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 15 / 19

  70. Comb associative operads Definition of CAs operads ◮ γ ≥ 1: a positive integer; ⊲ I CAs ( γ ) : the ideal generated by • • γ nodes γ nodes • − • • • ⊲ CAs ( γ ) := Mag / I CAs ( γ ) is called the γ -comb associative operad. ◮ For instance, ⊲ CAs (1) = K Mag , CAs (2) = As , CAs (3) is submitted to the relations generated by − C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 15 / 19

  71. Comb associative operads Definition of CAs operads ◮ γ ≥ 1: a positive integer; ⊲ I CAs ( γ ) : the ideal generated by • • γ nodes γ nodes • − • • • ⊲ CAs ( γ ) := Mag / I CAs ( γ ) is called the γ -comb associative operad. ◮ For instance, ⊲ CAs (1) = K Mag , CAs (2) = As , CAs (3) is submitted to the relations generated by − ◮ Objective of the section: show that CAs ( γ ) | γ ≥ 1 � CAs := � admits a lattice structure. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 15 / 19

  72. Comb associative operads The lattice of CAs operads ◮ � d : the restriction of � i to CAs ; ⊲ CAs ( γ ) � d CAs ( β ) is equivalent to • • ≡ I CAs ( γ ) β nodes β nodes • • • • C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 16 / 19

  73. Comb associative operads The lattice of CAs operads ◮ � d : the restriction of � i to CAs ; ⊲ CAs ( γ ) � d CAs ( β ) is equivalent to • • γ nodes γ nodes • • • • ⊲ using an orientation of ≡ I CAs ( γ ) : C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 16 / 19

  74. Comb associative operads The lattice of CAs operads ◮ � d : the restriction of � i to CAs ; ⊲ CAs ( γ ) � d CAs ( β ) is equivalent to • • ≡ I CAs ( γ ) β nodes β nodes • • • • ⊲ using an orientation of ≡ I CAs ( γ ) : CAs ( γ ) � d CAs ( β ) iff γ | β (with α := α − 1). C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 16 / 19

  75. Comb associative operads The lattice of CAs operads ◮ � d : the restriction of � i to CAs ; ⊲ CAs ( γ ) � d CAs ( β ) is equivalent to • • ≡ I CAs ( γ ) β nodes β nodes • • • • ⊲ using an orientation of ≡ I CAs ( γ ) : CAs ( γ ) � d CAs ( β ) iff γ | β (with α := α − 1). ◮ Let ∧ d , ∨ d : CAs × CAs → CAs defined by � gcd � γ,β � +1 � ⊲ CAs ( γ ) ∧ d CAs ( β ) := CAs ; � lcm � γ,β � +1 � ⊲ CAs ( γ ) ∨ d CAs ( β ) := CAs . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 16 / 19

  76. Comb associative operads The lattice of CAs operads ◮ � d : the restriction of � i to CAs ; ⊲ CAs ( γ ) � d CAs ( β ) is equivalent to • • ≡ I CAs ( γ ) β nodes β nodes • • • • ⊲ using an orientation of ≡ I CAs ( γ ) : CAs ( γ ) � d CAs ( β ) iff γ | β (with α := α − 1). ◮ Let ∧ d , ∨ d : CAs × CAs → CAs defined by � gcd � γ,β � +1 � ⊲ CAs ( γ ) ∧ d CAs ( β ) := CAs ; � lcm � γ,β � +1 � ⊲ CAs ( γ ) ∨ d CAs ( β ) := CAs . Theorem [C.-Cordero-Giraudo, 2018]. The tuple ( CAs , � d , ∧ d , ∨ d ) is a lattice. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 16 / 19

  77. Comb associative operads The lattice of CAs operads ◮ � d : the restriction of � i to CAs ; ⊲ CAs ( γ ) � d CAs ( β ) is equivalent to • • ≡ I CAs ( γ ) β nodes β nodes • • • • ⊲ using an orientation of ≡ I CAs ( γ ) : CAs ( γ ) � d CAs ( β ) iff γ | β (with α := α − 1). ◮ Let ∧ d , ∨ d : CAs × CAs → CAs defined by � gcd � γ,β � +1 � ⊲ CAs ( γ ) ∧ d CAs ( β ) := CAs ; � lcm � γ,β � +1 � ⊲ CAs ( γ ) ∨ d CAs ( β ) := CAs . Theorem [C.-Cordero-Giraudo, 2018]. The tuple ( CAs , � d , ∧ d , ∨ d ) is a lattice. Remark. ( CAs , � d , ∧ d , ∨ d ) does not embed into ( Q ( K Mag ) , � i , ∧ i , ∨ i ) as a sublattice: �≡ I CAs (3) ∧ i CAs (4) C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 16 / 19

  78. Comb associative operads The lattice of CAs operads ◮ � d : the restriction of � i to CAs ; ⊲ CAs ( γ ) � d CAs ( β ) is equivalent to • • ≡ I CAs ( γ ) β nodes β nodes • • • • ⊲ using an orientation of ≡ I CAs ( γ ) : CAs ( γ ) � d CAs ( β ) iff γ | β (with α := α − 1). ◮ Let ∧ d , ∨ d : CAs × CAs → CAs defined by � gcd � γ,β � +1 � ⊲ CAs ( γ ) ∧ d CAs ( β ) := CAs ; � lcm � γ,β � +1 � ⊲ CAs ( γ ) ∨ d CAs ( β ) := CAs . Theorem [C.-Cordero-Giraudo, 2018]. The tuple ( CAs , � d , ∧ d , ∨ d ) is a lattice. Remark. ( CAs , � d , ∧ d , ∨ d ) does not embed into ( Q ( K Mag ) , � i , ∧ i , ∨ i ) as a sublattice: ≡ I CAs (3) ∧ d CAs (4) since CAs (3) ∧ d CAs (4) = CAs (gcd(2 , 3)+1) = CAs (2) = As . C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 16 / 19

  79. Comb associative operads Completion of CAs operads ◮ The orientation of ≡ I CAs ( γ ) is not confluent: C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 17 / 19

  80. Comb associative operads Completion of CAs operads ◮ The orientation of ≡ I CAs ( γ ) is not confluent: ◮ Buchberger/Knuth-Bendix’s completion procedure applied to CAs (3) provides: ⊲ new rewrite rules for arities 5 , · · · , 8; C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 17 / 19

  81. Comb associative operads Completion of CAs operads ◮ The orientation of ≡ I CAs ( γ ) is not confluent: ◮ Buchberger/Knuth-Bendix’s completion procedure applied to CAs (3) provides: ⊲ new rewrite rules for arities 5 , · · · , 8; ⊲ no new rewrite rule for arities 9 , · · · , 14! C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 17 / 19

  82. Comb associative operads Completion of CAs operads ◮ The orientation of ≡ I CAs ( γ ) is not confluent: ◮ Buchberger/Knuth-Bendix’s completion procedure applied to CAs (3) provides: ⊲ new rewrite rules for arities 5 , · · · , 8; ⊲ no new rewrite rule for arities 9 , · · · , 14! Theorem [C.-Cordero-Giraudo, 2018]. The operad CAs (3) is presented by a finite Gröbner basis. C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 17 / 19

  83. Comb associative operads Completion of CAs operads ◮ The orientation of ≡ I CAs ( γ ) is not confluent: ◮ Buchberger/Knuth-Bendix’s completion procedure applied to CAs (3) provides: ⊲ new rewrite rules for arities 5 , · · · , 8; ⊲ no new rewrite rule for arities 9 , · · · , 14! Theorem [C.-Cordero-Giraudo, 2018]. The operad CAs (3) is presented by a finite Gröbner basis. Moreover, we have α n t n + � � ( n + 3) t n , H CAs (3) = n ≤ 10 n ≥ 11 where, value of n 2 3 4 5 6 7 8 9 10 value of α n 1 2 4 8 14 20 19 16 14 C.Chenavier, C.Cordero, S.Giraudo Quotients of the magmatic operad December 20, 2018 17 / 19

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