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 x m ≤ x n 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 x m ≤ x n for m � = n , m ≥ 1 , n ≥ 0 . Some known examples of these include contraction x ≤ x 2 , mingle x 2 ≤ 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 = x 2 y and a knotted axiom x m ≤ x n has the FEP. Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 4 / 16
Generalization Let’s start with xyx = x 2 y . Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16
Generalization Let’s start with xyx = x 2 y . A similar equality is xyx = yx 2 . Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16
Generalization Let’s start with xyx = x 2 y . A similar equality is xyx = yx 2 . The previous equalities can be represented by xyx = x a 0 yx a 1 , Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16
Generalization Let’s start with xyx = x 2 y . A similar equality is xyx = yx 2 . The previous equalities can be represented by xyx = x a 0 yx a 1 , where a 0 + a 1 = 2 and a 0 a 1 = 0. Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 5 / 16
Generalization We consider the generalization xy 1 xy 2 x · · · xy r x = x a 0 y 1 x a 1 y 2 x a 2 · · · x a r − 1 y r x a r , (1) Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 6 / 16
Generalization We consider the generalization xy 1 xy 2 x · · · xy r x = x a 0 y 1 x a 1 y 2 x a 2 · · · x a r − 1 y r x a r , (1) where at least one of the a i ’s is equal to 0 and the sum of the a i ’s is r + 1. Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 6 / 16
Generalization We consider the generalization xy 1 xy 2 x · · · xy r x = x a 0 y 1 x a 1 y 2 x a 2 · · · x a r − 1 y r x a r , (1) where at least one of the a i ’s is equal to 0 and the sum of the a i ’s is r + 1. Theorem For n > m ≥ 1 , r ≥ 1 , the variety V r of residuated lattices axiomatized by (1) and a knotted axiom x m ≤ x n 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 ∈ V r . 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 ∈ V r . Consider ( W , ◦ , 1), the submonoid of A generated by B . We define S W to be the set of unary linear polynomial (sections) of ( W , ◦ , 1). Elements of S W are of the form u ( ) = y ◦ ◦ w for y , w ∈ W . Let W ′ = S W × 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 ∈ V r . Consider ( W , ◦ , 1), the submonoid of A generated by B . We define S W to be the set of unary linear polynomial (sections) of ( W , ◦ , 1). Elements of S W are of the form u ( ) = y ◦ ◦ w for y , w ∈ W . Let W ′ = S W × 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 W A , 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 } γ N : P ( W ) → P ( W ), γ N ( 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 } γ N : P ( W ) → P ( W ), γ N ( X ) = X ⊲⊳ , is a closure operator. The Galois algebra of W A , B is W A , B + = ( γ N [ ℘ ( W )] , ∩ , ∪ γ N , ◦ γ N , \ , /, γ N ( { 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]. A , B and A belong to V k and the closed sets { ( u , b ) } ⊳ for Furthermore, W + u ∈ S W , 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 W ′ F 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 W ′ F 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 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 ordered. If � P , ≤� is well partially ordered, then it is known that for each k ∈ N , P k is well partially ordered under the direct product ordering. Furthermore, homomorphic images, finite disjoint unions, and subposets of well partially ordered 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 ordered. If � P , ≤� is well partially ordered, then it is known that for each k ∈ N , P k is well partially ordered under the direct product ordering. Furthermore, homomorphic images, finite disjoint unions, and subposets of well partially ordered sets are well partially ordered. Consider the poset � P , ≤� . An infinite sequence p 1 , p 2 , . . . of elements of P is called bad when i < j implies that p i �≤ p j . 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 C b = {{ ( u , b ) } ⊳ : u ∈ S W } . Riquelmi Cardona (DU) FEP for some noncommutative knotted RL BLAST 2013 12 / 16
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