Determining congruence n -permutability is hard ( n 3?) Jonah - PowerPoint PPT Presentation

Determining congruence n -permutability is hard ( n ≥ 3?) Jonah Horowitz Ryerson University May 30, 2015 Jonah Horowitz (Ryerson University) May 30, 2015 1 / 1

Outline Background 1 Proof of Main Result 2 Corollaries 3 Limitations 4 Questions 5 Jonah Horowitz (Ryerson University) May 30, 2015 2 / 1

Background Hagemann & Mitschke 1973 Given n ≥ 2, an algebra A generates a congruence n -permutable variety if and only if there exist ternary term operations d 0 , . . . , d n such that: d 0 ( x , y , z ) ≈ x , d n ( x , y , z ) ≈ z , and d i ( x , x , y ) ≈ d i + 1 ( x , y , y ) for all i < n . Jonah Horowitz (Ryerson University) May 30, 2015 3 / 1

Background Hagemann & Mitschke 1973 Given n ≥ 2, an algebra A generates a congruence n -permutable variety if and only if there exist ternary term operations d 0 , . . . , d n such that: d 0 ( x , y , z ) ≈ x , d n ( x , y , z ) ≈ z , and d i ( x , x , y ) ≈ d i + 1 ( x , y , y ) for all i < n . Freese & Valeriote 2009 G EN -C LO ′ : Given a finite set A , a finite set of operations F on A , and a unary operation h on A , is h ∈ �F� ? Jonah Horowitz (Ryerson University) May 30, 2015 3 / 1

Background Hagemann & Mitschke 1973 Given n ≥ 2, an algebra A generates a congruence n -permutable variety if and only if there exist ternary term operations d 0 , . . . , d n such that: d 0 ( x , y , z ) ≈ x , d n ( x , y , z ) ≈ z , and d i ( x , x , y ) ≈ d i + 1 ( x , y , y ) for all i < n . Freese & Valeriote 2009 G EN -C LO ′ : Given a finite set A , a finite set of operations F on A , and a unary operation h on A , is h ∈ �F� ? Bergman, Juedes & Slutzki 1999 G EN -C LO ′ is EXPTIME-complete. Jonah Horowitz (Ryerson University) May 30, 2015 3 / 1

Background Hagemann & Mitschke 1973 Given n ≥ 2, an algebra A generates a congruence n -permutable variety if and only if there exist ternary term operations d 0 , . . . , d n such that: d 0 ( x , y , z ) ≈ x , d n ( x , y , z ) ≈ z , and d i ( x , x , y ) ≈ d i + 1 ( x , y , y ) for all i < n . Freese & Valeriote 2009 G EN -C LO ′ : Given a finite set A , a finite set of operations F on A , and a unary operation h on A , is h ∈ �F� ? Bergman, Juedes & Slutzki 1999 G EN -C LO ′ is EXPTIME-complete. H 2013 Given g : A n → A , say that g is a Constant-Projection Blend (CPB) if there exist 0 ∈ A and i < n such that for every x ∈ A n , g ( x ) ∈ { 0 , x i } . Jonah Horowitz (Ryerson University) May 30, 2015 3 / 1

Background Hagemann & Mitschke 1973 Given n ≥ 2, an algebra A generates a congruence n -permutable variety if and only if there exist ternary term operations d 0 , . . . , d n such that: d 0 ( x , y , z ) ≈ x , d n ( x , y , z ) ≈ z , and d i ( x , x , y ) ≈ d i + 1 ( x , y , y ) for all i < n . Freese & Valeriote 2009 G EN -C LO ′ : Given a finite set A , a finite set of operations F on A , and a unary operation h on A , is h ∈ �F� ? Bergman, Juedes & Slutzki 1999 G EN -C LO ′ is EXPTIME-complete. H 2013 Given g : A n → A , say that g is a Constant-Projection Blend (CPB) if there exist 0 ∈ A and i < n such that for every x ∈ A n , g ( x ) ∈ { 0 , x i } . In this case say that g is CPB 0 (on coordinate i ). Jonah Horowitz (Ryerson University) May 30, 2015 3 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 For each operation g : A n → A define g ′ : B n → B such that g ′ | A = g and 3 ∈ A n . g ′ ( x ) = 0 whenever x / Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 For each operation g : A n → A define g ′ : B n → B such that g ′ | A = g and 3 ∈ A n . g ′ ( x ) = 0 whenever x / Let U be a finite set of idempotent CPB 0 operations on B . 4 Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 For each operation g : A n → A define g ′ : B n → B such that g ′ | A = g and 3 ∈ A n . g ′ ( x ) = 0 whenever x / Let U be a finite set of idempotent CPB 0 operations on B . 4 For each g ∈ U (with arity n ), define t g : B n + 1 → B to be 5 � g ( x 1 , . . . , x n ) if x 0 = h ′ ( x 1 ) t g ( x 0 , . . . , x n ) = 0 otherwise Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 For each operation g : A n → A define g ′ : B n → B such that g ′ | A = g and 3 ∈ A n . g ′ ( x ) = 0 whenever x / Let U be a finite set of idempotent CPB 0 operations on B . 4 For each g ∈ U (with arity n ), define t g : B n + 1 → B to be 5 � g ( x 1 , . . . , x n ) if x 0 = h ′ ( x 1 ) t g ( x 0 , . . . , x n ) = 0 otherwise Define Γ = { f ′ | f ∈ F} ∪ { t g | g ∈ U} . 6 Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 For each operation g : A n → A define g ′ : B n → B such that g ′ | A = g and 3 ∈ A n . g ′ ( x ) = 0 whenever x / Let U be a finite set of idempotent CPB 0 operations on B . 4 For each g ∈ U (with arity n ), define t g : B n + 1 → B to be 5 � g ( x 1 , . . . , x n ) if x 0 = h ′ ( x 1 ) t g ( x 0 , . . . , x n ) = 0 otherwise Define Γ = { f ′ | f ∈ F} ∪ { t g | g ∈ U} . 6 Prove that if h ∈ �F� then U ⊆ � Γ � . 7 Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 For each operation g : A n → A define g ′ : B n → B such that g ′ | A = g and 3 ∈ A n . g ′ ( x ) = 0 whenever x / Let U be a finite set of idempotent CPB 0 operations on B . 4 For each g ∈ U (with arity n ), define t g : B n + 1 → B to be 5 � g ( x 1 , . . . , x n ) if x 0 = h ′ ( x 1 ) t g ( x 0 , . . . , x n ) = 0 otherwise Define Γ = { f ′ | f ∈ F} ∪ { t g | g ∈ U} . 6 Prove that if h ∈ �F� then U ⊆ � Γ � . 7 Prove that if h / ∈ �F� then � Γ � has no idempotent operations which depend on 8 more than one variable. Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Proof Outline Main Result Determining if a finite algebra generates a congruence n -permutable variety (for fixed n ≥ 3) is EXPTIME-complete. Let F be a finite set of operations on a finite set A and let h be a unary operation 1 on A . Define B = A ∪ { 0 , 1 } where 0 , 1 / ∈ A . 2 For each operation g : A n → A define g ′ : B n → B such that g ′ | A = g and 3 ∈ A n . g ′ ( x ) = 0 whenever x / Let U be a finite set of idempotent CPB 0 operations on B . 4 For each g ∈ U (with arity n ), define t g : B n + 1 → B to be 5 � g ( x 1 , . . . , x n ) if x 0 = h ′ ( x 1 ) t g ( x 0 , . . . , x n ) = 0 otherwise Define Γ = { f ′ | f ∈ F} ∪ { t g | g ∈ U} . 6 Prove that if h ∈ �F� then U ⊆ � Γ � . 7 Prove that if h / ∈ �F� then � Γ � has no idempotent operations which depend on 8 more than one variable. Prove that generating a congruence n -permutable variety (for fixed n ≥ 3) is 9 satisfiable by CPB 0 operations. Jonah Horowitz (Ryerson University) May 30, 2015 4 / 1

Step 7 (Freese & Valeriote 2009) Lemma ′ distributes over functional composition. Jonah Horowitz (Ryerson University) May 30, 2015 5 / 1