Polynomial Completeness of Malcev Aichinger algebras Polynomials - PowerPoint PPT Presentation

Polynomial Completeness of Mal’cev algebras Erhard Polynomial Completeness of Mal’cev Aichinger algebras Polynomials Clones Description of Clones Mal’cev Erhard Aichinger Completeness Polynomial equivalence Department of Algebra DCC Theorems Johannes Kepler University Linz, Austria AAA79, Olomouc, Czech Republic

Polynomials Polynomial Definition Completeness of Mal’cev A = � A , F � an algebra, n ∈ N . Pol k ( A ) is the subalgebra of algebras Erhard A A k = �{ f : A k → A } , “ F pointwise” � Aichinger Polynomials that is generated by Clones Description of Clones ( x 1 , . . . , x k ) → x i ( i ∈ { 1 , . . . , k } ) Mal’cev Completeness ( x 1 , . . . , x k ) → a ( a ∈ A ) . Polynomial equivalence DCC Proposition Theorems A be an algebra, k ∈ N . Then p ∈ Pol k ( A ) iff there exists a term t in the language of A , ∃ m ∈ N , ∃ a 1 , a 2 , . . . , a m ∈ A such that p ( x 1 , x 2 , . . . , x k ) = t A ( a 1 , a 2 , . . . , a m , x 1 , x 2 , . . . , x k ) for all x 1 , x 2 , . . . , x k ∈ A .

Function algebras – Clones Polynomial Completeness of Mal’cev | f : A k → A } . algebras O ( A ) := � k ∈ N { f | | Erhard Aichinger Definition of Clone Polynomials C ⊆ O ( A ) is a clone on A iff Clones Description of Clones 1 ∀ k , i ∈ N with i ≤ k : ∈ C , � ( x 1 , . . . , x k ) �→ x i � Mal’cev Completeness 2 ∀ n ∈ N , m ∈ N , f ∈ C [ n ] , g 1 , . . . , g n ∈ C [ m ] : Polynomial equivalence f ( g 1 , . . . , g n ) ∈ C [ m ] . DCC Theorems C [ n ] . . . the n -ary functions in C . k ∈ N Pol k ( A ) is a clone on A . Pol ( A ) := �

Functional Description of Clones Polynomial Completeness of Mal’cev algebras Erhard Aichinger A algebra. Polynomials Clones Description of Clones Pol ( A ) . . . the smallest clone on A that contains all projections, all Mal’cev Completeness constant operations, all basic operations of A . Polynomial equivalence DCC Theorems Clo ( A ) . . . the smallest clone on A that contains all projections, and all basic operations of A = clone of term functions of A .

Clones vs. term functions Polynomial Completeness Proposition of Mal’cev algebras Every clone is the set of term functions of some algebra. Erhard Aichinger Proposition Polynomials Clones Let C be a clone on A . Define A := � A , C� . Then C = Clo ( A ) . Description of Clones Mal’cev Completeness Definition Polynomial A clone is constantive or a polynomial clone if it contains all unary equivalence DCC constant functions. Theorems Proposition Every constantive clone is the set of polynomial functions of some algebra.

Relational Description of Clones Polynomial Definition Completeness of Mal’cev I a finite set, ρ ⊆ A I , f : A n → A . f preserves ρ ( f ⊲ ρ ) if algebras ∀ v 1 , . . . , v n ∈ ρ : Erhard Aichinger � f ( v 1 ( i ) , . . . , v n ( i )) | | i ∈ I � ∈ ρ. | Polynomials Clones Description of Clones Mal’cev Remark Completeness ⇒ ρ is a subuniverse of � A , f � I . f ⊲ ρ ⇐ Polynomial equivalence DCC Theorems Definition (Polymorphisms) Let R be a set of finitary relations on A , ρ ∈ R . Pöl ( { ρ } ) { f ∈ O ( A ) | | | f ⊲ ρ } , := Pöl ( R ) := � ρ ∈ R Pöl ( { ρ } ) .

Relational Descriptions of Clones Polynomial Theorem Completeness of Mal’cev Let ρ be a finitary relation on A . Then Pöl ( { ρ } ) is a clone. algebras Erhard Aichinger Theorem (testing clone membership), [Pöschel and Kalužnin, 1979, Folgerung 1.1.18] Polynomials Let C be a clone on A , n ∈ N , f : A n → A . The set ρ := C [ n ] is a Clones Description of Clones subset of A A n , hence a relation on A with index set I := A n . Then Mal’cev Completeness f ∈ C ⇐ ⇒ f ⊲ ρ. Polynomial equivalence DCC Theorems Theorem (testing whether a relation is preserved) [Pöschel and Kalužnin, 1979, Satz 1.1.19] Let C be a clone on A , ρ a finitary relation on A with m elements. Then ⇒ ( ∀ c ∈ C [ m ] : c ⊲ ρ ) . ( ∀ c ∈ C : c ⊲ ρ ) ⇐

Finite Description of Clones Polynomial Completeness of Mal’cev algebras Definition Erhard Aichinger A clone is finitely generated if it is generated by a finite set of finitary functions. Polynomials Clones Definition Description of Clones Mal’cev A clone C is finitely related if there is a finite set of finitary Completeness relations R with C = Pöl ( R ) . Polynomial equivalence DCC Theorems Open and probably very hard Given a finite F ⊆ O ( A ) and a finitary relation ρ on A . Decide whether F generates Pöl ( { ρ } ) .

Mal’cev operations Polynomial Completeness A a set. A function d : A 3 → A is a Mal’cev operation if of Mal’cev algebras Erhard d ( a , a , b ) = d ( b , a , a ) = b for all a , b ∈ A . Aichinger Polynomials Clones Description of Clones Typical example: d ( x , y , z ) := x − y + z . Mal’cev Completeness Polynomial equivalence An algebra is a Mal’cev algebra if it has a Mal’cev operation in its DCC Theorems ternary term functions. (Algebra with a Mal’cev term should be used if the notion Mal’cev algebra causes confusion.) A clone is a Mal’cev clone if it has a Mal’cev operation in its ternary functions.

Polynomial Completeness of Mal’cev algebras Erhard Aichinger Polynomials Theorem [Mal’cev, 1954] Clones Description of Clones An algebra A is a Mal’cev algebra if for all B ∈ HSP A : Mal’cev ∀ α, β ∈ Con B : α ◦ β = β ◦ α . Completeness Polynomial equivalence DCC Theorems

A characterization of Mal’cev clones Polynomial Completeness of Mal’cev algebras Erhard Theorem ([Berman et al., 2010]) Aichinger Let A be a finite set, C a clone on A . For n ∈ N , let Polynomials i ( n ) := max {| X || | X is an independent subset of � A , C� n } . Clones | Description of Clones Mal’cev Then C is a Mal’cev clone if and only if ∃ α ∈ N such that Completeness Polynomial ∀ n ∈ N : i ( n ) ≤ 2 α n . equivalence DCC Theorems Note added: I have stated this Theorem incorrectly in my presentation at Olomouc.

Functionally complete algebras Polynomial Completeness of Mal’cev algebras Theorem (cf. [Hagemann and Herrmann, 1982]), forerunner in Erhard [Istinger et al., 1979] Aichinger Let A be a finite algebra, | A | ≥ 2. Then Pol ( A ) = O ( A ) if and only Polynomials if Pol 3 ( A ) contains a Mal’cev operation, and A is simple and Clones nonabelian. Description of Clones Mal’cev Completeness A is nonabelian iff [ 1 A , 1 A ] � = 0 A . Here, [ ., . ] is the term condition Polynomial equivalence commutator . DCC Theorems This describes finite algebras with Pol ( A ) = Pöl ( ∅ ) .

Affine complete algebras Polynomial Completeness Definition of affine completeness of Mal’cev algebras An algebra A is affine complete if Pol ( A ) = Pöl ( Con ( A )) . Erhard Aichinger Theorem [Hagemann and Herrmann, 1982, Polynomials Idziak and Słomczy´ nska, 2001, Aichinger, 2000] Clones Let A be a finite Mal’cev algebra. Then the following are Description of Clones Mal’cev equivalent: Completeness 1 Every B ∈ H ( A ) is affine complete. Polynomial equivalence DCC 2 For all α ∈ Con ( A ) , we have [ α, α ] = α . Theorems Open and probably still very hard Is affine completeness a decidable property of A = � A , F � (of finite type)?

Other concepts of polynomial completeness Polynomial Completeness of Mal’cev algebras Erhard Concepts of Polynomial completeness Aichinger Polynomials 1 weak polynomial richness: [Idziak and Słomczy´ nska, 2001], Clones [Aichinger and Mudrinski, 2009b] (expanded groups) Description of Clones Mal’cev 2 polynomial richness: [Idziak and Słomczy´ nska, 2001], Completeness [Aichinger and Mudrinski, 2009b] (expanded groups) Polynomial equivalence 3 “commutator-completeness”: every commutator-preserving DCC Theorems function is a polynomial function: [Your results, AAA80]

Conclusion about completeness properties Polynomial Completeness of Mal’cev algebras Erhard Aichinger Completeness provides relations Polynomials Completeness results often provide a finite set R of relations on A Clones such that Description of Clones Mal’cev Pol ( A ) = Pöl ( R ) . Completeness E.g., for every affine complete algebra, we have Polynomial equivalence DCC Pol ( A ) = Pöl ( Con ( A )) . Theorems

Polynomially equivalent algebras Polynomial Completeness of Mal’cev Definition algebras The algebras A and B are polynomially equivalent if A = B and Erhard Aichinger Pol ( A ) = Pol ( B ) . Polynomials Task Clones Description of Clones Classify finite algebras modulo polynomial equivalence. Mal’cev Completeness Polynomial Task equivalence DCC A = � A , F � algebra. Theorems Classify all expansions � A , F ∪ G � of A modulo polynomial equivalence. Determine all clones C with Pol ( A ) ⊆ C ⊆ O ( A ) .