On the relationship of maximal clones and maximal C-clones Mike - PowerPoint PPT Presentation

Galois theory C -clones Max clones On the relationship of maximal clones and maximal C-clones Mike Behrisch Institute of Computer Languages, Theory and Logic Group, Vienna University of Technology 21st June 2014 Warsaw, Poland Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Outline Galois theory for clones 1 Clausal relations and clausal clones 2 Maximal clones/ C -clones 3 Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Presenting joint work with. . . Edith Vargas-García Universidad Autónoma de la Ciudad de México Academia de Matemáticas && University of Leeds School of Mathematics. Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Outline Galois theory for clones 1 Clausal relations and clausal clones 2 Maximal clones/ C -clones 3 Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Galois theory for clones Basis Galois correspondence connecting finitary operations on D ← → finitary relations on D Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Galois theory for clones Basis Galois correspondence connecting finitary operations on D ← → finitary relations on D Finitary operations For k ∈ N + any f : D k − → D is a k -ary operation on D D := D D k set of k -ary operations on D O ( k ) k ∈ N + O ( k ) O D := � set of all finitary operations on D D Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Galois theory for clones Basis Galois correspondence connecting finitary operations on D ← → finitary relations on D Finitary operations For k ∈ N + any f : D k − → D is a k -ary operation on D D := D D k set of k -ary operations on D O ( k ) k ∈ N + O ( k ) O D := � set of all finitary operations on D D Finitary relations For m ∈ N + subsets ̺ ⊆ D m are m -ary relations on D R ( m ) := P ( D m ) set of m -ary relations on D D m ∈ N + R ( m ) R D := � set of all finitary relations on D D Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Preservation condition m , n ∈ N + , f ∈ O ( n ) D , ̺ ∈ R ( m ) D · · · x 0 , 0 x 0 , n − 1 . . ... . . . . · · · x m − 1 , 0 x m − 1 , n − 1 Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Preservation condition m , n ∈ N + , f ∈ O ( n ) D , ̺ ∈ R ( m ) D · · · x 0 , 0 x 0 , n − 1 . . ... . . . . · · · x m − 1 , 0 x m − 1 , n − 1 ∈ ̺ · · · ∈ ̺ Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Preservation condition m , n ∈ N + , f ∈ O ( n ) D , ̺ ∈ R ( m ) D · · · f ( x 0 , 0 x 0 , n − 1 ) . . ... . . . . f ( · · · ) x m − 1 , 0 x m − 1 , n − 1 ∈ ̺ · · · ∈ ̺ Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Preservation condition m , n ∈ N + , f ∈ O ( n ) D , ̺ ∈ R ( m ) D · · · f ( x 0 , 0 x 0 , n − 1 ) = . . ... . . . . f ( · · · ) = x m − 1 , 0 x m − 1 , n − 1 ∈ ̺ · · · ∈ ̺ Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Preservation condition m , n ∈ N + , f ∈ O ( n ) D , ̺ ∈ R ( m ) D · · · f ( x 0 , 0 x 0 , n − 1 ) = y 0 . . . ... . . . . . . f ( · · · ) = x m − 1 , 0 x m − 1 , n − 1 y m − 1 ∈ ̺ · · · ∈ ̺ Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Preservation condition m , n ∈ N + , f ∈ O ( n ) D , ̺ ∈ R ( m ) D · · · f ( x 0 , 0 x 0 , n − 1 ) = y 0 . . . ... . . . . . . f ( · · · ) = x m − 1 , 0 x m − 1 , n − 1 y m − 1 ∈ ̺ · · · ∈ ̺ ∈ ̺ Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Preservation condition m , n ∈ N + , f ∈ O ( n ) D , ̺ ∈ R ( m ) D · · · f ( x 0 , 0 x 0 , n − 1 ) = y 0 . . . ... . . . . . . f ( · · · ) = x m − 1 , 0 x m − 1 , n − 1 y m − 1 ∈ ̺ · · · ∈ ̺ ∈ ̺ Truth of this condition: f ⊲ ̺ Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Polymorphisms and invariant relations For F ⊆ O D and Q ⊆ R D : Inv D F := { ̺ ∈ R D | ∀ f ∈ F : f ⊲ ̺ } Pol D Q := { f ∈ O D | ∀ ̺ ∈ Q : f ⊲ ̺ } closure operators F �→ Pol D Inv D F Q �→ Inv D Pol D Q Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Connection to clones Lemma Q ⊆ R D = ⇒ Pol D Q is a clone. Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Connection to clones Lemma Q ⊆ R D = ⇒ Pol D Q is a clone. Theorem (Bodnarčuk, Kalužnin, Kotov, Romov 69, Geiger 68) D finite, F ⊆ O D a clone = ⇒ F = Pol D Q for Q = Inv D F . Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Connection to clones Lemma Q ⊆ R D = ⇒ Pol D Q is a clone. Theorem (Bodnarčuk, Kalužnin, Kotov, Romov 69, Geiger 68) D finite, F ⊆ O D a clone = ⇒ F = Pol D Q for Q = Inv D F . Consequence Every clone can be described by relations. Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Connection to clones Lemma Q ⊆ R D = ⇒ Pol D Q is a clone. Theorem (Bodnarčuk, Kalužnin, Kotov, Romov 69, Geiger 68) D finite, F ⊆ O D a clone = ⇒ F = Pol D Q for Q = Inv D F . Consequence Every clone can be described by relations. Idea Reduction of complexity by confining the allowed relations Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Outline Galois theory for clones 1 Clausal relations and clausal clones 2 Maximal clones/ C -clones 3 Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Clausal relations From now on D = { 0 , . . . , n − 1 } finite! Chain 0 ≤ 1 ≤ 2 ≤ · · · ≤ n − 1. Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Clausal relations From now on D = { 0 , . . . , n − 1 } finite! Chain 0 ≤ 1 ≤ 2 ≤ · · · ≤ n − 1. Definition (Clausal relation) p , q ∈ N + , a = ( a 1 , . . . , a p ) ∈ D p , b = ( b 1 , . . . , b q ) ∈ D q . � � p q � � � � R a ( x , y ) ∈ D p + q b := x i ≥ a i ∨ y j ≤ b j . � � � i = 1 j = 1 Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Clausal relations From now on D = { 0 , . . . , n − 1 } finite! Chain 0 ≤ 1 ≤ 2 ≤ · · · ≤ n − 1. Definition (Clausal relation) p , q ∈ N + , a = ( a 1 , . . . , a p ) ∈ D p , b = ( b 1 , . . . , b q ) ∈ D q . � � p q � � � � R a ( x , y ) ∈ D p + q b := x i ≥ a i ∨ y j ≤ b j . � � � i = 1 j = 1 Special case: binary clausal relation p = q = 1, a = ( a ) ∈ D 1 , b = ( b ) ∈ D 1 . R ( a ) � x ≥ a ∨ y ≤ b � ( x , y ) ∈ D 2 � � ( b ) = . Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones C -clones Definition ( C -clone) = every clone Pol D Q , where Q is a set of clausal relations. Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones C -clones Definition ( C -clone) = every clone Pol D Q , where Q is a set of clausal relations. C -clones form a lattice w.r.t. ⊆ . Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Lattice of C -clones for D = { 0 , 1 } O D M = Pol D ≤ � c 0 , c 1 , ∨� O D � c 0 , c 1 , ∧� O D C D Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Lattice of C -clones for D = { 0 , 1 } maximal clone in L D O D M = Pol D ≤ � c 0 , c 1 , ∨� O D � c 0 , c 1 , ∧� O D C D Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Lattice of C -clones for D = { 0 , . . . , n − 1 } , n ≥ 3 contains countably infinite descending chains Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Lattice of C -clones for D = { 0 , . . . , n − 1 } , n ≥ 3 contains countably infinite descending chains O D Pol D R ( 1 ) ( 1 ) Pol D R ( 11 ) ( 11 ) Pol D R ( 111 ) ( 111 ) Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Lattice of C -clones for D = { 0 , . . . , n − 1 } , n ≥ 3 contains countably infinite descending chains O D Pol D R ( 1 ) ( 1 ) Pol D R ( 11 ) ( 11 ) Pol D R ( 111 ) ( 111 ) no C -clone = a maximal clone [Beh,Var 2014, submitted] Mike Behrisch Relationship of max clones / C-clones

Galois theory C -clones Max clones Lattice of C -clones for D = { 0 , . . . , n − 1 } , n ≥ 3 contains countably infinite descending chains O D Pol D R ( 1 ) ( 1 ) Pol D R ( 11 ) ( 11 ) Pol D R ( 111 ) ( 111 ) no C -clone = a maximal clone [Beh,Var 2014, submitted] ⇒ every C -clone � = O D satisfies F � M for some = maximal clone M (as O D is finitely generated) Mike Behrisch Relationship of max clones / C-clones