CLOSED SETS OF FINITARY FUNCTIONS BETWEEN FINITE FIELDS OF COPRIME - PowerPoint PPT Presentation

CLOSED SETS OF FINITARY FUNCTIONS BETWEEN FINITE FIELDS OF COPRIME ORDER. Stefano Fioravanti September 2019, YRAC2019 Institute for Algebra Austrian Science Fund FWF P29931 0/11

( F p , F q ) -linear closed clonoids Definition Let p and q be powers of prime numbers. A ( F p , F q ) -linear closed clonoid is a F n non-empty subset C of � q such that: n ∈ N F p (1) if f, g ∈ C [ n ] then: f + p g ∈ C [ n ] ; (2) if f ∈ C [ m ] and A ∈ F m × n then: q g : ( x 1 , ..., x n ) �→ f ( A · q ( x 1 , ..., x n ) t ) is in C [ n ] . 1/11

Known results [1] E. Aichinger, P . Mayr, Polynomial clones on groups of order pq, in: Acta Mathematica Hungarica, Volume 114, Number 3, Page(s) 267-285, 2007. (All 17 clones containing ( Z p × Z q , + , (1 , 1)) ); [2] J. Bulín, A. Krokhin, and J. Opršal, Algebraic approach to promise constraint satisfaction, arXiv:1811.00970, 2018. [3] S. Kreinecker, Closed function sets on groups of prime order, Manuscript, arXiv:1810.09175, 2018. (All finitely many clones containing ( Z p , +) ). 1/11

( F p , F q ) -linear closed clonoids Proposition The intersection of ( F p , F q ) -linear closed clonoids is again a ( F p , F q ) -linear closed clonoid. Definition Let K be subset of n -ary function from F q to F p and A the set of all the ( F p , F q ) - linear closed clonoids. We define the ( F p , F q ) -linear closed clonoid generated by K as: C ( p,q ) ( K ) = � C C∈B where B = {C|C ∈ A , K ⊆ C} . 1/11

Definition Let f be an n -ary function from a group G 1 to a group G 2 . We say that f is 0 -preserving if: f (0 G 1 , ..., 0 G 1 ) = 0 G 2 . 2/11

Definition Let f be an n -ary function from a group G 1 to a group G 2 . We say that f is 0 -preserving if: f (0 G 1 , ..., 0 G 1 ) = 0 G 2 . Remark Let p and q be prime numbers. The 0 -preserving functions from F q to F p form a ( F p , F q ) -linear closed clonoid. 2/11

Other examples (1) The constant functions. 3/11

Other examples (1) The constant functions. (2) All the functions. 3/11

Other examples (1) The constant functions. (2) All the functions. Definition Let f be a function from F n q to F p . The function f is a star function if and only if for every vector w ∈ F n q there exists k ∈ F p such that for every λ ∈ F q − { 0 } : f ( λ w ) = k . 3/11

Other examples (1) The constant functions. (2) All the functions. Definition Let f be a function from F n q to F p . The function f is a star function if and only if for every vector w ∈ F n q there exists k ∈ F p such that for every λ ∈ F q − { 0 } : f ( λ w ) = k . (3) The star functions (functions constant on rays from the origin, but not in the origin). 3/11

A characterization Theorem (SF) Let p and q be powers of different primes. Then every ( F p , F q ) -linear closed clonoid C is generated by its unary functions. Thus C = C ( p,q ) ( C [1] ) . 4/11

A characterization Theorem (SF) Let p and q be powers of different primes. Then every ( F p , F q ) -linear closed clonoid C is generated by its unary functions. Thus C = C ( p,q ) ( C [1] ) . Corollary Let p and q be two distinct prime numbers. Then every ( F p , F q ) -linear closed clonoid has a set of finitely many unary functions as generators. Hence there are only finitely many distinct ( F p , F q ) -linear closed clonoids. 4/11

Example Matrix representation of a function f : Z 2 5 �→ Z 11 f ( i, j ) = a ij where ( a ij ) ∈ Z 5 × 5 11 4/11

Example Matrix representation of a function f : Z 2 5 �→ Z 11 f ( i, j ) = a ij where ( a ij ) ∈ Z 5 × 5 11   0 0 0 0 a 4 0 0 0 0  a 3      0 0 a 2 0 0     0 0 0 0 a 1     0 0 0 0 0 4/11

Example Matrix representation of a function f : Z 2 5 �→ Z 11 f ( i, j ) = a ij where ( a ij ) ∈ Z 5 × 5 11   0 0 0 0 a 4 0 0 0 0  a 3      0 0 a 2 0 0     0 0 0 0 a 1     0 0 0 0 0 The function f 1 : Z 5 �→ Z 11 defined as f 1 (0) = 0 , f 1 ( i ) = a i for i = 1 , . . . , 4 is in C ( { f } ) . 4/11

Example Matrix representation of a function f : Z 2 5 �→ Z 11 f ( i, j ) = a ij where ( a ij ) ∈ Z 5 × 5 11   0 0 0 0 a 4 0 0 0 0  a 3      0 0 a 2 0 0     0 0 0 0 a 1     0 0 0 0 0 The function f 1 : Z 5 �→ Z 11 defined as f 1 (0) = 0 , f 1 ( i ) = a i for i = 1 , . . . , 4 is in C ( { f } ) . How to generate f from f 1 in a ( F p , F q ) -linear closed clonoid? 4/11

Example Let us define the function s : Z 2 5 �→ Z 11 : s ( x, y ) = f 1 ( y )   a 4 a 4 a 4 a 4 a 4  a 3 a 3 a 3 a 3 a 3      a 2 a 2 a 2 a 2 a 2     a 1 a 1 a 1 a 1 a 1     0 0 0 0 0 4/11

Example Let us define the function s : Z 2 5 �→ Z 11 : s ( x, y ) = f 1 ( y ) + f 1 ( y − x )   2 a 4 a 3 + a 4 a 2 + a 4 a 1 + a 4 a 4  2 a 3 a 2 + a 3 a 1 + a 3 a 3 a 3 + a 4      2 a 2 a 1 + a 2 a 2 + a 4 a 2 + a 3 a 2     2 a 1 a 1 a 1 + a 4 a 1 + a 3 a 1 + a 2     0 a 4 a 3 a 2 a 1 4/11

Example Let us define the function s : Z 2 5 �→ Z 11 : s ( x, y ) = f 1 ( y ) + f 1 ( y − x ) + f 1 ( y − 2 x )   3 a 4 a 2 + a 3 + a 4 a 2 + a 4 a 1 + a 3 + a 4 a 1 + a 4  3 a 3 a 1 + a 2 + a 3 a 1 + a 3 + a 4 a 2 + a 3 a 3 + a 4      3 a 2 a 1 + a 2 a 2 + a 3 a 1 + a 2 + a 4 a 2 + a 3 + a 4     3 a 1 a 1 + a 4 a 1 + a 2 + a 4 a 1 + a 3 a 1 + a 2 + a 3     0 a 3 + a 4 a 1 + a 3 a 2 + a 4 a 1 + a 2 4/11

Example Let us define the function s : Z 2 5 �→ Z 11 : s ( x, y ) = f 1 ( y ) + f 1 ( y − x ) + f 1 ( y − 2 x ) + f 1 ( y − 3 x ) + f 1 ( y − 4 x )   5 a 4 λ λ λ λ 5 a 3  λ λ λ λ      5 a 2 λ λ λ λ     5 a 1 λ λ λ λ     0 λ λ λ λ where λ = a 1 + a 2 + a 3 + a 4 4/11

Example Let us define the function s : Z 2 5 �→ Z 11 : q − 1 � s ( x, y ) = f 1 ( y ) + f 1 ( y − x ) + f 1 ( y − 2 x ) + f 1 ( y − 3 x ) + f 1 ( y − 4 x ) − f 1 ( kx ) k =1   5 a 4 0 0 0 0  5 a 3 0 0 0 0      5 a 2 0 0 0 0     5 a 1 0 0 0 0     0 0 0 0 0 4/11

Example Let n ∈ N be s.t n ∗ 5 ≡ 11 1 . Hence n ∗ s is the function:   a 4 0 0 0 0 0 0 0 0  a 3      a 2 0 0 0 0     a 1 0 0 0 0     0 0 0 0 0 4/11

Example for all x, y ∈ Z 5 : f ( x, y ) = n ∗ s ( x − y, y )   0 0 0 0 a 4  0 0 0 a 3 0      0 0 0 0 a 2     0 a 1 0 0 0     0 0 0 0 0 4/11

Definition Let F q and F p be finite fields and let f : F q → F p be a unary function. Let α be a generator of the multiplicative subgroup F × q of F q . We define the α -vector encoding of f as the vector v ∈ F q p such that: = f ( α i ) v i +1 for 0 ≤ i ≤ q − 2 , v 0 = f (0) . 5/11

How to describe the unary functions Proposition Let C be a ( F p , F q ) -linear closed clonoid, and let α be a generator of the multi- plicative subgroup F × q of F q . Then the set S of all the α -vector encodings of unary functions in C is a subspace of F q p and it satisfies: ( x 0 , x k , x k +1 , . . . , x q − 1 , x 1 , . . . , x k − 1 ) ∈ S (1) and ( x 0 , . . . , x 0 ) ∈ S (2) for all ( x 0 , . . . , x q − 1 ) ∈ S and k ∈ { 1 , . . . , q − 1 } . 6/11

A characterization We denote by A ( p, q ) and C ( p, q ) respectively the linear transformations of F q p defined as: A ( p, q )(( v 0 , . . . , v q − 1 )) = ( v 0 , v k , v k +1 , . . . , x q − 1 , v 1 , . . . , v k − 1 ) C ( p, q )(( v 0 , . . . , v q − 1 )) = ( v 0 , . . . , v 0 ) 7/11

A characterization Definition An invariant subspace of a linear operator T on some vector space V is a subspace W of V that is preserved by T ; that is, T ( W ) ⊆ W . 8/11

A characterization Definition An invariant subspace of a linear operator T on some vector space V is a subspace W of V that is preserved by T ; that is, T ( W ) ⊆ W . Theorem (SF) Let p and q be powers of distinct prime numbers. Then the lattice of all ( F p , F q ) - linear closed clonoids L ( p, q ) is isomorphic to the lattice L ( A ( p, q ) , C ( p, q )) of all the ( A ( p, q ) , C ( p, q )) -invariant subspaces of F q p . 8/11

Lattice of the ( F p , F q ) -linear closed clonoids s ❍ 1 ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ s ❍ 0 P ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ s ❍ ❍ C ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ s { 0 } 9/11

Lattice of the ( F p , F q ) -linear closed clonoids Let us denote by 2 the two-elements chain and, in general, by C k the chain with k elements. 10/11