Commutator Definition ◮ The modular commutator can be equivalently defined with either 1. the term condition, or 2. properties of a relation, usually called ∆. Definition (Term Condition) Let A be an algebra and take α, β, δ ∈ Con( A ). We say that α centralizes β modulo δ when the following condition is met: ◮ For all t ∈ Pol( A ) and a 0 ≡ α b 0 and a 1 ≡ β b 1 with | a 0 | + | a 1 | = σ ( t ) , � � t ( a 0 , a 1 ) ≡ δ t ( a 0 , b 1 ) = ⇒ t ( b 0 , a 0 ) ≡ δ t ( b 0 , b 1 )
Commutator Definition ◮ The modular commutator can be equivalently defined with either 1. the term condition, or 2. properties of a relation, usually called ∆. Definition (Term Condition) Let A be an algebra and take α, β, δ ∈ Con( A ). We say that α centralizes β modulo δ when the following condition is met: ◮ For all t ∈ Pol( A ) and a 0 ≡ α b 0 and a 1 ≡ β b 1 with | a 0 | + | a 1 | = σ ( t ) , � � t ( a 0 , a 1 ) ≡ δ t ( a 0 , b 1 ) = ⇒ t ( b 0 , a 0 ) ≡ δ t ( b 0 , b 1 ) We write C TC ( α, β ; δ ) whenever this is true.
Commutator Definition ◮ The modular commutator can be equivalently defined with either 1. the term condition, or 2. properties of a relation, usually called ∆. Definition (Term Condition) Let A be an algebra and take α, β, δ ∈ Con( A ). We say that α centralizes β modulo δ when the following condition is met: ◮ For all t ∈ Pol( A ) and a 0 ≡ α b 0 and a 1 ≡ β b 1 with | a 0 | + | a 1 | = σ ( t ) , � � t ( a 0 , a 1 ) ≡ δ t ( a 0 , b 1 ) = ⇒ t ( b 0 , a 0 ) ≡ δ t ( b 0 , b 1 ) We write C TC ( α, β ; δ ) whenever this is true. ◮ The term condition may be described as a condition that is quantified over a certain invariant relation of A which is called the algebra of ( α, β )-matrices and is denoted M ( α, β ).
Matrices ◮ A square is the graph � 2 2 ; E � , where two functions f , g ∈ 2 2 are connected by an edge if and only if their outputs differ in exactly one argument.
Matrices ◮ A square is the graph � 2 2 ; E � , where two functions f , g ∈ 2 2 are connected by an edge if and only if their outputs differ in exactly one argument. (0 , 1) (1 , 1) (0 , 0) (1 , 0) ◮ We say that a relation R on a set A is 2-dimensional if R ⊆ A 2 2 ( R is a set of squares whos vertices are labeled by elements of A .)
Matrices ◮ A square is the graph � 2 2 ; E � , where two functions f , g ∈ 2 2 are connected by an edge if and only if their outputs differ in exactly one argument. (0 , 1) (1 , 1) (0 , 0) (1 , 0) ◮ We say that a relation R on a set A is 2-dimensional if R ⊆ A 2 2 ( R is a set of squares whos vertices are labeled by elements of A .) ◮ M ( α, β ) is the subalgebra of A 2 2 with generators �� x � � �� y � � � y y : x ≡ α y : x ≡ β y x y x x
Matrices For δ ∈ Con( A ) we have that α centralizes β modulo δ if the implication c d c d → δ a b a b β α holds for all ( α, β )-matrices. This condition is abbreviated C TC ( α, β ; δ ).
Matrices Similarly, we have that β centralizes α modulo δ if the implication δ c d c d → a b a b β α holds for all ( α, β )-matrices. This condition is abbreviated C TC ( β, α ; δ ).
Matrices ◮ The binary commutator is defined to be � [ α, β ] TC = { δ : C ( α, β ; δ ) }
Matrices ◮ The notions of matrices and centrality for three congruences are defined similarly.
Matrices ◮ The notions of matrices and centrality for three congruences are defined similarly. ◮ A cube is the graph � 2 3 ; E � , where two functions f , g ∈ 2 3 are connected by an edge if and only if their outputs differ in exactly one argument.
Matrices ◮ The notions of matrices and centrality for three congruences are defined similarly. ◮ A cube is the graph � 2 3 ; E � , where two functions f , g ∈ 2 3 are connected by an edge if and only if their outputs differ in exactly one argument. (0 , 1 , 0) (1 , 1 , 0) (0 , 1 , 1) (1 , 1 , 1) (0 , 0 , 0) (1 , 0 , 0) (0 , 0 , 1) (1 , 0 , 1) ◮ We say that a relation R on a set A is 3-dimensional if R ⊆ A 3 2 ( R is a set of cubes whos vertices are labeled by elements of A .)
Matrices ◮ The notions of matrices and centrality for three congruences are defined similarly. ◮ A cube is the graph � 2 3 ; E � , where two functions f , g ∈ 2 3 are connected by an edge if and only if their outputs differ in exactly one argument. (0 , 1 , 0) (1 , 1 , 0) (0 , 1 , 1) (1 , 1 , 1) (0 , 0 , 0) (1 , 0 , 0) (0 , 0 , 1) (1 , 0 , 1) ◮ We say that a relation R on a set A is 3-dimensional if R ⊆ A 3 2 ( R is a set of cubes whos vertices are labeled by elements of A .)
Matrices ◮ For congruences θ 0 , θ 1 , θ 2 ∈ Con( A ), set M ( θ 0 , θ 1 , θ 2 ) ≤ A 2 3 to be the subalgebra generated by the following labeled cubes: x y y y x x x y y y y y θ 1 θ 0 x y x x x x x y x x y y θ 2 M ( θ 0 , θ 1 , θ 2 ) is called the algebra of ( θ 0 , θ 1 , θ 2 )-matrices.
Centrality ◮ For δ ∈ Con( A ), we say that θ 0 , θ 1 centralize θ 2 modulo δ if the following implication holds for all ( θ 0 , θ 1 , θ 2 )-matrices:
Centrality ◮ For δ ∈ Con( A ), we say that θ 0 , θ 1 centralize θ 2 modulo δ if the following implication holds for all ( θ 0 , θ 1 , θ 2 )-matrices: c d θ 1 g h θ 0 a b θ 2 e f
Centrality ◮ For δ ∈ Con( A ), we say that θ 0 , θ 1 centralize θ 2 modulo δ if the following implication holds for all ( θ 0 , θ 1 , θ 2 )-matrices: c d δ θ 1 g h θ 0 a b θ 2 e f
Centrality ◮ For δ ∈ Con( A ), we say that θ 0 , θ 1 centralize θ 2 modulo δ if the following implication holds for all ( θ 0 , θ 1 , θ 2 )-matrices: c d δ θ 1 g h θ 0 a b θ 2 e f
Centrality ◮ For δ ∈ Con( A ), we say that θ 0 , θ 1 centralize θ 2 modulo δ if the following implication holds for all ( θ 0 , θ 1 , θ 2 )-matrices: c d δ θ 1 g h θ 0 a b θ 2 e f ◮ This condition is abbreviated C TC ( θ 0 , θ 1 , θ 2 ; δ ).
Centrality ◮ Here is a picture of C TC ( θ 1 , θ 2 , θ 0 ; δ ): δ c d 1 g h 0 a b 2 e f
Matrices ◮ For congruences θ 0 , θ 1 , θ 2 we set � [ θ 0 , θ 1 , θ 2 ] TC = { δ : C TC ( θ 0 , θ 1 , θ 2 ; δ ) }
Matrices ◮ For congruences θ 0 , θ 1 , θ 2 we set � [ θ 0 , θ 1 , θ 2 ] TC = { δ : C TC ( θ 0 , θ 1 , θ 2 ; δ ) } ◮ Higher centrality and the commutator for arity ≥ 4 are similarly defined.
Matrices ◮ An n -dimensional hypercube is the graph H n = � 2 n ; E � , where two functions f , g ∈ 2 n are connected by an edge if and only if their outputs differ in exactly one argument.
Matrices ◮ An n -dimensional hypercube is the graph H n = � 2 n ; E � , where two functions f , g ∈ 2 n are connected by an edge if and only if their outputs differ in exactly one argument. ◮ We say that a relation R on a set A is n -dimensional if R ⊆ A 2 n
Matrices ◮ An n -dimensional hypercube is the graph H n = � 2 n ; E � , where two functions f , g ∈ 2 n are connected by an edge if and only if their outputs differ in exactly one argument. ◮ We say that a relation R on a set A is n -dimensional if R ⊆ A 2 n ◮ Observation: The term condition definition of centrality involving n -many congruences θ 0 , . . . , θ n − 1 is a condition that is quantified over ( θ 0 , . . . , θ n − 1 ) -matrices , which are certain n -dimensional invariant relations M ( θ 0 , . . . , θ n − 1 ) ≤ A 2 n that have generators of the form ( n − 1)-dimensional cube f ∈ 2 n such that f ( i ) = 0 x θ i y f ∈ 2 n such that f ( i ) = 1
◮ Consider the n -dimensional hypercube H n = � 2 n ; E � . For any coordinate i ∈ n , there are two ( n − 1)-dimensional hyperfaces that are ‘perpendicular’ to i :
◮ Consider the n -dimensional hypercube H n = � 2 n ; E � . For any coordinate i ∈ n , there are two ( n − 1)-dimensional hyperfaces that are ‘perpendicular’ to i : i = �{ f ∈ 2 n : f ( i ) = 0 } ; E � and 1. ( H n ) 0
◮ Consider the n -dimensional hypercube H n = � 2 n ; E � . For any coordinate i ∈ n , there are two ( n − 1)-dimensional hyperfaces that are ‘perpendicular’ to i : i = �{ f ∈ 2 n : f ( i ) = 0 } ; E � and 1. ( H n ) 0 i = �{ f ∈ 2 n : f ( i ) = 1 } ; E � . 2. ( H n ) 1
◮ Consider the n -dimensional hypercube H n = � 2 n ; E � . For any coordinate i ∈ n , there are two ( n − 1)-dimensional hyperfaces that are ‘perpendicular’ to i : i = �{ f ∈ 2 n : f ( i ) = 0 } ; E � and 1. ( H n ) 0 i = �{ f ∈ 2 n : f ( i ) = 1 } ; E � . 2. ( H n ) 1 (0 , 1 , 0 , 0) (1 , 1 , 0 , 0) (0 , 1 , 0 , 1) (1 , 1 , 0 , 1) (0 , 1 , 1 , 0) (1 , 1 , 1 , 0) (0 , 1 , 1 , 1) (1 , 1 , 1 , 1) (0 , 0 , 0 , 1) (1 , 0 , 0 , 1) (0 , 0 , 1 , 1) (1 , 0 , 1 , 1) (0 , 0 , 0 , 0) (1 , 0 , 0 , 0) (0 , 0 , 1 , 0) (1 , 0 , 1 , 0)
◮ Consider the n -dimensional hypercube H n = � 2 n ; E � . For any coordinate i ∈ n , there are two ( n − 1)-dimensional hyperfaces that are ‘perpendicular’ to i : i = �{ f ∈ 2 n : f ( i ) = 0 } ; E � and 1. ( H n ) 0 i = �{ f ∈ 2 n : f ( i ) = 1 } ; E � . 2. ( H n ) 1 (0 , 1 , 0 , 0) (1 , 1 , 0 , 0) (0 , 1 , 0 , 1) (1 , 1 , 0 , 1) (0 , 1 , 1 , 0) (1 , 1 , 1 , 0) (0 , 1 , 1 , 1) (1 , 1 , 1 , 1) (0 , 0 , 0 , 1) (1 , 0 , 0 , 1) (0 , 0 , 1 , 1) (1 , 0 , 1 , 1) (0 , 0 , 0 , 0) (1 , 0 , 0 , 0) (0 , 0 , 1 , 0) (1 , 0 , 1 , 0) ( H n ) 0 3 and ( H n ) 1 3
◮ Consider the n -dimensional hypercube H n = � 2 n ; E � . For any coordinate i ∈ n , there are two ( n − 1)-dimensional hyperfaces that are ‘perpendicular’ to i : i = �{ f ∈ 2 n : f ( i ) = 0 } ; E � and 1. ( H n ) 0 i = �{ f ∈ 2 n : f ( i ) = 1 } ; E � . 2. ( H n ) 1 (0 , 1 , 0 , 0) (1 , 1 , 0 , 0) (0 , 1 , 0 , 1) (1 , 1 , 0 , 1) (0 , 1 , 1 , 0) (1 , 1 , 1 , 0) (0 , 1 , 1 , 1) (1 , 1 , 1 , 1) (0 , 0 , 0 , 1) (1 , 0 , 0 , 1) (0 , 0 , 1 , 1) (1 , 0 , 1 , 1) (0 , 0 , 0 , 0) (1 , 0 , 0 , 0) (0 , 0 , 1 , 0) (1 , 0 , 1 , 0) ( H n ) 0 0 and ( H n ) 1 0
◮ Take h ∈ A 2 n . We consider h as a vertex labeled n -dimensional hypercube. For any coordinate i ∈ n , there are two ( n − 1)-dimensional vertex labeled hyperfaces that are perpendicular to i , which we denote
◮ Take h ∈ A 2 n . We consider h as a vertex labeled n -dimensional hypercube. For any coordinate i ∈ n , there are two ( n − 1)-dimensional vertex labeled hyperfaces that are perpendicular to i , which we denote 1. h 0 i and
◮ Take h ∈ A 2 n . We consider h as a vertex labeled n -dimensional hypercube. For any coordinate i ∈ n , there are two ( n − 1)-dimensional vertex labeled hyperfaces that are perpendicular to i , which we denote 1. h 0 i and 2. h 1 i .
◮ Take h ∈ A 2 n . We consider h as a vertex labeled n -dimensional hypercube. For any coordinate i ∈ n , there are two ( n − 1)-dimensional vertex labeled hyperfaces that are perpendicular to i , which we denote 1. h 0 i and 2. h 1 i . c d l k g s o p i j m n a b h ∈ A 2 n e f
◮ Take h ∈ A 2 n . We consider h as a vertex labeled n -dimensional hypercube. For any coordinate i ∈ n , there are two ( n − 1)-dimensional vertex labeled hyperfaces that are perpendicular to i , which we denote 1. h 0 i and 2. h 1 i . c d l k g s o p i j m n a b h ∈ A 2 n e f h 0 3 and h 1 3
◮ Take h ∈ A 2 n . We consider h as a vertex labeled n -dimensional hypercube. For any coordinate i ∈ n , there are two ( n − 1)-dimensional vertex labeled hyperfaces that are perpendicular to i , which we denote 1. h 0 i and 2. h 1 i . c d l k g s o p i j m n a b h ∈ A 2 n e f h 0 1 and h 1 1
◮ For R ⊆ A 2 n , set R i = {� h 0 i , h 1 i � : h ∈ R } .
◮ For R ⊆ A 2 n , set R i = {� h 0 i , h 1 i � : h ∈ R } . ◮ Fact: Suppose A is a member of a permutable variety, and take ( θ 0 , . . . , θ n − 1 ) ∈ Con( A ) n . Then, M ( θ 0 , . . . , θ n − 1 ) i is a congruence relation, for all i ∈ n .
◮ For R ⊆ A 2 n , set R i = {� h 0 i , h 1 i � : h ∈ R } . ◮ Fact: Suppose A is a member of a permutable variety, and take ( θ 0 , . . . , θ n − 1 ) ∈ Con( A ) n . Then, M ( θ 0 , . . . , θ n − 1 ) i is a congruence relation, for all i ∈ n . ◮ This leads to a nice characterization of the commutator for permutable varieties.
Theorem (Binary Commutator) Let V be a permutable variety and let A ∈ V . For α, β ∈ Con( A ) , the following are equivalent:
Theorem (Binary Commutator) Let V be a permutable variety and let A ∈ V . For α, β ∈ Con( A ) , the following are equivalent: 1. � x , y � ∈ [ α, β ] TC
Theorem (Binary Commutator) Let V be a permutable variety and let A ∈ V . For α, β ∈ Con( A ) , the following are equivalent: 1. � x , y � ∈ [ α, β ] TC � x � y 2. ∈ M ( α, β ) x x
Theorem (Binary Commutator) Let V be a permutable variety and let A ∈ V . For α, β ∈ Con( A ) , the following are equivalent: 1. � x , y � ∈ [ α, β ] TC � x � y 2. ∈ M ( α, β ) x x � a � y 3. ∈ M ( α, β ) for some a ∈ A a x
Theorem (Binary Commutator) Let V be a permutable variety and let A ∈ V . For α, β ∈ Con( A ) , the following are equivalent: 1. � x , y � ∈ [ α, β ] TC � x � y 2. ∈ M ( α, β ) x x � a � y 3. ∈ M ( α, β ) for some a ∈ A a x � x � y 4. ∈ M ( α, β ) for some b ∈ A. b b
◮ Let V be a modular variety and let A ∈ V . For α, β ∈ Con( A ), define ∆ α,β to be the transitive closure of M ( α, β ) 0 .
◮ Let V be a modular variety and let A ∈ V . For α, β ∈ Con( A ), define ∆ α,β to be the transitive closure of M ( α, β ) 0 . a e 0 e 1 e 2 e n − 1 e n c b f 0 f 1 f 2 f n − 1 f n d a c 1 ∈ ∆ α,β 0 b d
◮ Let V be a modular variety and let A ∈ V . For α, β ∈ Con( A ), define ∆ α,β to be the transitive closure of M ( α, β ) 0 . a e 0 e 1 e 2 e n − 1 e n c b f 0 f 1 f 2 f n − 1 f n d a c 1 ∈ ∆ α,β 0 b d ◮ Fact: Both (∆ α,β ) 0 and (∆ α,β ) 1 are congruence relations.
Theorem (Binary Commutator) Let V be a modular variety and let A ∈ V . For α, β ∈ Con( A ) , the following are equivalent: 1. � x , y � ∈ [ α, β ] TC � x � y 2. ∈ ∆ α,β x x � a � y 3. ∈ ∆ α,β for some a ∈ A a x � x � y 4. ∈ ∆ α,β for some b ∈ A. b b
Theorem: Let V be a permutable variety. Take θ 0 , θ 1 , θ 2 ∈ Con( A ) for A ∈ V . The following are equivalent: (1) � x, y � ∈ [ θ 0 , θ 1 , θ 2 ] x x (2) x y ∈ M ( θ 0 , θ 1 , θ 2 ) x x x x There exist elements of A such that b a h x (3) (5) c y h y ∈ M ( θ 0 , θ 1 , θ 2 ) ∈ M ( θ 0 , θ 1 , θ 2 ) b a i j θ 1 c x i j d d θ 0 (4) ∈ M ( θ 0 , θ 1 , θ 2 ) x y θ 2 e e f f
Theorem: Let V be a modular variety. Take θ 0 , θ 1 , θ 2 ∈ Con( A ) for A ∈ V . The following are equivalent: (1) � x, y � ∈ [ θ 0 , θ 1 , θ 2 ] x x (2) x y ∈ ∆ θ 0 ,θ 1 ,θ 2 x x x x There exist elements of A such that b a h x (3) c y (5) ∈ ∆ θ 0 ,θ 1 ,θ 2 h y ∈ ∆ θ 0 ,θ 1 ,θ 2 b a i j θ 1 c x i j d d θ 0 (4) ∈ ∆ θ 0 ,θ 1 ,θ 2 x y θ 2 e e f f
Higher Dimensional Congruence Relations Definition Let R ⊆ A 2 n be an n -dimensional relation on some set A . R is called an n -dimensional equivalence relation if for all i ∈ n , each R i is an equivalence relation.
Higher Dimensional Congruence Relations Definition Let R ⊆ A 2 n be an n -dimensional relation on some set A . R is called an n -dimensional equivalence relation if for all i ∈ n , each R i is an equivalence relation. Definition Let A be an algebra with underlying set A . Let R ∈ A 2 n be an n -dimensional equivalence relation. R is called an n -dimensional congruence if R is preserved by the basic operations of A .
Higher Dimensional Congruence Relations Definition Let R ⊆ A 2 n be an n -dimensional relation on some set A . R is called an n -dimensional equivalence relation if for all i ∈ n , each R i is an equivalence relation. Definition Let A be an algebra with underlying set A . Let R ∈ A 2 n be an n -dimensional equivalence relation. R is called an n -dimensional congruence if R is preserved by the basic operations of A . ◮ Fix n ≥ 1. The collection of all n -dimensional congruences of an algebra A is an algebraic lattice, which we denote by Con n ( A ).
Higher Dimensional Congruence Relations Definition Let R ⊆ A 2 n be an n -dimensional relation on some set A . R is called an n -dimensional equivalence relation if for all i ∈ n , each R i is an equivalence relation. Definition Let A be an algebra with underlying set A . Let R ∈ A 2 n be an n -dimensional equivalence relation. R is called an n -dimensional congruence if R is preserved by the basic operations of A . ◮ Fix n ≥ 1. The collection of all n -dimensional congruences of an algebra A is an algebraic lattice, which we denote by Con n ( A ). ◮ There are n distinct embeddings from Con 1 ( A ) into Con n ( A ).
Con 1 ( A )
α φ 0 2 Con 1 ( A ) Con 2 ( A )
α φ 0 2 Con 1 ( A ) Con 2 ( A ) � � x � � y φ 0 2 ( α ) = : � x, y � ∈ α x y
φ 1 2 α β φ 0 2 Con 1 ( A ) Con 2 ( A ) � � x � � y φ 0 2 ( α ) = : � x, y � ∈ α x y � � x � � x φ 1 2 ( β ) = : � x, y � ∈ β y y
Define ∆ α,β = φ 0 2 ( α ) ∨ φ 1 2 ( β ) ∆ α,β φ 1 2 α β φ 0 2 Con 1 ( A ) Con 2 ( A ) � � x � � y φ 0 2 ( α ) = : � x, y � ∈ α x y � � x � � x φ 1 2 ( β ) = : � x, y � ∈ β y y
Higher Dimensional Congruence Relations ◮ Fix a dimension n and take i ∈ n . For a pair � x , y � ∈ A 2 , let Cube i ( � x , y � ) ∈ A 2 n be such that
Higher Dimensional Congruence Relations ◮ Fix a dimension n and take i ∈ n . For a pair � x , y � ∈ A 2 , let Cube i ( � x , y � ) ∈ A 2 n be such that � � 0 1. Cube i ( � x , y � ) i is the ( n − 1)-dimensional cube with each vertex labeled by x .
Higher Dimensional Congruence Relations ◮ Fix a dimension n and take i ∈ n . For a pair � x , y � ∈ A 2 , let Cube i ( � x , y � ) ∈ A 2 n be such that � � 0 1. Cube i ( � x , y � ) i is the ( n − 1)-dimensional cube with each vertex labeled by x .
Higher Dimensional Congruence Relations ◮ Fix a dimension n and take i ∈ n . For a pair � x , y � ∈ A 2 , let Cube i ( � x , y � ) ∈ A 2 n be such that � � 0 1. Cube i ( � x , y � ) i is the ( n − 1)-dimensional cube with each vertex labeled by x . � � 1 2. Cube i ( � x , y � ) i is the ( n − 1)-dimensional cube with each vertex labeled by y . ◮ Define φ i n : Con 1 ( A ) → Con n ( A ) by φ i n ( α ) = { Cube i ( � x , y � ) : � x , y � ∈ α }
i φ i Define ∆ θ 0 ,...,θ n − 1 = � n ( θ i ) ∆ θ 0 ,...,θ n − 1 φ n − 1 n θ 0 θ n − 1 φ 0 n Con 1 ( A ) Con n ( A )
Characterizing Joins ◮ Let A be an algebra and let θ be an equivalence relation on A . Then, θ is an admissible relation if and only if θ is compatible with the unary polynomials of A .
Characterizing Joins ◮ Let A be an algebra and let θ be an equivalence relation on A . Then, θ is an admissible relation if and only if θ is compatible with the unary polynomials of A . ◮ This generalizes to: Theorem Let A be an algebra and let n ≥ 1 . An n-dimensional equivalence relation θ is admissible if and only if θ is compatible with the n-ary polynomials of A .
Proof Idea a 0 a 1 a 2 a 3 b 0 b 1 b 2 b 3 ∈ θ Take d 0 , d 1 , d 2 , c 0 c 1 c 2 c 3 d 3
Proof Idea a 0 a 1 a 2 a 3 b 0 b 1 b 2 b 3 ∈ θ Take Then, d 0 , d 1 , d 2 , c 0 c 1 c 2 c 3 d 3 a 1 a 1 a 0 b 0 b 0 b 0 b 0 b 1 b 1 b 1 c 0 a 1 a 1 d 0 b 1 b 1 b 1 d 0 d 0 d 0 c 0 d 0 d 0 c 1 c 1 d 0 d 0 d 1 d 1 d 1 c 1 c 0 c 1 d 0 d 0 d 0 d 0 d 1 d 1 d 1 c 1 c 0 c 1 d 0 d 0 d 0 d 1 d 1 d 1 d 0 ∈ θ a 2 a 2 a 2 a 3 a 3 a 3 a 3 b 2 b 2 b 3 a 2 a 3 a 3 a 3 a 3 a 2 a 2 b 2 b 2 b 3 a 2 a 2 a 3 a 3 a 3 a 3 a 2 b 2 b 2 b 3 c 2 c 2 a 3 a 3 a 3 a 3 c 2 d 2 d 2 b 3 c 2 c 2 c 2 c 3 c 3 c 3 c 3 d 3 d 2 d 2
Proof Idea a 0 a 1 a 2 a 3 b 0 b 1 b 2 b 3 Take ∈ θ Then, d 0 , d 1 , d 2 , c 0 c 1 c 2 c 3 d 3 a 1 a 1 a 0 b 0 b 1 b 1 b 0 b 0 b 0 b 1 c 0 a 1 a 1 d 0 d 0 d 0 d 0 b 1 b 1 b 1 c 0 c 1 c 1 d 0 d 0 d 0 d 0 d 1 d 1 d 1 c 0 c 1 d 0 d 0 d 0 d 0 c 1 d 1 d 1 d 1 c 1 c 1 c 0 d 0 d 0 d 0 d 1 d 1 d 0 d 1 ∈ θ a 2 a 2 a 2 a 3 a 3 a 3 a 3 b 2 b 2 b 3 Compatibility a 3 a 2 a 2 a 2 b 2 a 3 a 3 a 3 b 3 b 2 with binary polynomials is a 3 a 2 a 2 a 2 b 2 a 3 a 3 a 3 b 2 b 3 sufficient to show compatibility c 2 c 2 a 3 a 3 a 3 a 3 c 2 b 3 d 2 d 2 with a 4-ary operation. c 2 c 2 c 2 c 3 c 3 c 3 c 3 d 2 d 2 d 3
Characterizing Joins ◮ ∆ θ 0 ,...,θ n − 1 = � i φ i n ( θ i ) is therefore obtained by 1. Closing � φ i n ( θ i ) under all n -ary polynomials and then
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