The finite embeddability property for some noncommutative knotted - - PowerPoint PPT Presentation

The finite embeddability property for some noncommutative knotted extensions of RL.

Riquelmi Cardona

University of Denver

BLAST 2013

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 1 / 16

Preliminaries

Finite embeddability property

A class of algebras K has the finite embeddability property (FEP) if for every A ∈ K, every finite partial subalgebra B of A can be embedded in a finite D ∈ K.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 2 / 16

Preliminaries

Finite embeddability property

A class of algebras K has the finite embeddability property (FEP) if for every A ∈ K, every finite partial subalgebra B of A can be embedded in a finite D ∈ K. A residuated lattice, is an algebra L = (L, ∧, ∨, ·, \, /, 1) such that (L, ∧, ∨) is a lattice, (L, ·, 1) is a monoid and for all a, b, c ∈ L, ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b. RL denotes the variety of residuated lattices.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 2 / 16

Knotted axioms

A (non-trivial) knotted axiom is an inequality of the form xm ≤ xn for m = n, m ≥ 1, n ≥ 0.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 3 / 16

Knotted axioms

A (non-trivial) knotted axiom is an inequality of the form xm ≤ xn for m = n, m ≥ 1, n ≥ 0. Some known examples of these include contraction x ≤ x2, mingle x2 ≤ x, and integrality x ≤ 1.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 3 / 16

Some Results

Theorem (Van Alten)

The variety of commutative residuated lattices axiomatized by a knotted axiom has the FEP.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 4 / 16

Some Results

Theorem (Van Alten)

The variety of commutative residuated lattices axiomatized by a knotted axiom has the FEP.

Theorem

The variety of residuated lattices axiomatized by xyx = x2y and a knotted axiom xm ≤ xn has the FEP.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 4 / 16

Generalization

Let’s start with xyx = x2y.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16

Generalization

Let’s start with xyx = x2y. A similar equality is xyx = yx2.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16

Generalization

Let’s start with xyx = x2y. A similar equality is xyx = yx2. The previous equalities can be represented by xyx = xa0yxa1,

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16

Generalization

Let’s start with xyx = x2y. A similar equality is xyx = yx2. The previous equalities can be represented by xyx = xa0yxa1, where a0 + a1 = 2 and a0a1 = 0.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16

Generalization

We consider the generalization xy1xy2x · · · xyrx = xa0y1xa1y2xa2 · · · xar−1yrxar , (1)

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 6 / 16

Generalization

We consider the generalization xy1xy2x · · · xyrx = xa0y1xa1y2xa2 · · · xar−1yrxar , (1) where at least one of the ai’s is equal to 0 and the sum of the ai’s is r + 1.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 6 / 16

Generalization

We consider the generalization xy1xy2x · · · xyrx = xa0y1xa1y2xa2 · · · xar−1yrxar , (1) where at least one of the ai’s is equal to 0 and the sum of the ai’s is r + 1.

Theorem

For n > m ≥ 1, r ≥ 1, the variety Vr of residuated lattices axiomatized by (1) and a knotted axiom xm ≤ xn has the FEP.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 6 / 16

Residuated frames

Let B be a finite partial subalgebra of A ∈ Vr. Consider (W , ◦, 1), the submonoid of A generated by B.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 7 / 16

Residuated frames

Let B be a finite partial subalgebra of A ∈ Vr. Consider (W , ◦, 1), the submonoid of A generated by B. We define SW to be the set of unary linear polynomial (sections) of (W , ◦, 1). Elements of SW are of the form u( ) = y ◦

• w for y, w ∈ W .

Let W ′ = SW × B, and define xN(u, b) iff u(x) ≤A b

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 7 / 16

Residuated frames

Let B be a finite partial subalgebra of A ∈ Vr. Consider (W , ◦, 1), the submonoid of A generated by B. We define SW to be the set of unary linear polynomial (sections) of (W , ◦, 1). Elements of SW are of the form u( ) = y ◦

• w for y, w ∈ W .

Let W ′ = SW × B, and define xN(u, b) iff u(x) ≤A b We define y (u, b) = {(u(y ◦ ), b)} and (u, b) y = {(u( ◦ y), b)}. The relation N is a nuclear relation, because it satisfies the condition (x ◦ y)Nz ⇔ yN(x z) ⇔ xN(z y) Then WA,B = (W , W ′, N, ◦, , , {1}) is a unital residuated frame.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 7 / 16

Galois algebra

For X ⊆ W and Y ⊆ W ′ we define X ⊲ = {b ∈ W ′ : xNb, for all x ∈ X} Y ⊳ = {a ∈ W : aNy, for all y ∈ Y }

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 8 / 16

Galois algebra

For X ⊆ W and Y ⊆ W ′ we define X ⊲ = {b ∈ W ′ : xNb, for all x ∈ X} Y ⊳ = {a ∈ W : aNy, for all y ∈ Y } γN : P(W ) → P(W ), γN(X) = X ⊲⊳, is a closure operator.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 8 / 16

Galois algebra

For X ⊆ W and Y ⊆ W ′ we define X ⊲ = {b ∈ W ′ : xNb, for all x ∈ X} Y ⊳ = {a ∈ W : aNy, for all y ∈ Y } γN : P(W ) → P(W ), γN(X) = X ⊲⊳, is a closure operator. The Galois algebra of WA,B is WA,B+ = (γN[℘(W )], ∩, ∪γN, ◦γN, \, /, γN({1})), which is a complete residuated lattice.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 8 / 16

The embedding

The map b → {(id, b)}⊳ is an embedding of the partial subalgebra B of A into W+

A,B [Galatos,Jipsen].

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 9 / 16

The embedding

The map b → {(id, b)}⊳ is an embedding of the partial subalgebra B of A into W+

A,B [Galatos,Jipsen].

Furthermore, W+

A,B and A belong to Vk and the closed sets {(u, b)}⊳ for

u ∈ SW , b ∈ B form a basis for W+

A,B.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 9 / 16

The setting

W W′ N

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 10 / 16

The setting

F W W′ h N F is a pomonoid and h is an order preserving homomorphism.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 10 / 16

The setting

F W W′ h N F is a pomonoid and h is an order preserving homomorphism. Furthermore, h is surjective and F is a well partially ordered set.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 10 / 16

Well partially ordered sets

A poset is said to be well partially ordered if it has no infinite antichains and no infinite descending chains. For instance, N, ≤ is well partially

• rdered.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 11 / 16

Well partially ordered sets

A poset is said to be well partially ordered if it has no infinite antichains and no infinite descending chains. For instance, N, ≤ is well partially

• rdered.

If P, ≤ is well partially ordered, then it is known that for each k ∈ N, Pk is well partially ordered under the direct product ordering. Furthermore, homomorphic images, finite disjoint unions, and subposets of well partially

• rdered sets are well partially ordered.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 11 / 16

Well partially ordered sets

A poset is said to be well partially ordered if it has no infinite antichains and no infinite descending chains. For instance, N, ≤ is well partially

• rdered.

If P, ≤ is well partially ordered, then it is known that for each k ∈ N, Pk is well partially ordered under the direct product ordering. Furthermore, homomorphic images, finite disjoint unions, and subposets of well partially

• rdered sets are well partially ordered.

Consider the poset P, ≤. An infinite sequence p1, p2, . . . of elements of P is called bad when i < j implies that pi ≤ pj. Note that an infinitely descending chain or antichain would be a bad sequence. A poset is well partially ordered if and only if it has no bad sequences.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 11 / 16

The proof

Assume that we have F and h satisfy the given conditions.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Lemma

For each b ∈ B, Cb, ⊇ is well partially ordered.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Lemma

For each b ∈ B, Cb, ⊇ is well partially ordered.

Proof.

Cb, ⊇ is a homomorphic image of F 2, ≤F.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Lemma

For each b ∈ B, Cb, ⊇ is well partially ordered.

Proof.

Cb, ⊇ is a homomorphic image of F 2, ≤F. Define ϕ : F 2 → Cb by ϕ(y, w) = {(h(y) ◦

• h(w), b)}⊳. ϕ is surjective.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Lemma

For each b ∈ B, Cb, ⊇ is well partially ordered.

Proof.

Cb, ⊇ is a homomorphic image of F 2, ≤F. Define ϕ : F 2 → Cb by ϕ(y, w) = {(h(y) ◦

• h(w), b)}⊳. ϕ is surjective.

Let (y1, w1), (y2, w2) ∈ F 2 such that (y1, w1) ≤F (y2, w2)

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Lemma

For each b ∈ B, Cb, ⊇ is well partially ordered.

Proof.

Cb, ⊇ is a homomorphic image of F 2, ≤F. Define ϕ : F 2 → Cb by ϕ(y, w) = {(h(y) ◦

• h(w), b)}⊳. ϕ is surjective.

Let (y1, w1), (y2, w2) ∈ F 2 such that (y1, w1) ≤F (y2, w2). For all x ∈ F, y1 ·F x ·F w1 ≤F y2 ·F x ·F w2 and h(y1) ◦ h(x) ◦ h(w1) ≤ h(y2) ◦ h(x) ◦ h(w2).

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Lemma

For each b ∈ B, Cb, ⊇ is well partially ordered.

Proof.

Cb, ⊇ is a homomorphic image of F 2, ≤F. Define ϕ : F 2 → Cb by ϕ(y, w) = {(h(y) ◦

• h(w), b)}⊳. ϕ is surjective.

Let (y1, w1), (y2, w2) ∈ F 2 such that (y1, w1) ≤F (y2, w2). For all x ∈ F, y1 ·F x ·F w1 ≤F y2 ·F x ·F w2 and h(y1) ◦ h(x) ◦ h(w1) ≤ h(y2) ◦ h(x) ◦ h(w2). Now if z ∈ {(h(y2) ◦

• h(w2), b)}⊳, then h(y2) ◦ h(x) ◦ h(w2) ≤ b. Hence

h(y1) ◦ h(x) ◦ h(w1) ≤ b and z = h(x) ∈ {(h(y1) ◦

• h(w1), b)}⊳.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Assume that we have F and h satisfy the given conditions. For each b ∈ B, define Cb = {{(u, b)}⊳ : u ∈ SW }.

Lemma

For each b ∈ B, Cb, ⊇ is well partially ordered.

Proof.

Cb, ⊇ is a homomorphic image of F 2, ≤F. Define ϕ : F 2 → Cb by ϕ(y, w) = {(h(y) ◦

• h(w), b)}⊳. ϕ is surjective.

Let (y1, w1), (y2, w2) ∈ F 2 such that (y1, w1) ≤F (y2, w2). For all x ∈ F, y1 ·F x ·F w1 ≤F y2 ·F x ·F w2 and h(y1) ◦ h(x) ◦ h(w1) ≤ h(y2) ◦ h(x) ◦ h(w2). Now if z ∈ {(h(y2) ◦

• h(w2), b)}⊳, then h(y2) ◦ h(x) ◦ h(w2) ≤ b. Hence

h(y1) ◦ h(x) ◦ h(w1) ≤ b and z = h(x) ∈ {(h(y1) ◦

• h(w1), b)}⊳. So

{(h(y1) ◦

• h(w1), b)}⊳ ⊇ {(h(y2) ◦
• h(w2), b)}⊳.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb. For each i ∈ Z+, choose wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb. For each i ∈ Z+, choose wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. If for some i < j we have wi ≤A wj, then wi ∈ {(uj, b)}⊳

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb. For each i ∈ Z+, choose wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. If for some i < j we have wi ≤A wj, then wi ∈ {(uj, b)}⊳. From {(ui+1, b)}⊳ ⊇ {(uj, b)}⊳, it follows that wi ∈ {(ui+1, b)}⊳

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb. For each i ∈ Z+, choose wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. If for some i < j we have wi ≤A wj, then wi ∈ {(uj, b)}⊳. From {(ui+1, b)}⊳ ⊇ {(uj, b)}⊳, it follows that wi ∈ {(ui+1, b)}⊳. However, this contradicts the fact that wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb. For each i ∈ Z+, choose wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. If for some i < j we have wi ≤A wj, then wi ∈ {(uj, b)}⊳. From {(ui+1, b)}⊳ ⊇ {(uj, b)}⊳, it follows that wi ∈ {(ui+1, b)}⊳. However, this contradicts the fact that wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. We conclude that i < j = ⇒ wi ≤A wj and w1, w2, . . . is a bad sequence in W

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb. For each i ∈ Z+, choose wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. If for some i < j we have wi ≤A wj, then wi ∈ {(uj, b)}⊳. From {(ui+1, b)}⊳ ⊇ {(uj, b)}⊳, it follows that wi ∈ {(ui+1, b)}⊳. However, this contradicts the fact that wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. We conclude that i < j = ⇒ wi ≤A wj and w1, w2, . . . is a bad sequence in W , which contradicts the fact that F is well partially ordered.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The proof

Cb, ⊇ has no infinite antichains or descending chains.

Lemma

Cb, ⊇ has no infinite ascending chains.

Proof.

Assume there exists an infinite chain {(u1, b)}⊳ ⊃ {(u2, b)}⊳ ⊃ . . . in Cb. For each i ∈ Z+, choose wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. If for some i < j we have wi ≤A wj, then wi ∈ {(uj, b)}⊳. From {(ui+1, b)}⊳ ⊇ {(uj, b)}⊳, it follows that wi ∈ {(ui+1, b)}⊳. However, this contradicts the fact that wi ∈ {(ui, b)}⊳ \ {(ui+1, b)}⊳. We conclude that i < j = ⇒ wi ≤A wj and w1, w2, . . . is a bad sequence in W , which contradicts the fact that F is well partially ordered. Cb is finite for every b ∈ B. Thus, there are finitely many sets {(u, b)}⊳.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 13 / 16

The pomonoid F

Need to choose a good representation to obtain a well ordered set.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 14 / 16

The pomonoid F

Need to choose a good representation to obtain a well ordered set. We represent elements as (exponents, order of vars).

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 14 / 16

The pomonoid F

Need to choose a good representation to obtain a well ordered set. We represent elements as (exponents, order of vars). For instance, in the monoid on generators {z1, z2, z3, z4, z5}, z3

5z4 1z2 3 will

be represented by ((4, 0, 2, 0, 3), 513)

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 14 / 16

The pomonoid F

Need to choose a good representation to obtain a well ordered set. We represent elements as (exponents, order of vars). For instance, in the monoid on generators {z1, z2, z3, z4, z5}, z3

5z4 1z2 3 will

be represented by ((4, 0, 2, 0, 3), 513) For that purpose we need to look at the defining equation and obtain information out of it.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 14 / 16

Example

For instance, consider the equation xy1xy2xy3xy4xy5x = x2y1y2y3x3y4y5x.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 15 / 16

Example

For instance, consider the equation xy1xy2xy3xy4xy5x = x2y1y2y3x3y4y5x. We can use it rewrite expressions like xy1xy2xy3xy4xy5xy6xy7xy8xy9x = (x2y1y2y3x3y4y5x)y6xy7xy8xy9x

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 15 / 16

Example

For instance, consider the equation xy1xy2xy3xy4xy5x = x2y1y2y3x3y4y5x. We can use it rewrite expressions like xy1xy2xy3xy4xy5xy6xy7xy8xy9x = (x2y1y2y3x3y4y5x)y6xy7xy8xy9x = x6y1y2y3x3y4y5y6y7y8y9x In general, we can gather generators together when we have enough of them.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 15 / 16

Example

For instance, consider the equation xy1xy2xy3xy4xy5x = x2y1y2y3x3y4y5x. We can use it rewrite expressions like xy1xy2xy3xy4xy5xy6xy7xy8xy9x = (x2y1y2y3x3y4y5x)y6xy7xy8xy9x = x6y1y2y3x3y4y5y6y7y8y9x = xx8y1y2y3y4y5y6y7y8y9x In general, we can gather generators together when we have enough of them.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 15 / 16

Further work

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 16 / 16

Further work

Thank you for your attention.

Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 16 / 16