[PPT] - Plan Discrete paths as Heyting algebras Discrete paths as PowerPoint Presentation

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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 paths1

Luigi Santocanale LIS (team LIRICA), Aix-Marseille Universit´ e, France S´ eminaire tutor´ e, M2 IMD, November 5, 2020

1Based on L.S. WORDS 2019 Thanks to: M. J. Gouveia (ULisboa), S. Brlek (UQAM), A. Joyal (UQAM), C. Muresan (UCagliari, UBucarest) 1/33

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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

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Binary words as discrete paths

Let P(n, m) := { w ∈ { x, y }∗ | |w|x = n and |w|y = m } .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Binary words as discrete paths

Let P(n, m) := { w ∈ { x, y }∗ | |w|x = n and |w|y = m } . A word w ∈ P(n, m) is a discrete path from (0, 0) to (n, m).

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Binary words as discrete paths

Let P(n, m) := { w ∈ { x, y }∗ | |w|x = n and |w|y = m } . A word w ∈ P(n, m) is a discrete path from (0, 0) to (n, m).

0, 0 5, 4

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Binary words as discrete paths

Let P(n, m) := { w ∈ { x, y }∗ | |w|x = n and |w|y = m } . A word w ∈ P(n, m) is a discrete path from (0, 0) to (n, m).

0, 0 5, 4

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Binary words as discrete paths

Let P(n, m) := { w ∈ { x, y }∗ | |w|x = n and |w|y = m } . A word w ∈ P(n, m) is a discrete path from (0, 0) to (n, m).

0, 0 5, 4 x y y x x x y y x

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

π1 ≤ π2

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

π3 = π1 ∨ π2

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

π3 = π1 ∧ π2

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is a lattice

Remark

P(n, m) is also called a binomial lattice since |P(n, m)| =

n+m

m

.

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

The fundamental bijection

Notation: [n] := { 1, . . . , n } , [n]+ := { 0, 1, . . . , n } .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

The fundamental bijection

Notation: [n] := { 1, . . . , n } , [n]+ := { 0, 1, . . . , n } . Then: P(n, m)

height

≃

− − − − − → { f : [n] − → [m]+ | f is monotone } ≃ { f : [n]+ − → [m]+ | f is monotone, f(0) = 0 } = { f : [n]+ − → [m]+ | f is preserves sups } = SLatt( [n]+, [m]+ ) .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

The fundamental bijection

Notation: [n] := { 1, . . . , n } , [n]+ := { 0, 1, . . . , n } . Then: P(n, m)

height

≃

− − − − − → { f : [n] − → [m]+ | f is monotone } ≃ { f : [n]+ − → [m]+ | f is monotone, f(0) = 0 } = { f : [n]+ − → [m]+ | f is preserves sups } = SLatt( [n]+, [m]+ ) . heightπ(1) = heightπ(2) = heightπ(3) = 1 heightπ(4) = 2 heightπ(5) = 4

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is an Heyting algebra

Proposition

P(n, m) is a distributive lattice. Car (f ∨ g)(i) = max(f(i), g(i)) , (f ∧ g)(i) = min(f(i), g(i)) .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is an Heyting algebra

Proposition

P(n, m) is a distributive lattice. Car (f ∨ g)(i) = max(f(i), g(i)) , (f ∧ g)(i) = min(f(i), g(i)) . Heyting algebra: algebraic model of propositional intuitionisitic logic. Signature: ⊤, ∧, ⊥, ∨, →

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

P(n, m) is an Heyting algebra

Proposition

P(n, m) is a distributive lattice. Car (f ∨ g)(i) = max(f(i), g(i)) , (f ∧ g)(i) = min(f(i), g(i)) . Heyting algebra: algebraic model of propositional intuitionisitic logic. Signature: ⊤, ∧, ⊥, ∨, →

Proposition

P(n, m) is an Heyting algebra. Car every finite distributive lattice is an Heyting algebra.

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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 }

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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.

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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 π1 → π2 as follows:

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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 π1 → π2 as follows:

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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 π1 → π2 as follows:

7/33

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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 π1 → π2 as follows:

7/33

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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 π1 → π2 as follows:

7/33

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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 π1 → π2 as follows:

7/33

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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 π1 → π2 as follows:

7/33

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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 π1 → π2 as follows:

7/33

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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 π1 → π2 as follows:

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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?

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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) |=

i=1,2,3

((pi →

ji

pj) →

ji

pj) .

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Plan

Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

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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)

≃

⊗

? ≃

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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).

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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 y x y xx y y zz y y z

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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 y x y xx y y zz y y z

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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 y x y xx y x xzz xx z y zz y y z

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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

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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

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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) .

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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).

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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?

[n]+ [m]+ [k]+

f g h

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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.

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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

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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

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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

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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

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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

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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 i k i

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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 + m n m + k k

=

m

i=0

n + k + m − i n + k n i k i

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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): n + m n m + k k

=

m

i=0

n + k + m − i n + k n i k i

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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

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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:
i∈I

wi ⊗

j∈J

uj =

(i,j)∈I×J

wi ⊗ uj .

We have

w ⊸ u :=

{ v | w ⊗ v ≤ u } ,

u w :=

{ v | v ⊗ w ≤ u } .

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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:
i∈I

wi ⊗

j∈J

uj =

(i,j)∈I×J

wi ⊗ uj .

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).

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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:
i∈I

wi ⊗

j∈J

uj =

(i,j)∈I×J

wi ⊗ uj .

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.

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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.

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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 x x y y x x y y x y

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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 x x y y x x y y x y

Reflection (exchange of xs and ys) yields a linear negation:

( · )⋆ : P(n, m) − − − − → P(m, n)

such that w ⊗ u ≤ z iff u ⊗ z⋆ ≤ w⋆ iff z⋆ ⊗ w ≤ u⋆ .

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The dual product ⊕

Recall: xyxyxxy ⊗ yzzyyz =x|xzz|xx|z .

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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 .

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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⋆)⋆ .

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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 . . .

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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.

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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.

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Plan

Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

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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.

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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.

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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.

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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.

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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 { [yi, xi] | i ∈ I } of pairwise disjoint intervals of L such that

{ yi | i ∈ I } closed under infs,
{ xi | 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.

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

y0 x0 y1 x1 y2 x2

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

y0 x0 y1 x1 y2 x2

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

y0 x0 y1 x1 y2 x2

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

y0 x0 y1 x1 y2 x2

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

y0 x0 y1 x1 y2 x2

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = n

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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 = y0 ≤ x0 < y1≤ x1 < . . . yk ≤ xk = 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.

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting idempotents

Pn := P(n, n) , On := { f : [n] − → [n] | f monotone } ≃ { w ∈ Pn | heightw(1) ≥ 1 } .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting idempotents

Pn := P(n, n) , On := { f : [n] − → [n] | f monotone } ≃ { w ∈ Pn | heightw(1) ≥ 1 } .

Howie 1971, Laradji and Umar 2006:

|E(On)| = Fib2n , |{ f ∈ E(On) | f(n) = n }| = Fib2n−1 .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting idempotents

Pn := P(n, n) , On := { f : [n] − → [n] | f monotone } ≃ { w ∈ Pn | heightw(1) ≥ 1 } .

Howie 1971, Laradji and Umar 2006:

|E(On)| = Fib2n , |{ f ∈ E(On) | f(n) = n }| = Fib2n−1 .

Combinatorial interpretation/bijective proof:

Proposition

The number of upper zigzags in Pn equals Fib2n+1.
The number of upper zigzags in Pn with y as a first step

equals Fib2n.

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Fibonacci sequences

Recursive formulas: Fib2n+2 = Fib2n+1 + Fib2n , Fib2n+3 = Fib2n+2 + Fib2n+1 .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Fibonacci sequences

Recursive formulas: Fib2n+2 = Fib2n+1 + Fib2n , Fib2n+3 = Fib2n+2 + Fib2n+1 . Whence we need to show that:

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, I

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, I

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, I

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, I

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, II

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, II

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, II

|E(On+1)| = |E(Pn)| + |E(On)| , |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, III

|E(On+1)| = |E(Pn)| + |E(On)| |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, III

|E(On+1)| = |E(Pn)| + |E(On)| |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, III

|E(On+1)| = |E(Pn)| + |E(On)| |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, III

|E(On+1)| = |E(Pn)| + |E(On)| |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Counting upper zigzags, III

|E(On+1)| = |E(Pn)| + |E(On)| |E(Pn+1)| = |E(On+1)| + |E(Pn)| .

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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

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Open problems

A topic across different domains:

discrete geometry, logic, algebra, (enumerative) combinatorics.

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Open problems

A topic across different domains:

discrete geometry, logic, algebra, (enumerative) combinatorics.

Category and order theory:

axiomatization of the category SLatt,
quantaloids in autonomous categories other than SLatt.

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Open problems

A topic across different domains:

discrete geometry, logic, algebra, (enumerative) combinatorics.

Category and order theory:

axiomatization of the category SLatt,
quantaloids in autonomous categories other than SLatt.

Algebraic logic:

equational theories of the P(n, m) as Heyting algebras,
equational theories of the P(n, n) as residuated lattices.

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Open problems

A topic across different domains:

discrete geometry, logic, algebra, (enumerative) combinatorics.

Category and order theory:

axiomatization of the category SLatt,
quantaloids in autonomous categories other than SLatt.

Algebraic logic:

equational theories of the P(n, m) as Heyting algebras,
equational theories of the P(n, n) as residuated lattices.

Combinatorics on words:

lattice theoretic approach to 3d Christoffel words,
discrete approximations of 3 (or >3) dimensional curves.

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Discrete paths as Heyting algebras Discrete paths as categories Discrete paths as quantales Counting idempotents Conclusions

Thank you !

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