dualizable algebras with parallelogram terms
play

Dualizable Algebras with Parallelogram Terms gnes Szendrei CU - PowerPoint PPT Presentation

Feb 01, 2024 •306 likes •1.47k views

Dualizable Algebras with Parallelogram Terms gnes Szendrei CU Boulder/U Szeged Joint work with Keith Kearnes AAA88 Warsaw, Poland, June 1922, 2014 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 1 / 19 Motivating

  1. The Dualizing Structure If A = ( A ; R ) dualizes A = ( A ; F ) , then R has to contain ‘enough’ compatible relations of A : For any B ∈ SP ( A ) and continuous map f : B ∂ → A , f preserves each ρ ∈ R ⇔ f ∈ Hom ( B ∂ , A ) A dualizes A ⇒ f is an evaluation map ⇒ f preserves every compatible relation of A . In particular, for B := F ( k ) = F V ( A ) ( k ) ( k ≥ 1), F ( k ) ∂ = Hom ( F ( k ) , A ) ∼ = A k , A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 4 / 19

  2. The Dualizing Structure If A = ( A ; R ) dualizes A = ( A ; F ) , then R has to contain ‘enough’ compatible relations of A : For any B ∈ SP ( A ) and continuous map f : B ∂ → A , f preserves each ρ ∈ R ⇔ f ∈ Hom ( B ∂ , A ) A dualizes A ⇒ f is an evaluation map ⇒ f preserves every compatible relation of A . In particular, for B := F ( k ) = F V ( A ) ( k ) ( k ≥ 1), F ( k ) ∂ = Hom ( F ( k ) , A ) ∼ = A k , Hom ( F ( k ) ∂ , A ) ∼ = Hom ( A k , A ) = { k -ary ops preserving each ρ ∈ R } ; A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 4 / 19

  3. The Dualizing Structure If A = ( A ; R ) dualizes A = ( A ; F ) , then R has to contain ‘enough’ compatible relations of A : For any B ∈ SP ( A ) and continuous map f : B ∂ → A , f preserves each ρ ∈ R ⇔ f ∈ Hom ( B ∂ , A ) A dualizes A ⇒ f is an evaluation map ⇒ f preserves every compatible relation of A . In particular, for B := F ( k ) = F V ( A ) ( k ) ( k ≥ 1), F ( k ) ∂ = Hom ( F ( k ) , A ) ∼ = A k , Hom ( F ( k ) ∂ , A ) ∼ = Hom ( A k , A ) = { k -ary ops preserving each ρ ∈ R } ; for every f : A k → A , f preserves each ρ ∈ R ⇒ f preserves every compatible relation of A . A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 4 / 19

  4. The Dualizing Structure If A = ( A ; R ) dualizes A = ( A ; F ) , then R has to contain ‘enough’ compatible relations of A : For any B ∈ SP ( A ) and continuous map f : B ∂ → A , f preserves each ρ ∈ R ⇔ f ∈ Hom ( B ∂ , A ) A dualizes A ⇒ f is an evaluation map ⇒ f preserves every compatible relation of A . In particular, for B := F ( k ) = F V ( A ) ( k ) ( k ≥ 1), F ( k ) ∂ = Hom ( F ( k ) , A ) ∼ = A k , Hom ( F ( k ) ∂ , A ) ∼ = Hom ( A k , A ) = { k -ary ops preserving each ρ ∈ R } ; for every f : A k → A , f preserves each ρ ∈ R ⇒ f preserves every compatible relation of A . Corollary. A is dualizable ⇔ A is dualized by A = ( A ; { all compatible relations of A } ) . A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 4 / 19

  5. Two Galois Connections A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 5 / 19

  6. Two Galois Connections Let A be a finite algebra, let R be the set of all (finitary) relations on A . R A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 5 / 19

  7. Two Galois Connections Let A be a finite algebra, let R be the set of all (finitary) relations on A . F = { f : B ∂ cont → A | B ∈ SP ( A ) } ✲ R ✛ A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 5 / 19

  8. Two Galois Connections Let A be a finite algebra, let R be the set of all (finitary) relations on A . F = { f : B ∂ cont → A | B ∈ SP ( A ) } ✲ F 0 = { f : F ( k ) ∂ R ✛ → A | k = 0 , 1 , 2 , . . . } � �� � ∼ A k A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 5 / 19

  9. Two Galois Connections Let A be a finite algebra, let R be the set of all (finitary) relations on A . F = { f : B ∂ cont → A | B ∈ SP ( A ) } ✲ F 0 = { f : F ( k ) ∂ R ✛ → A | k = 0 , 1 , 2 , . . . } � �� � ∼ A k The compatibility of a function with a relation determines a Galois connection between R and F 0 , and between R and F . A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 5 / 19

  10. Two Galois Connections Let A be a finite algebra, let R be the set of all (finitary) relations on A . F = { f : B ∂ cont → A | B ∈ SP ( A ) } ✲ F 0 = { f : F ( k ) ∂ R ✛ → A | k = 0 , 1 , 2 , . . . } � �� � ∼ A k The compatibility of a function with a relation determines a Galois connection between R and F 0 , and between R and F . Definition. For R ⊆ R and γ ∈ R , R | = c γ if every f ∈ F 0 preserving each ρ ∈ R also preserves γ, R | = d γ if every f ∈ F preserving each ρ ∈ R also preserves γ. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 5 / 19

  11. Clone Entailment ( | = c ) vs. Duality entailment ( | = d ) Definition. For R ⊆ R and γ ∈ R , R | = c γ if every f ∈ F 0 preserving each ρ ∈ R also preserves γ, R | = d γ if every f ∈ F preserving each ρ ∈ R also preserves γ. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 6 / 19

  12. Clone Entailment ( | = c ) vs. Duality entailment ( | = d ) Definition. For R ⊆ R and γ ∈ R , R | = c γ if every f ∈ F 0 preserving each ρ ∈ R also preserves γ, R | = d γ if every f ∈ F preserving each ρ ∈ R also preserves γ. R | = d γ ⇒ R | = c γ . ( A ; R ) dualizes A ⇒ R | = d every compatible relation of A . A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 6 / 19

  13. Clone Entailment ( | = c ) vs. Duality entailment ( | = d ) Definition. For R ⊆ R and γ ∈ R , R | = c γ if every f ∈ F 0 preserving each ρ ∈ R also preserves γ, R | = d γ if every f ∈ F preserving each ρ ∈ R also preserves γ. R | = d γ ⇒ R | = c γ . ( A ; R ) dualizes A ⇒ R | = d every compatible relation of A . The difference between | = c and | = d is identified by: A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 6 / 19

  14. Clone Entailment ( | = c ) vs. Duality entailment ( | = d ) Definition. For R ⊆ R and γ ∈ R , R | = c γ if every f ∈ F 0 preserving each ρ ∈ R also preserves γ, R | = d γ if every f ∈ F preserving each ρ ∈ R also preserves γ. R | = d γ ⇒ R | = c γ . ( A ; R ) dualizes A ⇒ R | = d every compatible relation of A . The difference between | = c and | = d is identified by: Theorem. [BKKR] R | = c γ ⇔ γ is constructible from R using = , permutation of coordinates, product, intersection and projection onto a subset of coordinates. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 6 / 19

  15. Clone Entailment ( | = c ) vs. Duality entailment ( | = d ) Definition. For R ⊆ R and γ ∈ R , R | = c γ if every f ∈ F 0 preserving each ρ ∈ R also preserves γ, R | = d γ if every f ∈ F preserving each ρ ∈ R also preserves γ. R | = d γ ⇒ R | = c γ . ( A ; R ) dualizes A ⇒ R | = d every compatible relation of A . The difference between | = c and | = d is identified by: Theorem. [BKKR] R | = c γ ⇔ γ is constructible from R using = , permutation of coordinates, product, intersection and projection onto a subset of coordinates. Theorem. [Z, DHP] R | = d γ ⇔ γ is constructible from R using = , permutation of coordinates, product, intersection and bijective projection onto a subset of coordinates. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 6 / 19

  16. Finitely Related Algebras A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 7 / 19

  17. Finitely Related Algebras Theorem. [Willard, Zádori] Assume that R is a finite set of compatible relations of A . If R | = d every compatible relation of A , then A is dualizable. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 7 / 19

  18. Finitely Related Algebras Theorem. [Willard, Zádori] Assume that R is a finite set of compatible relations of A . If R | = d every compatible relation of A , then A is dualizable. Definition. Call A finitely related if there is a finite set R of compatible relations of A such that R | = c every compatible relation of A . A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 7 / 19

  19. Finitely Related Algebras Theorem. [Willard, Zádori] Assume that R is a finite set of compatible relations of A . If R | = d every compatible relation of A , then A is dualizable. Definition. Call A finitely related if there is a finite set R of compatible relations of A such that R | = c every compatible relation of A . Theorem above concerns finitely related algebras only. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 7 / 19

  20. Finitely Related Algebras Theorem. [Willard, Zádori] Assume that R is a finite set of compatible relations of A . If R | = d every compatible relation of A , then A is dualizable. Definition. Call A finitely related if there is a finite set R of compatible relations of A such that R | = c every compatible relation of A . Theorem above concerns finitely related algebras only. Most algebras that are known to be dualizable are finitely related. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 7 / 19

  21. Dualizable vs. Finitely Related Algebras A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 8 / 19

  22. Dualizable vs. Finitely Related Algebras In general, ‘dualizable’ and ‘finitely related’ are independent properties. A dualizable A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 8 / 19

  23. Dualizable vs. Finitely Related Algebras In general, ‘dualizable’ and ‘finitely related’ are independent properties. A dualizable There exist dualizable algebras that ✛ are not finitely related. [Pitkethly] A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 8 / 19

  24. Dualizable vs. Finitely Related Algebras In general, ‘dualizable’ and ‘finitely related’ are independent properties. A dualizable There exist dualizable algebras that ✛ are not finitely related. [Pitkethly] implication alg ❄ ✲ A 2-element algebra is dualizable iff BA it is finitely related. [Clark–Davey] ✻ lattice A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 8 / 19

  25. Dualizable vs. Finitely Related Algebras In general, ‘dualizable’ and ‘finitely related’ are independent properties. A dualizable There exist dualizable algebras that ✛ are not finitely related. [Pitkethly] implication alg ❄ ✲ A 2-element algebra is dualizable iff BA it is finitely related. [Clark–Davey] ✻ lattice Nonabelian nilpotent groups are ✛ finitely related, but not dualizable. [Quackenbush–Szabó] A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 8 / 19

  26. Parallelogram Terms A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 9 / 19

  27. Parallelogram Terms ✁ ✁ ✁ -terms generalize Maltsev terms and near unanimity (NU) terms. ✁ A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 9 / 19

  28. Parallelogram Terms ✁ ✁ ✁ -terms generalize Maltsev terms and near unanimity (NU) terms. ✁ Definition. A k - ✁ ✁ ✁ ✁ -term for an algebra A is a ( k + 3 ) -ary term t ( k ≥ 2) s.t.     x x y · · · · · · z y y y y y y . . . . ...  .    . . . . . . .         x x y y y z y y y y     A | = t = .     y x x y y y z y y y         . . . . ...     . . . . . . . .     y y · · · y y · · · y z y y x x A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 9 / 19

  29. Parallelogram Terms ✁ ✁ ✁ -terms generalize Maltsev terms and near unanimity (NU) terms. ✁ Definition. A k - ✁ ✁ ✁ ✁ -term for an algebra A is a ( k + 3 ) -ary term t ( k ≥ 2) s.t.     x x y · · · · · · z y y y y y y . . . . ...  .    . . . . . . .         x x y y y z y y y y     A | = t = .     y x x y y y z y y y         . . . . ...     . . . . . . . .     y y · · · y y · · · y z y y x x A Maltsev term for A is a k - ✁ ✁ ✁ ✁ -term independent of its last k variables. Example: xy − 1 z for any group. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 9 / 19

  30. Parallelogram Terms ✁ ✁ ✁ -terms generalize Maltsev terms and near unanimity (NU) terms. ✁ Definition. A k - ✁ ✁ ✁ ✁ -term for an algebra A is a ( k + 3 ) -ary term t ( k ≥ 2) s.t.     x x y · · · · · · z y y y y y y . . . . ...  .    . . . . . . .         x x y y y z y y y y     A | = t = .     y x x y y y z y y y         . . . . ...     . . . . . . . .     y y · · · y y · · · y z y y x x A Maltsev term for A is a k - ✁ ✁ ✁ ✁ -term independent of its last k variables. Example: xy − 1 z for any group. A k -NU term for A is a k - ✁ ✁ ✁ ✁ -term independent of its first 3 variables. Example: ( x ∧ y ) ∨ ( y ∧ z ) ∨ ( z ∧ x ) for any lattice. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 9 / 19

  31. Dualizable vs. Finitely Related Algebras in CD Varieties A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 10 / 19

  32. Dualizable vs. Finitely Related Algebras in CD Varieties V ( A ) CD A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 10 / 19

  33. Dualizable vs. Finitely Related Algebras in CD Varieties Theorem. (1) ⇔ (2) for any finite algebra A . (1) (a) A is finitely related & (b) V ( A ) is CD. A has (2) A has an NU term. NU term V ( A ) CD [(2) ⇒ (b): Mitschke; (2) ⇒ (1)(a): Baker–Pixley; (1) ⇒ (2): Barto; A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 10 / 19

  34. Dualizable vs. Finitely Related Algebras in CD Varieties Theorem. (1) ⇔ (2) ⇔ (3) for any A dualizable finite algebra A . (1) (a) A is finitely related & (b) V ( A ) is CD. A has (2) A has an NU term. NU term (3) (a) A is dualizable & V ( A ) CD (b) V ( A ) is CD. [(2) ⇒ (b): Mitschke; (2) ⇒ (1)(a): Baker–Pixley; (1) ⇒ (2): Barto; (2) ⇒ (3)(a): Davey–Werner; A finitely related (3) ⇒ (2): Davey–Heindorf–McKenzie. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 10 / 19

  35. Dualizable vs. Finitely Related Algebras in CM Varieties A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 11 / 19

  36. Dualizable vs. Finitely Related Algebras in CM Varieties V ( A ) CD V ( A ) CM A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 11 / 19

  37. Dualizable vs. Finitely Related Algebras in CM Varieties Theorem. [1] ⇔ [2] for any finite algebra A . [1] (a) A is finitely related & (b) V ( A ) is CM. [2] A has a ✁ ✁ ✁ ✁ -term. A has NU term A has V ( A ) CD ✁ ✁ ✁ ✁ -term [[2] ⇒ (b): Kearnes–Sz & BIMMVW; [1] ⇒ [2]: Barto; [2] ⇒ [1](a): Aichinger–Mayr–McK; V ( A ) CM A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 11 / 19

  38. Dualizable vs. Finitely Related Algebras in CM Varieties Theorem. [1] ⇔ [2] ⇐ [3] for any A dualizable finite algebra A . [1] (a) A is finitely related & (b) V ( A ) is CM. [2] A has a ✁ ✁ ✁ ✁ -term. A has NU term [3] (a) A is dualizable & (b) V ( A ) is CM. A has V ( A ) CD ✁ ✁ ✁ ✁ -term [[2] ⇒ (b): Kearnes–Sz & BIMMVW; [1] ⇒ [2]: Barto; [2] ⇒ [1](a): Aichinger–Mayr–McK; [3] ⇒ [2]: Moore.] V ( A ) CM A finitely related A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 11 / 19

  39. Dualizable vs. Finitely Related Algebras in CM Varieties Theorem. [1] ⇔ [2] ⇐ [3] for any A dualizable finite algebra A . [1] (a) A is finitely related & (b) V ( A ) is CM. [2] A has a ✁ ✁ ✁ ✁ -term. A has NU term [3] (a) A is dualizable & (b) V ( A ) is CM. A has V ( A ) CD ✁ ✁ ✁ ✁ -term [[2] ⇒ (b): Kearnes–Sz & BIMMVW; [1] ⇒ [2]: Barto; [2] ⇒ [1](a): Aichinger–Mayr–McK; [3] ⇒ [2]: Moore.] V ( A ) CM A finitely related [2] �⇒ [3](a) (e.g., nonabelian nilpotent groups) A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 11 / 19

  40. Dualizability vs. Residual Smallness A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  41. Dualizability vs. Residual Smallness Definition. A variety V is residually small (RS) if there is a cardinal κ such that every B ∈ V embeds in a product of algebras of size < κ . (I.e., every SI in V has size < κ ). A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  42. Dualizability vs. Residual Smallness Definition. A variety V is residually small (RS) if there is a cardinal κ such that every B ∈ V embeds in a product of algebras of size < κ . (I.e., every SI in V has size < κ ). A dualizable ??? ⇒ V ( A ) RS A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  43. Dualizability vs. Residual Smallness Definition. A variety V is residually small (RS) if there is a cardinal κ such that every B ∈ V embeds in a product of algebras of size < κ . (I.e., every SI in V has size < κ ). A dualizable A dualizable ??? ⇒ V ( A ) RS V ( A ) RS These properties are independent, A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  44. Dualizability vs. Residual Smallness Definition. A variety V is residually small (RS) if there is a cardinal κ such that every B ∈ V embeds in a product of algebras of size < κ . (I.e., every SI in V has size < κ ). A dualizable A dualizable ??? ⇒ V ( A ) RS V ( A ) RS These properties are independent, even for expanded groups. A has ✁ ✁ ✁ ✁ -term A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  45. Dualizability vs. Residual Smallness Definition. A variety V is residually small (RS) if there is a cardinal κ such that every B ∈ V embeds in a product of algebras of size < κ . (I.e., every SI in V has size < κ ). A dualizable A dualizable ??? ⇒ V ( A ) RS V ( A ) RS These properties are independent, even for expanded groups. ✛ � � Z 4 ; + , ( xy ) 2 , 0 , 1 [Davey–Pitkethly–Willard] A has ✁ ✁ ✁ ✁ -term A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  46. Dualizability vs. Residual Smallness Definition. A variety V is residually small (RS) if there is a cardinal κ such that every B ∈ V embeds in a product of algebras of size < κ . (I.e., every SI in V has size < κ ). A dualizable A dualizable ??? ⇒ V ( A ) RS V ( A ) RS These properties are independent, even for expanded groups. ✛ � � Z 4 ; + , ( xy ) 2 , 0 , 1 [Davey–Pitkethly–Willard] For a group A , ✛ A dualizable ⇔ V ( A ) RS (discussed later) A has ✁ ✁ ✁ ✁ -term A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  47. Dualizability vs. Residual Smallness Definition. A variety V is residually small (RS) if there is a cardinal κ such that every B ∈ V embeds in a product of algebras of size < κ . (I.e., every SI in V has size < κ ). A dualizable A dualizable ??? ⇒ V ( A ) RS V ( A ) RS These properties are independent, even for expanded groups. ✛ � � Z 4 ; + , ( xy ) 2 , 0 , 1 [Davey–Pitkethly–Willard] For a group A , ✛ A dualizable ⇔ V ( A ) RS (discussed later) ✛ S 3 expanded by all constants [Idziak] A has ✁ ✁ ✁ ✁ -term A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 12 / 19

  48. Main Theorem A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 13 / 19

  49. Main Theorem Theorem. Let A be a finite algebra such that � � (i) A has a ✁ ✁ ✁ ✁ -term ⇔ A fin rel & V ( A ) CM , and � � ⇔ Con ( S ( A )) | (ii) V ( A ) is RS = x ∧ [ y , y ] ≈ [ x ∧ y , y ] . A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 13 / 19

  50. Main Theorem Theorem. Let A be a finite algebra such that � � (i) A has a ✁ ✁ ✁ ✁ -term ⇔ A fin rel & V ( A ) CM , and � � ⇔ Con ( S ( A )) | (ii) V ( A ) is RS = x ∧ [ y , y ] ≈ [ x ∧ y , y ] . A is dualizable if the following split centralizer condition holds in every subalgebra S of A : A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 13 / 19

  51. Main Theorem Theorem. Let A be a finite algebra such that � � (i) A has a ✁ ✁ ✁ ✁ -term ⇔ A fin rel & V ( A ) CM , and � � ⇔ Con ( S ( A )) | (ii) V ( A ) is RS = x ∧ [ y , y ] ≈ [ x ∧ y , y ] . A is dualizable if the following split centralizer condition holds in every subalgebra S of A : 1 ∀ δ ≺ θ s.t. s Con ( S ) δ is ∧ -irred. and [ θ, θ ] ≤ δ θ s δ s s 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 13 / 19

  52. Main Theorem Theorem. Let A be a finite algebra such that � � (i) A has a ✁ ✁ ✁ ✁ -term ⇔ A fin rel & V ( A ) CM , and � � ⇔ Con ( S ( A )) | (ii) V ( A ) is RS = x ∧ [ y , y ] ≈ [ x ∧ y , y ] . A is dualizable if the following split centralizer condition holds in every subalgebra S of A : 1 ∀ δ ≺ θ s.t. s Con ( S ) ν δ is ∧ -irred. and [ θ, θ ] ≤ δ s for ν = ( δ : θ ) θ s δ s s 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 13 / 19

  53. Main Theorem Theorem. Let A be a finite algebra such that � � (i) A has a ✁ ✁ ✁ ✁ -term ⇔ A fin rel & V ( A ) CM , and � � ⇔ Con ( S ( A )) | (ii) V ( A ) is RS = x ∧ [ y , y ] ≈ [ x ∧ y , y ] . A is dualizable if the following split centralizer condition holds in every subalgebra S of A : 1 ∀ δ ≺ θ s.t. s Con ( S ) ν δ is ∧ -irred. and [ θ, θ ] ≤ δ s for ν = ( δ : θ ) θ � RS,CM � ⇒ [ ν, ν ] ≤ δ ] s δ s s 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 13 / 19

  54. Main Theorem Theorem. Let A be a finite algebra such that � � (i) A has a ✁ ✁ ✁ ✁ -term ⇔ A fin rel & V ( A ) CM , and � � ⇔ Con ( S ( A )) | (ii) V ( A ) is RS = x ∧ [ y , y ] ≈ [ x ∧ y , y ] . A is dualizable if the following split centralizer condition holds in every subalgebra S of A : 1 ∀ δ ≺ θ s.t. s Con ( S ) ν δ is ∧ -irred. and [ θ, θ ] ≤ δ s ❅ for ν = ( δ : θ ) ❅ θ � RS,CM � ❅ ⇒ [ ν, ν ] ≤ δ ] s δ ❅ s ❅ ∃ κ ∈ Q - Con ( S ) , Q = SP ( A ) β s α s ❅ � � ∃ β ≤ δ ❅ � ❅ � ∃ α with [ α, α ] ≤ κ s.t. ❅ ❅ � α ∨ β = ν κ ❝ α ∧ β = κ s 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 13 / 19

  55. Idziak’s Example A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  56. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  57. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  58. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  59. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. The split centralizer condition fails for A : A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  60. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. The split centralizer condition fails for A : S = A A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  61. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. The split centralizer condition fails for A : S = A Con ( A ) r r r A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  62. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. The split centralizer condition fails for A : S = A θ = ν Con ( A ) r δ r r A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  63. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. The split centralizer condition fails for A : S = A θ = ν Con ( A ) r δ r r κ = 0; A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  64. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. The split centralizer condition fails for A : S = A θ = ν Con ( A ) r δ r r κ = 0; α, β ≤ δ , A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  65. Idziak’s Example Example (Idziak) : For A = ( S 3 ; all constants ) , A has a Maltsev term, V ( A ) is RS, but A is not dualizable. The split centralizer condition fails for A : S = A θ = ν Con ( A ) r δ r r κ = 0; α, β ≤ δ , α ∨ β � = ν A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 14 / 19

  66. Applications: 1. Algebras with NU Terms A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 15 / 19

  67. Applications: 1. Algebras with NU Terms 1. [Davey–Werner] A has an NU term ⇒ A is dualizable. Proof: 1 s Con ( S ) θ s δ s r 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 15 / 19

  68. Applications: 1. Algebras with NU Terms 1. [Davey–Werner] A has an NU term ⇒ A is dualizable. Proof: 1 s Con ( S ) No δ ≺ θ to check. θ s δ s r 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 15 / 19

  69. Applications: 2. Modules A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 16 / 19

  70. Applications: 2. Modules 2. [NEW] A is a module ⇒ A is dualizable. Proof: 1 = ν = α s Con ( S ) θ s δ s r ❞ 0 = κ = β A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 16 / 19

  71. Applications: 3. Groups A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 17 / 19

  72. Applications: 3. Groups 3. [Nickodemus] A is a group whose Sylow subgroups of A are abelian ( ⇔ V ( A ) is RS) ⇒ A is dualizable. A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 17 / 19

  73. Applications: 3. Groups 3. [Nickodemus] A is a group whose Sylow subgroups of A are abelian ( ⇔ V ( A ) is RS) ⇒ A is dualizable. ( ⇐ by [Quackenbush–Szabó]) A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 17 / 19

  74. Applications: 3. Groups 3. [Nickodemus] A is a group whose Sylow subgroups of A are abelian ( ⇔ V ( A ) is RS) ⇒ A is dualizable. ( ⇐ by [Quackenbush–Szabó]) Proof ( ⇒ ): 1 s ν Con ( S ) s θ s δ s s 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 17 / 19

  75. Applications: 3. Groups 3. [Nickodemus] A is a group whose Sylow subgroups of A are abelian ( ⇔ V ( A ) is RS) ⇒ A is dualizable. ( ⇐ by [Quackenbush–Szabó]) Proof ( ⇒ ): 1 s ν Con ( S ) s θ s δ s s [ ν, ν ] s 0 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 17 / 19

Recommend

Parallelogram This is a four sided figure where opposite sides are equal and parallel. The

Parallelogram This is a four sided figure where opposite sides are equal and parallel. The

D AY 113 D IAGONALS OF A PARALLELOGRAM I NTRODUCTION We are not yet done with all the properties of a parallelogram. In the previous lessons, our focus was first on the boundaries of a parallelogram. We then came to the inside, to discuss

655 views • 12 slides
Tense MV-algebras and related functions Jan Paseka Michal Botur Department of Mathematics and

Tense MV-algebras and related functions Jan Paseka Michal Botur Department of Mathematics and

Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Tense MV-algebras

1.59k views • 121 slides
The perimeter and area of both a rhombus and a parallelogram can be found by applying the

The perimeter and area of both a rhombus and a parallelogram can be found by applying the

D AY 118 P ERIMETER AND AREA OF A RHOMBUS AND A PARALLELOGRAM ON XY - PLANE I NTRODUCTION The perimeter and area of rhombi and parallelograms have been dealt with in earlier lessons. Both the area and perimeter of these quadrilaterals can

437 views • 15 slides
Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March 2018 1 We consider finiteness conditions and ques- tions of growth of noncommutative algebras, known as A con s. They appear in M.Wemyss work on

214 views • 17 slides
Varieties of positive interior algebras: operation . The two presentations produce

Varieties of positive interior algebras: operation . The two presentations produce

Modal algebras form the algebraic semantics of normal modal logics. They are Boolean algebras with a unary operation s.t. 1 1 and ( x y ) x y . Modal algebras can be presented also as BAs with a unary

186 views • 5 slides
Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras Tanusree

Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras Tanusree

Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras Tanusree Khandai Indian Institute of Science Education and Research, Mohali Interactions of quantum affine algebras with cluster algebras, current algebras

327 views • 28 slides
(Pre-)Algebras for Linguistics 2. Introducing Preordered Algebras Carl Pollard Linguistics 680:

(Pre-)Algebras for Linguistics 2. Introducing Preordered Algebras Carl Pollard Linguistics 680:

(Pre-)Algebras for Linguistics 2. Introducing Preordered Algebras Carl Pollard Linguistics 680: Formal Foundations Autumn 2010 Carl Pollard (Pre-)Algebras for Linguistics Algebras A (one-sorted) algebra is a set with one or more operations

497 views • 38 slides
Spirals Zarathustra Brady Clone-minimal algebras A reduct of A is an algebra with the same

Spirals Zarathustra Brady Clone-minimal algebras A reduct of A is an algebra with the same

Spirals Zarathustra Brady Clone-minimal algebras A reduct of A is an algebra with the same underlying set as A and basic operations a subset of the terms of A . A reduct of A is proper if it is not term equivalent to A , and nontrivial if at

1.93k views • 64 slides
The Calabi-Yau property of Hopf algebras and braided Hopf algebras Xiaolan YU joint work with

The Calabi-Yau property of Hopf algebras and braided Hopf algebras Xiaolan YU joint work with

The Calabi-Yau property of Hopf algebras and braided Hopf algebras Xiaolan YU joint work with Yinhuo Zhang Hangzhou Normal University September 16th, 2011 1 / 35 Outline Motivation Braided Hopf algebras Calabi-Yau algebras The Calabi-Yau

639 views • 63 slides
Lattices from Octonion Algebras Octonion Algebras Lattices via Octonion Algebras Nelson G.

Lattices from Octonion Algebras Octonion Algebras Lattices via Octonion Algebras Nelson G.

Nelson, Cintya, Sueli Lattices Lattices from Octonion Algebras Octonion Algebras Lattices via Octonion Algebras Nelson G. Brasil Junior 1 Cintya W. O. Benedito 2 Examples Sueli I. R. Costa 1 1 Institute of Mathematics, Statistics and

650 views • 6 slides
Spinning Parallelogram Operator (SPO) for Light Field Depth Estimation 4D Light Field Benchmark

Spinning Parallelogram Operator (SPO) for Light Field Depth Estimation 4D Light Field Benchmark

Spinning Parallelogram Operator (SPO) for Light Field Depth Estimation 4D Light Field Benchmark Challenge at 2 nd Workshop on LF4CV, CVPR 2017 Shuo Zhang 1 , Hao Sheng 1 , Chao Li 1 , Jun Zhang 2 , Zhang Xiong 1 1 Beihang University 2

419 views • 12 slides
Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations Riccardo Biagioli

Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations Riccardo Biagioli

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations Riccardo Biagioli (Universit e Lyon 1) joint work with M. Bousquet-M

809 views • 59 slides
Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh*

Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh*

Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh* Noncommutative and non-associative structures, braces and applications Malta 2018 Throughout, k is an algebraically closed field. Definition Let a , b , c

574 views • 20 slides
On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar Venjakob Mathematisches

On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar Venjakob Mathematisches

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras and SK 1 Application On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar Venjakob Mathematisches Institut

1.07k views • 16 slides
III. Staying in a Hotel - Terms of Accommodation Contract Headings: Terms Standard

III. Staying in a Hotel - Terms of Accommodation Contract Headings: Terms Standard

III. Staying in a Hotel - Terms of Accommodation Contract Headings: Terms Standard terms Implied terms Noise Privacy and data protection Terms and Conditions of Use These slides are intended for use as a teaching

670 views • 46 slides
Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Notre Dame 2015 Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c University of Zagreb, Croatia Supported by CSF, grant. no. 2634 Conference on Lie algebras, vertex operator algebras and related

586 views • 39 slides
Relation algebras Robin Hirsch and Ian Hodkinson Thanks to the organisers for inviting us! And

Relation algebras Robin Hirsch and Ian Hodkinson Thanks to the organisers for inviting us! And

Relation algebras Robin Hirsch and Ian Hodkinson Thanks to the organisers for inviting us! And Happy New Year! Workshop outline 1. Introduction to relation algebras 2. Games 3. Monk algebras: completions, canonicity 4. Rainbow construction:

599 views • 37 slides
Homogeneous orthocomplete effect algebras are covered by MV-algebras Josef Niederle and Jan

Homogeneous orthocomplete effect algebras are covered by MV-algebras Josef Niederle and Jan

Homogeneous orthocomplete effect algebras are covered by MV-algebras Josef Niederle and Jan Paseka Department of Mathematics and Statistics Masaryk University Brno, Czech Republic niederle,paseka@math.muni.cz Supported by TACL 2011 July

951 views • 80 slides
Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La

Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La

Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La Trobe University Victorian Algebra Conference 2015 Chris Taylor Algebras of incidence structures VAC 2015 1 / 24 Chris Taylor Algebras of

967 views • 69 slides
Quaternion Algebras Properties and Applications Rob Eimerl 1 Department of Mathematics

Quaternion Algebras Properties and Applications Rob Eimerl 1 Department of Mathematics

Quaternion Algebras Applications of Quaternion Algebras Quaternion Algebras Properties and Applications Rob Eimerl 1 Department of Mathematics University of Puget Sound May 5th, 2015 Quaternion Algebras Applications of Quaternion Algebras

563 views • 20 slides
Lecture 1: Group C*-algebras and Actions of Finite 1129 July 2016 Groups on C*-Algebras

Lecture 1: Group C*-algebras and Actions of Finite 1129 July 2016 Groups on C*-Algebras

The Second Summer School on Operator Algebras and Noncommutative Geometry 2016 East China Normal University, Shanghai Lecture 1: Group C*-algebras and Actions of Finite 1129 July 2016 Groups on C*-Algebras Lecture 1 (11 July 2016): Group

107 views • 7 slides
Multi derivation Maurer-Cartan algebras and sh-Lie-Rinehart algebras Johannes Huebschmann USTL,

Multi derivation Maurer-Cartan algebras and sh-Lie-Rinehart algebras Johannes Huebschmann USTL,

Multi derivation Maurer-Cartan algebras and sh-Lie-Rinehart algebras Johannes Huebschmann USTL, UFR de Math ematiques CNRS-UMR 8524 Labex CEMPI (ANR-11-LABX-0007-01) 59655 Villeneuve dAscq Cedex, France

605 views • 23 slides
Weak Quasi-Hopf Algebras and Vertex Operator Algebras Sebastiano Carpi University of Chieti and

Weak Quasi-Hopf Algebras and Vertex Operator Algebras Sebastiano Carpi University of Chieti and

Weak Quasi-Hopf Algebras and Vertex Operator Algebras Sebastiano Carpi University of Chieti and Pescara Dubrovnik, June 28, 2019 Based on a joint work with Sergio Ciamprone and Claudia Pinzari (in preparation) 1 Introduction Rational QFTs,

374 views • 19 slides
Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for dendriform algebras Yuqun

Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for dendriform algebras Yuqun

Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for dendriform algebras Yuqun Chen (Joint work with L.A. Bokut) South China Normal University, China The 4th International Workshop on Differential Algebra and Related Topics,

1.21k views • 87 slides

More recommend