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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Algebra and logic of discrete paths 1 Luigi Santocanale LIS (team LIRICA), Aix-Marseille Universit e, France S

  1. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Heyting implication of paths In every finite distributive lattice  ⊤ , x ≤ y , �  { z | z ∧ x ≤ y } = ∗  x → y =   y , y < x ,   * : if the lattice is a chain. We can compute π 1 → π 2 as follows: 7/33

  2. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Heyting implication of paths In every finite distributive lattice  ⊤ , x ≤ y , �  { z | z ∧ x ≤ y } = ∗  x → y =   y , y < x ,   * : if the lattice is a chain. We can compute π 1 → π 2 as follows: 7/33

  3. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Heyting implication of paths In every finite distributive lattice  ⊤ , x ≤ y , �  { z | z ∧ x ≤ y } = ∗  x → y =   y , y < x ,   * : if the lattice is a chain. We can compute π 1 → π 2 as follows: 7/33

  4. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The equational theory of the P ( n , m ) ? Problem: characterize the equational theory of the P ( n , m ) . That is : what are the IPC (Intuitionistic Propositonal Calculus) formulas that are true in this kind of models? 8/33

  5. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The equational theory of the P ( n , m ) ? Problem: characterize the equational theory of the P ( n , m ) . That is : what are the IPC (Intuitionistic Propositonal Calculus) formulas that are true in this kind of models? An example, Gabbay/De Jong formula(s): for each n , m , � � � P ( n , m ) | = (( p i → p j ) → p j ) . i = 1 , 2 , 3 j � i j � i 8/33

  6. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Plan Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions 9/33

  7. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Composition of paths/words? SLatt is a category, can we characterize composition? ≃ SLatt ([ n ] + , [ m ] + ) × SLatt ([ m ] + , [ k ] + ) P( n , m ) × P( m , k ) ? ◦ ⊗ ≃ SLatt ([ n ] + , [ k ] + ) P( n , k ) 10/33

  8. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The product ⊗ on binary words as synchronisation By example: xyxyxxy ∈ P x , y ( 4 , 3 ) and yzzyyz ∈ P y , z ( 3 , 3 ) . 11/33

  9. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The product ⊗ on binary words as synchronisation By example: xyxyxxy ∈ P x , y ( 4 , 3 ) and yzzyyz ∈ P y , z ( 3 , 3 ) . y y y x x xx y y y zz z 11/33

  10. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The product ⊗ on binary words as synchronisation By example: xyxyxxy ∈ P x , y ( 4 , 3 ) and yzzyyz ∈ P y , z ( 3 , 3 ) . y y y x x xx y y y zz z 11/33

  11. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The product ⊗ on binary words as synchronisation By example: xyxyxxy ∈ P x , y ( 4 , 3 ) and yzzyyz ∈ P y , z ( 3 , 3 ) . y y y x x xx x xzz xx z y y y zz z 11/33

  12. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The product ⊗ on binary words as synchronisation By example: xyxyxxy ∈ P x , y ( 4 , 3 ) and yzzyyz ∈ P y , z ( 3 , 3 ) . y y y x xzz xx z y y y 11/33

  13. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The product ⊗ on binary words as synchronisation By example: xyxyxxy ∈ P x , y ( 4 , 3 ) and yzzyyz ∈ P y , z ( 3 , 3 ) . x xzz xx z 11/33

  14. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The product ⊗ on binary words as synchronisation By example: xyxyxxy ∈ P x , y ( 4 , 3 ) and yzzyyz ∈ P y , z ( 3 , 3 ) . x xzz xx z xyxyxxy ⊗ yzzyyz = xxzzxxz ∈ P x , z ( 4 , 3 ) . 11/33

  15. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The category P of discrete words/paths • Objects: natural numbers 0 , 1 , . . . , n , . . . • An arrow from n to m is a word w ∈ P ( n , m ) , • Composition is ⊗ (up to renaming of letters). 12/33

  16. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations Given f ∈ SLatt ([ n ] + , [ k ] + ) and m ≥ 0, how many g ∈ SLatt ([ n ] + , [ m ] + ) and h ∈ SLatt ([ m ] + , [ k ] + ) are there such that f = h ◦ g ? [ m ] + g h [ n ] + [ k ] + f 13/33

  17. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. 14/33

  18. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzzxzzzxx 14/33

  19. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzz | xzzz | xx 14/33

  20. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzz | xz | zz | x | x 14/33

  21. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzz | xz | zz | x | x yielding xyxyyxyx zzyzyzzyy 14/33

  22. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzzxz | zzx | x yielding xyxyyxyx zzyzyzzyy � n + k + m − i � n + k 14/33

  23. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzzxz | zzx | x yielding xyxyyxyx zzyzyzzyy � n + k + m − i �� n �� k � n + k i i 14/33

  24. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzzxz | zzx | x yielding xyxyyxyx zzyzyzzyy m � n + m �� m + k � � n + k + m − i �� n �� k � � = n + k n k i i i = 0 14/33

  25. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting factorizations by NE-turns Guess right places where to put m vertical bars. Here xzzxzzzxx has i = 2 NE-turns, we want to insert m = 4 bars. xzzxz | zzx | x yielding xyxyyxyx zzyzyzzyy Worpitzky style identity (Dzhumadil` daev, 2011): m � n + m �� m + k � � n + k + m − i �� n �� k � � = n + k n k i i i = 0 14/33

  26. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Plan Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions 15/33

  27. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions P ( n , n ) as a quantale • For each n , m P ( n , m ) is a poset. • Actually, it is a (complete and distributive) lattice. • ⊗ distributes with suprema: � � � w i ⊗ u j = w i ⊗ u j . i ∈ I j ∈ J ( i , j ) ∈ I × J • We have � � w ⊸ u := { v | w ⊗ v ≤ u } , u � w := { v | v ⊗ w ≤ u } . 16/33

  28. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions P ( n , n ) as a quantale • For each n , m P ( n , m ) is a poset. • Actually, it is a (complete and distributive) lattice. • ⊗ distributes with suprema: � � � w i ⊗ u j = w i ⊗ u j . i ∈ I j ∈ J ( i , j ) ∈ I × J • We have � � w ⊸ u := { v | w ⊗ v ≤ u } , u � w := { v | v ⊗ w ≤ u } . A quantale is a monoid object in SLatt (a model of non-commutative intuitionistic propositional linear logic). 16/33

  29. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions P ( n , n ) as a quantale • For each n , m P ( n , m ) is a poset. • Actually, it is a (complete and distributive) lattice. • ⊗ distributes with suprema: � � � w i ⊗ u j = w i ⊗ u j . i ∈ I j ∈ J ( i , j ) ∈ I × J • We have � � w ⊸ u := { v | w ⊗ v ≤ u } , u � w := { v | v ⊗ w ≤ u } . A quantale is a monoid object in SLatt (a model of non-commutative intuitionistic propositional linear logic). Proposition For each n ≥ 1, P ( n , n ) is a quantale. 16/33

  30. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Quantales and residuated lattices Not exactly the same thing, but strictly related. A residuated lattice is an algebra for the signature ⊤ , ∧ , ⊥ , ∨ , 1 , ⊗ , ⊸ , � , satisfying a bunch of axioms. 17/33

  31. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Reflection as linear negation y x x y y x x y y x x y y x x y y x 18/33

  32. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Reflection as linear negation y x x y y x x y y x x y y x x y y x Reflection (exchange of x s and y s) yields a linear negation: ( · ) ⋆ : P ( n , m ) − → P ( m , n ) − − − such that u ⊗ z ⋆ ≤ w ⋆ z ⋆ ⊗ w ≤ u ⋆ . w ⊗ u ≤ z iff iff 18/33

  33. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The dual product ⊕ Recall: xyxyxxy ⊗ yzzyyz = x | xzz | xx | z . 19/33

  34. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The dual product ⊕ Recall: xyxyxxy ⊗ yzzyyz = x | xzz | xx | z . We can also define xyxyxxy ⊕ yzzyyz := x | zzx | xx | z . 19/33

  35. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions The dual product ⊕ Recall: xyxyxxy ⊗ yzzyyz = x | xzz | xx | z . We can also define xyxyxxy ⊕ yzzyyz := x | zzx | xx | z . Then, we have w ⊕ u = ( u ⋆ ⊗ w ⋆ ) ⋆ . 19/33

  36. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions P ( n , n ) as a Girard quantale A Girard quantale (or involutive residuated lattice ) is quantale coming with a “good” involution . . . 20/33

  37. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions P ( n , n ) as a Girard quantale A Girard quantale (or involutive residuated lattice ) is quantale coming with a “good” involution . . . That is, it is • a model of non-commutative classical cyclic propositional linear logic, or • a non-symmetric ∗ -autonomous poset. 20/33

  38. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions P ( n , n ) as a Girard quantale A Girard quantale (or involutive residuated lattice ) is quantale coming with a “good” involution . . . That is, it is • a model of non-commutative classical cyclic propositional linear logic, or • a non-symmetric ∗ -autonomous poset. Proposition For each n ≥ 1, P ( n , n ) is a Girard quantale. 20/33

  39. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Plan Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions 21/33

  40. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Why idempotents? • As in standard algebra, idempotents play an important role. • Conguences of residuated lattices: analogous to normal subgroups of a group. 22/33

  41. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Why idempotents? • As in standard algebra, idempotents play an important role. • Conguences of residuated lattices: analogous to normal subgroups of a group. • f is idempotent : f ◦ f = f , • f is central : f ◦ g = g ◦ f , for each g . 22/33

  42. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Why idempotents? • As in standard algebra, idempotents play an important role. • Conguences of residuated lattices: analogous to normal subgroups of a group. • f is idempotent : f ◦ f = f , • f is central : f ◦ g = g ◦ f , for each g . Lemma If L is a finite residuated lattice, then the central idempotent elements below the unit are in bijective correspondence with the congruences. 22/33

  43. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Why idempotents? • As in standard algebra, idempotents play an important role. • Conguences of residuated lattices: analogous to normal subgroups of a group. • f is idempotent : f ◦ f = f , • f is central : f ◦ g = g ◦ f , for each g . Lemma If L is a finite residuated lattice, then the central idempotent elements below the unit are in bijective correspondence with the congruences. Proposition For each n , P ( n , n ) is simple (=no nontrivial congruences). For each complete lattice L , SLatt ( L , L ) is simple. 22/33

  44. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Idempotents as emmentalers Definition Let L be a complete lattice. An emmentaler of L is a collection { [ y i , x i ] | i ∈ I } of pairwise disjoint intervals of L such that • { y i | i ∈ I } closed under infs, • { x i | i ∈ I } closed under sups. Proposition Let • L be a complete lattice, • f : L − → L sup-preserving and idempotent, • g be the right-adjoint of f . Then { [ f ( x ) , g ( f ( x ))] | x ∈ L } is an emmentaler of L . This sets up a bijective correspondence between idempotents and emmentalers. 23/33

  45. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n 24/33

  46. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n 24/33

  47. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n x 2 y 2 x 1 y 1 x 0 y 0 24/33

  48. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n x 2 y 2 x 1 y 1 x 0 y 0 24/33

  49. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n x 2 y 2 x 1 y 1 x 0 y 0 24/33

  50. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n x 2 y 2 x 1 y 1 x 0 y 0 24/33

  51. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n x 2 y 2 x 1 y 1 x 0 y 0 24/33

  52. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n 24/33

  53. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n 24/33

  54. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Emmentalers of [ n ] + as (upper) zigzags An emmentaler becomes an alternating sequence 0 = y 0 ≤ x 0 < y 1 ≤ x 1 < . . . y k ≤ x k = n Definition A path w is an upper zigzag if every NE-turn is above y = x + 1 2 , every EN-turn is below this line. 24/33

  55. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting idempotents P n := P ( n , n ) , O n := { f : [ n ] − → [ n ] | f monotone } ≃ { w ∈ P n | height w ( 1 ) ≥ 1 } . 25/33

  56. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting idempotents P n := P ( n , n ) , O n := { f : [ n ] − → [ n ] | f monotone } ≃ { w ∈ P n | height w ( 1 ) ≥ 1 } . Howie 1971, Laradji and Umar 2006: | E ( O n ) | = Fib 2 n , |{ f ∈ E ( O n ) | f ( n ) = n }| = Fib 2 n − 1 . 25/33

  57. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting idempotents P n := P ( n , n ) , O n := { f : [ n ] − → [ n ] | f monotone } ≃ { w ∈ P n | height w ( 1 ) ≥ 1 } . Howie 1971, Laradji and Umar 2006: | E ( O n ) | = Fib 2 n , |{ f ∈ E ( O n ) | f ( n ) = n }| = Fib 2 n − 1 . Combinatorial interpretation/bijective proof: Proposition • The number of upper zigzags in P n equals Fib 2 n + 1 . • The number of upper zigzags in P n with y as a first step equals Fib 2 n . 25/33

  58. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Fibonacci sequences Recursive formulas: Fib 2 n + 2 = Fib 2 n + 1 + Fib 2 n , Fib 2 n + 3 = Fib 2 n + 2 + Fib 2 n + 1 . 26/33

  59. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Fibonacci sequences Recursive formulas: Fib 2 n + 2 = Fib 2 n + 1 + Fib 2 n , Fib 2 n + 3 = Fib 2 n + 2 + Fib 2 n + 1 . Whence we need to show that: | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 26/33

  60. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, I | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 27/33

  61. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, I | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 27/33

  62. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, I | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 27/33

  63. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, I | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 27/33

  64. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, II | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 28/33

  65. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, II | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 28/33

  66. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, II | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | , | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 28/33

  67. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, III | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 29/33

  68. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, III | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 29/33

  69. Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions Counting upper zigzags, III | E ( O n + 1 ) | = | E ( P n ) | + | E ( O n ) | | E ( P n + 1 ) | = | E ( O n + 1 ) | + | E ( P n ) | . 29/33

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