Associativity, preassociativity, and string functions Erkko Lehtonen - PowerPoint PPT Presentation

Associativity, preassociativity, and string functions Erkko Lehtonen Centro de Álgebra da Universidade de Lisboa Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa erkko@campus.ul.pt joint work with Jean-Luc Marichal (University of Luxembourg) Bruno Teheux (University of Luxembourg) AAA88 Warsaw, 20–22 June 2014 E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 1 / 18

Associative functions Let X be a nonempty set. F : X 2 → X is associative if F ( F ( a , b ) , c ) = F ( a , F ( b , c )) . on X = R F ( a , b ) = a + b Examples: on a semilattice X F ( a , b ) = a ∧ b E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 2 / 18

Associative functions F ( F ( a , b ) , c ) = F ( a , F ( b , c )) . E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 3 / 18

Associative functions Extension to 3 -ary functions F ( F ( a , b ) , c ) = F ( a , F ( b , c )) . F ( F ( a , b , c ) , d , e ) = F ( a , F ( b , c , d ) , e ) = F ( a , b , F ( c , d , e )) E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 3 / 18

Associative functions Extension to n -ary functions F ( F ( a , b ) , c ) = F ( a , F ( b , c )) . F ( F ( a , b , c ) , d , e ) = F ( a , F ( b , c , d ) , e ) = F ( a , b , F ( c , d , e )) F : X n → X is associative if F ( F ( a 1 , . . . , a n ) , a n + 1 , . . . , a 2 n − 1 ) = F ( a 1 , F ( a 2 , . . . , a n + 1 ) , a n + 2 , . . . , a 2 n − 1 ) = · · · = F ( a 1 , . . . , a i , F ( a i + 1 , . . . , a i + n ) , a i + n + 1 , . . . , a 2 n − 1 ) = · · · = F ( a 1 , . . . , a n − 1 , F ( a n , . . . , a 2 n − 1 )) . E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 3 / 18

Associative functions with indefinite arity Let X ∗ = � X n . n ∈ N Disclaimer: When we write F : X ∗ → X , we mean a map X n → X � F : n ≥ 1 that is extended into a map X ∗ → X ∪ { ε } by setting F ( ε ) = ε. E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 4 / 18

Associative functions with indefinite arity Let X ∗ = � X n . n ∈ N F : X ∗ → X is associative if F ( x 1 , . . . , x p , y 1 , . . . , y q , z 1 , . . . , z r ) = F ( x 1 , . . . , x p , F ( y 1 , . . . , y q ) , z 1 , . . . , z r ) . on X = R F ( x 1 , . . . , x n ) = x 1 + · · · + x n Examples: on a semilattice X F ( x 1 , . . . , x n ) = x 1 ∧ · · · ∧ x n E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 4 / 18

Notation We regard n -tuples x ∈ X n as n-strings over X . 0-string: ε 1-strings: x , y , z , . . . n -strings: x , y , z , . . . X ∗ is endowed with concatenation. x y z ∈ X n + 1 + m x ∈ X n , y ∈ X , z ∈ X m Example: = ⇒ | x | = length of x F ( x ) = ε ⇐ ⇒ x = ε E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 5 / 18

Associative functions with indefinite arity F : X ∗ → X is associative if ∀ x , y , z ∈ X ∗ . F ( xyz ) = F ( x F ( y ) z ) E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 6 / 18

Associative functions with indefinite arity F : X ∗ → X is associative if ∀ x , y , z ∈ X ∗ . F ( xyz ) = F ( x F ( y ) z ) Equivalent definitions ∀ x , y , z ∈ X ∗ . F ( F ( xy ) z ) = F ( x F ( yz )) ∀ x , y ∈ X ∗ . F ( xy ) = F ( F ( xy )) E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 6 / 18

Associative functions with indefinite arity F : X ∗ → X is associative if ∀ x , y , z ∈ X ∗ . F ( xyz ) = F ( x F ( y ) z ) Theorem (Marichal, Teheux) We can assume that | xz | ≤ 1 in the definition above. That is, F : X ∗ → X is associative if and only if F ( y ) = F ( F ( y )) F ( x y ) = F ( xF ( y )) F ( y z ) = F ( F ( y ) z ) E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 6 / 18

Associative functions with indefinite arity F n = F | X n n ≥ 2 F n ( x 1 · · · x n ) = F 2 ( F n − 1 ( x 1 · · · x n − 1 ) x n ) Thus, associative functions are completely determined by their unary and binary parts. Theorem (Marichal) Let F : X ∗ → X and G : X ∗ → X be two associative functions such that F 1 = G 1 and F 2 = G 2 . Then F = G. E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 7 / 18

Preassociative functions Let Y be a nonempty set. F : X ∗ → Y is preassociative if F ( y ) = F ( y ′ ) F ( xyz ) = F ( xy ′ z ) = ⇒ and F ( x ) = F ( ε ) ⇐ ⇒ x = ε. E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 8 / 18

Preassociative functions Let Y be a nonempty set. F : X ∗ → Y is string-preassociative if F ( y ) = F ( y ′ ) F ( xyz ) = F ( xy ′ z ) . = ⇒ E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 8 / 18

Preassociative functions F : X ∗ → Y is preassociative if F ( y ) = F ( y ′ ) F ( xyz ) = F ( xy ′ z ) = ⇒ and F ( x ) = F ( ε ) ⇐ ⇒ x = ε. F ( x ) = x 2 1 + · · · + x 2 ( X = Y = R ) Examples: n ( X arbitrary, Y = N ) F ( x ) = | x | E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 9 / 18

Preassociative functions F : X ∗ → Y is preassociative if F ( y ) = F ( y ′ ) F ( xyz ) = F ( xy ′ z ) = ⇒ and F ( x ) = F ( ε ) ⇐ ⇒ x = ε. Fact: If F : X ∗ → X is associative, then it is preassociative. Proof. Suppose F ( y ) = F ( y ′ ) . Then F ( xyz ) = F ( x F ( y ) z ) = F ( x F ( y ′ ) z ) = F ( xy ′ z ) . The second condition holds by definition. E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 9 / 18

Preassociative functions F : X ∗ → Y is preassociative if F ( y ) = F ( y ′ ) F ( xyz ) = F ( xy ′ z ) = ⇒ and F ( x ) = F ( ε ) ⇐ ⇒ x = ε. Proposition (Marichal, Teheux) F : X ∗ → X is associative if and only if it is preassociative and F 1 ( F ( x )) = F ( x ) . Proof. (Necessity) Clear. (Sufficiency) We have F ( y ) = F ( F ( y )) . Hence, by preassociativity, F ( xyz ) = F ( x F ( y ) z ) . E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 9 / 18

Preassociative functions Proposition (Marichal, Teheux) If F : X ∗ → Y is preassociative, then so is the function x 1 · · · x n �→ F n ( g ( x 1 ) · · · g ( x n )) for every function g : X → X. F n ( x ) = x 6 1 + · · · + x 6 ( X = Y = R ) Example: n E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 10 / 18

Preassociative functions Proposition (Marichal, Teheux) If F : X ∗ → Y is preassociative, then so is g ◦ F for every function g : Y → Y such that g | Im F is injective. F n ( x ) = exp ( x 6 1 + · · · + x 6 ( X = Y = R ) Example: n ) E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 10 / 18

String functions A string function is a map F : X ∗ → X ∗ . F : X ∗ → X ∗ is associative if F ( xyz ) = F ( x F ( y ) z ) and F ( x ) = F ( ε ) ⇐ ⇒ x = ε. (the same formula as before!) E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 11 / 18

String functions A string function is a map F : X ∗ → X ∗ . F : X ∗ → X ∗ is string-associative if F ( xyz ) = F ( x F ( y ) z ) . (the same formula as before!) E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 11 / 18

Associative string functions Examples: identity function sorting data in alphabetical order F ( mathematics ) = aacehimmstt F ( warszawa ) = aaarswwz removing occurrences of a given letter, say, of a F ( mathematics ) = mthemtics F ( warszawa ) = wrszw removing duplicates, keeping only the first occurrence of each letter F ( mathematics ) = matheics F ( warszawa ) = warsz E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 12 / 18

Associative string functions Examples: identity function sorting data in alphabetical order F ( mathematics ) = aacehimmstt F ( warszawa ) = aaarswwz removing occurrences of a given letter, say, of a F ( mathematics ) = mthemtics string-associative not associative F ( warszawa ) = wrszw F ( aaa ) = ε = F ( ε ) removing duplicates, keeping only the first occurrence of each letter F ( mathematics ) = matheics F ( warszawa ) = warsz E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 12 / 18

Associative string functions F : X ∗ → X ∗ is string-associative if F ( xyz ) = F ( x F ( y ) z ) . Proposition Assume F : X ∗ → X ∗ satisfies F ( ε ) = ε . The following are equivalent: F is string-associative. 1 F ( x F ( y ) z ) = F ( x ′ F ( y ′ ) z ′ ) for all x , y , z , x ′ , y ′ , z ′ ∈ X ∗ such that 2 xyz = x ′ y ′ z ′ . F ( F ( xy ) z ) = F ( x F ( yz )) for all x , y , z ∈ X ∗ . 3 F ( xy ) = F ( F ( x ) F ( y )) for all x , y ∈ X ∗ . 4 E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 13 / 18

Associative string functions F : X ∗ → X ∗ is string-associative if F ( xyz ) = F ( x F ( y ) z ) . Proposition F : X ∗ → X ∗ is string-associative if and only if F ( xyz ) = F ( x F ( y ) z ) for any x , y , z ∈ X ∗ such that | xy | ≤ 1 . E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 13 / 18

Associative string functions F : X ∗ → X ∗ is string-associative if F ( xyz ) = F ( x F ( y ) z ) . Fact A string-associative function F : X ∗ → X ∗ satisfies n ≥ 1 , F ( x 1 · · · x n ) = F ( F ( x 1 · · · x n − 1 ) x n ) , or, equivalently, n ≥ 1 . F ( x 1 · · · x n ) = F ( F ( · · · F ( F ( x 1 ) x 2 ) · · · ) x n ) , E. Lehtonen (CAUL) Associativity, preassociativity, . . . AAA88 13 / 18