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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

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