Residuated lattices
Nikolaos Galatos University of Denver
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #1 Outline
Part I: Motivation, examples and basic theory (congruences) Title Outline
RL examples Part II: Subvariety lattice (atoms and joins) Congruences Subvariety lattice (atoms)
Subvariety lattice (joins)
Part III: Representation, Logic, Decidability Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #2 Title Outline
RL examples Boolean algebras Algebras of relations Relation algebras ℓ-groups RL examples Powerset of a monoid Ideals of a ring Residuated lattices Properties Properties (proofs) Lattice/monoid properties Linguistics (verbs) Linguistics (adverbs)
Congruences
Subvariety lattice (atoms)
Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #3 Boolean algebras
A Boolean algebra is a structure A =(A, ∧, ∨, →, 0, 1) such Title Outline that (we define ¬a = a → 0) [a → b = ¬a → b] RL examples ■ Boolean algebras (A, ∧, ∨, 0, 1) is a bounded lattice, Algebras of relations ■ Relation algebras for all a,b,c ∈ A, ℓ-groups Powerset of a monoid Ideals of a ring a ∧ b ≤ c ⇔ b ≤ a → c (∧-residuation) Residuated lattices Properties Properties (proofs) ■ for all a ∈ A, ¬¬a = a (alt. a ∨¬a =1). Lattice/monoid properties Linguistics (verbs) Linguistics (adverbs)
Exercise. Distributivity (of ∧ over ∨) and complementation Congruences
follow from the above conditions. Also, ∧-residuation can be Subvariety lattice (atoms)
written equationally. Subvariety lattice (joins)
Logic Boolean algebras provide algebraic semantics for classical Representation - Frames propositional logic. Applications of frames Heyting algebras are defined without the third condition and Undecidability are algebraic semantics for intuitionistic propositional logic. References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #4 Algebras of relations
Let X be a set and Rel(X) = P(X × X) be the set of all Title Outline binary relations on X. RL examples Boolean algebras Algebras of relations For relations R, and S, we denote by Relation algebras ■ − ∪ ℓ-groups R the complement and by R the converse of R Powerset of a monoid ■ Ideals of a ring ∆ is the equality/diagonal relation on X Residuated lattices Properties ■ R ; S the relational composition of R and S Properties (proofs) Lattice/monoid properties ■ R\S =(R ; S−)− and S/R =(S− ; R)− Linguistics (verbs) Linguistics (adverbs) ■ − − − R → S =(R ∩ S ) = R ∪ S Congruences
Subvariety lattice (atoms)
We have Subvariety lattice (joins) 2 ■ (Rel(X), ∩, ∪, →, ∅, X ) is a Boolean algebra Logic ■ (Rel(X), ; , ∆) is a monoid Representation - Frames ■ for all R,S,T ∈ Rel(X), Applications of frames Undecidability
R ; S ⊆ T ⇔ S ⊆ R\T ⇔ R ⊆ T/S. References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #5 Relation algebras
A Relation algebra is a structure Title Outline A =(A, ∧, ∨, ; , \,/, 0, 1, (_)−) such that (0=1−) RL examples ■ − Boolean algebras (A, ∧, ∨, ⊥, ⊤, (_) ) is a Boolean algebra Algebras of relations (we define ⊥ =1 ∧ 1− and ⊤ =1 ∨ 1−), Relation algebras ℓ-groups ■ Powerset of a monoid (A, ; , 1) is a monoid Ideals of a ring ■ Residuated lattices for all a,b,c ∈ A, Properties Properties (proofs) Lattice/monoid properties a ; b ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b (residuation) Linguistics (verbs) Linguistics (adverbs) ■ for all a ∈ A, ¬¬a = a (we define ¬a = a\0=0/a) Congruences ■ ¬(a−)=(¬a)− and ¬(¬x ; ¬y)=(x− ; y−)−. Subvariety lattice (atoms) Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #6 ℓ-groups
A lattice-ordered group is a lattice with a compatible group Title Outline structure. Alternatively, a lattice-ordered group is an algebra RL examples L =(L, ∧, ∨, ·, \,/, 1) such that Boolean algebras Algebras of relations ■ (L, ∧, ∨) is a lattice, Relation algebras ℓ-groups ■ Powerset of a monoid (L, ·, 1) is a monoid Ideals of a ring ■ Residuated lattices for all a,b,c ∈ L, Properties Properties (proofs) Lattice/monoid properties ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b. Linguistics (verbs) Linguistics (adverbs) ■ −1 −1 for all a ∈ L, a · a =1 (we define x = x\1=1/x). Congruences
Subvariety lattice (atoms)
Example. The set of real numbers under the usual order, Subvariety lattice (joins)
addition and subtraction. Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #7 Powerset of a monoid
Let M =(M, ·,e) be a monoid and X,Y ⊆ M. Title Outline We define X · Y = {x · y : x ∈ X, y ∈ Y }, RL examples X\Y = {z ∈ M : X ·{z}⊆ Y }, Boolean algebras Algebras of relations Y/X = {z ∈ M : {z}· X ⊆ Y }. Relation algebras ℓ-groups Powerset of a monoid Ideals of a ring For the powerset P(M), we have Residuated lattices ■ (P(M), ∩, ∪) is a lattice Properties Properties (proofs) ■ Lattice/monoid properties (P(M), ·, {e}) is a monoid Linguistics (verbs) ■ for all X,Y,Z ⊆ M, Linguistics (adverbs) Congruences
X · Y ⊆ Z ⇔ Y ⊆ X\Z ⇔ X ⊆ Z/Y . Subvariety lattice (atoms)
Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #8 Ideals of a ring
Let R be a ring with unit and let I(R) be the set of all Title Outline (two-sided) ideals of R. RL examples For I,J ∈I(R), we write IJ = { ij : i ∈ I, j ∈ J} Boolean algebras Pfin Algebras of relations I\J = {k : Ik ⊆ J}, Relation algebras ℓ-groups J/I = {k : kI ⊆ J}. Powerset of a monoid Ideals of a ring Residuated lattices R Properties For the powerset I( ), we have Properties (proofs) ■ R is a lattice Lattice/monoid properties (I( ), ∩, ∪) Linguistics (verbs) ■ (I(R), ·, R) is a monoid Linguistics (adverbs) Congruences ■ for all ideals I, J, K of R, Subvariety lattice (atoms)
I · J ⊆ K ⇔ J ⊆ I\K ⇔ I ⊆ K/J. Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #9 Residuated lattices
A residuated lattice, or residuated lattice-ordered monoid, is Title Outline an algebra L =(L, ∧, ∨, ·, \,/, 1) such that RL examples ■ Boolean algebras (L, ∧, ∨) is a lattice, Algebras of relations ■ Relation algebras (L, ·, 1) is a monoid and ℓ-groups ■ Powerset of a monoid for all a,b,c ∈ L, Ideals of a ring Residuated lattices Properties ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b. Properties (proofs) Lattice/monoid properties Linguistics (verbs) (We think of x\y and y/x as x → y, when they are equal.) Linguistics (adverbs) A pointed residuated lattice an extension of a residuated Congruences lattice with a new constant 0. (∼x = x\0 and −x =0/x.) Subvariety lattice (atoms) Subvariety lattice (joins)
A (pointed) residuated lattice is called Logic ■ commutative, if (L, ·, 1) is commutative (xy = yx). Representation - Frames ■ distributive, if (L, ∧, ∨) is distibutive Applications of frames ■ integral, if it satisfies x ≤ 1 Undecidability References ■ contractive, if it satisfies x ≤ x2 ■ involutive, if it satisfies ∼−x = x = −∼x.
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #10 Properties
1. x(y ∨ z)= xy ∨ xz and (y ∨ z)x = yx ∨ zx Title Outline
2. x\(y ∧ z)=(x\y) ∧ (x\z) and (y ∧ z)/x =(y/x) ∧ (z/x) RL examples Boolean algebras 3. x/(y ∨ z)=(x/y) ∧ (x/z) and (y ∨ z)\x =(y\x) ∧ (z\x) Algebras of relations Relation algebras 4. (x/y)y ≤ x and y(y\x) ≤ x ℓ-groups Powerset of a monoid Ideals of a ring 5. x(y/z) ≤ (xy)/z and (z\y)x ≤ z\(yx) Residuated lattices Properties 6. (x/y)/z = x/(zy) and z\(y\x)=(yz)\x Properties (proofs) Lattice/monoid properties 7. x\(y/z)=(x\y)/z; Linguistics (verbs) Linguistics (adverbs) 8. x/1= x =1\x Congruences
9. 1 ≤ x/x and 1 ≤ x\x Subvariety lattice (atoms)
10. x ≤ y/(x\y) and x ≤ (y/x)\y Subvariety lattice (joins) 11. y/((y/x)\y)= y/x and (y/(x\y))\y = x\y Logic 12. x/(x\x)= x and (x/x)\x = x; Representation - Frames Applications of frames 13. and (z/y)(y/x) ≤ z/x (x\y)(y\z) ≤ x\z Undecidability
Multiplication is order preserving in both coordinates. Each References division operation is order preserving in the numerator and order reversing in the denominator.
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #11 Properties (proofs)
Title x(y ∨ z) ≤ w ⇔ y ∨ z ≤ x\w Outline
RL examples ⇔ y,z ≤ x\w Boolean algebras Algebras of relations ⇔ xy,xz ≤ w Relation algebras ℓ-groups ⇔ xy ∨ xz ≤ w Powerset of a monoid Ideals of a ring Residuated lattices Properties x/y ≤ x/y ⇒ (x/y)y ≤ x Properties (proofs) Lattice/monoid properties Linguistics (verbs) Linguistics (adverbs) x(y/z)z ≤ xy ⇒ x(y/z) ≤ (xy)/z Congruences
Subvariety lattice (atoms)
[(x/y)/z](zy) ≤ x ⇒ (x/y)/z ≤ x/(zy) Subvariety lattice (joins) [x/(zy)]zy ≤ x ⇒ x/(zy) ≤ (x/y)/z Logic Representation - Frames
w ≤ x\(y/z) ⇔ xw ≤ y/z Applications of frames ⇔ xwz ≤ y Undecidability ⇔ wz ≤ x\y References ⇔ w ≤ (x\y)/z
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #12 Lattice/monoid properties
Title (z/y)(y/x)x ≤ (z/y)y ≤ z ⇒ (z/y)(y/x) ≤ z/x Outline
RL examples Boolean algebras Algebras of relations Relation algebras RL’s satisfy no special purely lattice-theoretic or ℓ-groups Powerset of a monoid monoid-theoretic property. Ideals of a ring Residuated lattices Properties Properties (proofs) Every lattice can be embedded in a (cancellative) residuated Lattice/monoid properties lattice. Linguistics (verbs) Linguistics (adverbs) Every monoid can be embedded in a (distributive) residuated Congruences lattice. Subvariety lattice (atoms) Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #13 Linguistics (verbs)
We want to assign (a limited number of) linquistic types to Title Outline English words, as well as to phrases, in such a way that we RL examples will be able to tell if a given phrase is a (syntacticly correct) Boolean algebras Algebras of relations sentence. Relation algebras ℓ-groups Powerset of a monoid We will use n for ‘noun phrase’ and s for ‘sentence’. Ideals of a ring Residuated lattices For phrases we use the rule: if A : a and B : b, then AB : ab. Properties Properties (proofs) Lattice/monoid properties We write C : a\b if A : a implies AC : b, for all A. Linguistics (verbs) Linguistics (adverbs)
Likewise, C : b/a if A : a implies CA : b, for all A. Congruences We assign type n to ‘John.’ Clearly, ‘plays’ has type n\s, as Subvariety lattice (atoms) all intransitive verbs. Subvariety lattice (joins) John plays Logic n(n\s) ≤ s n n\s Representation - Frames Applications of frames Some words may have more than one type. We write a ≤ b if Undecidability every word with type a has also type b. References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #14 Linguistics (adverbs)
Title (John plays) here Outline [n(n\s)](s\s) ≤ s(s\s) ≤ s n n\s s\s RL examples Boolean algebras Algebras of relations John (plays here) Relation algebras s\s ≤ (n\s)\(n\s) ℓ-groups n n\s (n\s)\(n\s) Powerset of a monoid Ideals of a ring Residuated lattices Properties Note that ‘plays’ is also a transitive verb, so it has type Properties (proofs) Lattice/monoid properties (n\s)/n. Linguistics (verbs) Linguistics (adverbs) John (plays football) [n((n\s)/n)]n ≤ s Congruences
n (n\s)/n n Subvariety lattice (atoms)
(John plays) football (n\s)/n ≤ n\(s/n) Subvariety lattice (joins) n n\(s/n) n n[(n\(s/n))n] ≤ s Logic Representation - Frames Also, for ‘John definitely plays football’, note that we need to Applications of frames have s\s ≤ (n\s)/(n\s). Undecidability References Q: Can we decide (in)equations in residuated lattices?
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #15 Title Outline
RL examples
Congruences Congruences G, B Congruences R, M Congruences Congruences and sets Correspondence CNM to CNS CNS to congruence CNS to congruence Lattice isomorphism Compositions Generation Generation of CNM
Subvariety lattice (atoms)
Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #16 Congruences G, B
Definition. A congruence on an algebra A is an equivalence Title Outline relation on A that is compatible with the operations of A. RL examples (Alt.the kernel of a homomorphism out of A.) Congruences Congruences G, B Congruences in groups correspond to normal subgroups. Congruences R, M Congruences and sets Correspondence Given a congruence θ on a group G, the congruence class CNM to CNS CNS to congruence [1]θ of 1 is a normal subgroup. CNS to congruence Lattice isomorphism Given a normal subgroup N of a group G, the relation θN is a Compositions Generation congruence, where (a, b) ∈ θN iff a\b ∈ N iff {a\b, b\a}⊆ N. Generation of CNM
Subvariety lattice (atoms) Congruences in Boolean algebras correspond to filters. Subvariety lattice (joins)
Given a congruence θ on a Boolean algebra A, the Logic congruence class [1]θ of 1 is a filter of A. Representation - Frames
Applications of frames Given a filter F of a Boolean algebra A, θF is a congruence, Undecidability where (a, b) ∈ θF iff a ↔ b ∈ F iff {a → b, b → a}⊆ F . Note that a filter is a subset of A closed under {∧, ∨, →, 1} References that is convex (x ≤ y ≤ z and x,z ∈ F implies y ∈ F ).
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #17 Congruences R, M
Congruences on rings correspond to ideals. Title Outline
RL examples
Congruences on ℓ-groups correspond to convex ℓ-subgroups. Congruences Congruences G, B Congruences R, M Congruences and sets Congruences on monoids do not correspond to any Correspondence CNM to CNS particular kind of subset. CNS to congruence CNS to congruence Lattice isomorphism Compositions Generation Do congruences on residuated lattices correspond to certain Generation of CNM
subsets? Subvariety lattice (atoms)
Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #18 Congruences and sets
Let A be a residuated lattice and a,x ∈ A. We define the Title Outline conjugates λ (x)=[a\(xa)] ∧ 1 and ρ (x)= ax/a ∧ 1. a a RL examples
Congruences An iterated conjugate is a composition γa1 (γa2 (...γan (x))), Congruences G, B where n ∈ ω, a1,a2,...,an ∈ A and γai ∈{λai , ρai }, for all i. Congruences R, M Congruences and sets Correspondence X ⊆ A is called normal, if it is closed under conjugates. CNM to CNS CNS to congruence CNS to congruence We will be considering correspondences between: Lattice isomorphism Compositions ■ A Generation Congruences on Generation of CNM ■ Convex, normal subalgebras (CNSs) of A Subvariety lattice (atoms) ■ Convex , normal (in A) submonoids (CNMs) of A− =↓ 1 Subvariety lattice (joins) ■ Deductive filters of A: F ⊆ A Logic ◆ ↑ 1 ⊆ F Representation - Frames ◆ a,a\b ∈ F implies b ∈ F (eqv. ↑ F = F ) Applications of frames ◆ a ∈ F implies a ∧ 1 ∈ F (eqv. F is ∧-closed) Undecidability ◆ a ∈ F implies b\ab, ba/b ∈ F References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #19 Correspondence
A A− A Title If S is a CNS of , M a SNM of , θ a congruence on Outline and a DF of A, then F RL examples 1. −, − and − are SNMs Congruences Ms(S)= S Mc(θ) = [1]θ Mf (F )= F Congruences G, B A− Congruences R, M of , Congruences and sets − Correspondence 2. Sm(M)=Ξ(M), Sc(θ) = [1]θ and Sf (F )=Ξ(F ) are CNM to CNS CNS to congruence CNSs of A, CNS to congruence Lattice isomorphism 3. Fs(S)= ↑ S, Fm(M)= ↑ M, and Fc(θ)= ↑[1]θ are DFs of Compositions Generation A. Generation of CNM
Subvariety lattice (atoms) 4. Θs(S)= {(a, b)|a ↔ b ∈ S}, Θm(M)= {(a, b)|a ↔ b ∈ M} Subvariety lattice (joins) and Θf (F )= {(a, b)|a ↔ b ∈ F } = {(a, b)|a\b, b\a ∈ F } are congruences of A. Logic Representation - Frames
a ↔ b = a\b ∧ b\a ∧ 1 Applications of frames
Undecidability Ξ(X) = {a ∈ A : x ≤ a ≤ x\1, for some x ∈ X}. References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #20 CNM to CNS
Ξ(M)= {a ∈ A|x ≤ a ≤ x\1, for some x ∈ M} is a CNS. Title Outline
Claim: a ∈ Ξ(M) iff ∃y,z ∈ M such that y ≤ a ≤ z\1. RL examples
Congruences Indeed, yz ≤ y ≤ a ≤ z\1 ≤ yz\1 and yz ∈ M. Congruences G, B Congruences R, M Convexity: If a, b ∈ Ξ(M), then ∃x, y ∈ M such that Congruences and sets Correspondence x ≤ a ≤ x\1 and y ≤ b ≤ y\1. CNM to CNS CNS to congruence If a ≤ c ≤ b, then x ≤ a ≤ c ≤ b ≤ y\1, so c ∈ Ξ(M). CNS to congruence Lattice isomorphism Subalg.: xy ≤ x ∧ y ≤ a ∧ b ≤ x\1 ∧ y\1=(x ∨ y)\1 ≤ x\1 Compositions Generation Generation of CNM x ≤ x ∨ y ≤ a ∨ b ≤ x\1 ∨ y\1 ≤ (x ∧ y)\1 ≤ (xy)\1 Subvariety lattice (atoms) xy ≤ ab ≤ (x\1)(y\1) ≤ x\(y\1)=(yx)\1 Subvariety lattice (joins) Logic
λa(yx) ≤ a\yxa ≤ a\[y/(x\1)]a ≤ a\[b/a]a ≤ a\b ≤ x\(y\1) = yx\Representation1 - Frames
xy ≤ x/(y\1) ≤ a/b ≤ (x\1)/y ≤ [xρ(x\1)/y(y)]\1 Applications of frames Undecidability (for u =(x\1)/y we have xρu(y)u ≤ x{uy/u}u ≤ xuy ≤ 1) References Normality: As λc(x)λc(x\1) ≤ c\x(x\1)c ∧ 1 ≤ c\c ∧ 1=1,
λc(x) ≤ λc(a) ≤ λc(x\1) ≤ λc(x)\1
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #21 CNS to congruence
Θ (S)= {(a, b)|a ↔ b ∈ S} is a congruence. Title s Outline a ↔ b = a\b ∧ b\a ∧ 1 RL examples
Equivalance: Θs(S) is reflexive and symmetric. If Congruences Congruences G, B a ↔ b, b ↔ c ∈ S, we have Congruences R, M Congruences and sets Correspondence (a ↔ b)(b ↔ c) ∧ (b ↔ c)(a ↔ b) ≤ CNM to CNS CNS to congruence CNS to congruence ≤ (a\b)(b\c) ∧ (c\b)(b\a) ∧ 1 ≤ (a ↔ c) ≤ 1. Lattice isomorphism Compositions Generation Comptibility: Assume a ↔ b ∈ S and c ∈ A. Generation of CNM Subvariety lattice (atoms)
a\b ≤ ca\cb implies a ↔ b ≤ ca ↔ cb ≤ 1 Subvariety lattice (joins)
Logic λc(a ↔ b) ≤ c\(a\b)c ∧ c\(b\a)c ∧ e ≤ ac ↔ bc ≤ 1 Representation - Frames
(a ∧ c) · (a ↔ b) ≤ a(a ↔ b) ∧ c(a ↔ b) ≤ b ∧ c implies Applications of frames
a ↔ b ≤ (a ∧ c)\(b ∧ c). Likewise, a ↔ b ≤ (b ∧ c)\(a ∧ c). So, Undecidability
a ↔ b ≤ (a ∧ c) ↔(b ∧ c) ≤ 1 References a\b ≤ (c\a)\(c\b) and b\a ≤ (c\b)\(c\a) imply a ↔ b ≤ (c\a) ↔(c\b) ≤ 1
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #22 CNS to congruence
a\b ≤ (a\c)/(b\c) and b\a ≤ (b\c)/(a\c) imply Title Outline ′ a ↔ b ≤ (a\c)↔ (b\c) ≤ 1 RL examples where a ↔′ b = a/b ∧ b/a ∧ 1. Congruences Congruences G, B Congruences R, M ′ Congruences and sets So, (a\c) ↔ (b\c) ∈ S and (a\c) ↔(b\c) ∈ S. Correspondence CNM to CNS CNS to congruence ′ CNS to congruence Claim: a ↔ b ∈ S iff a ↔ b ∈ S. Lattice isomorphism Compositions ′ Generation λb(a↔ b)= b\[a/b ∧ b/a ∧ 1]b ∧ 1 ≤ b\a ∧ 1 Generation of CNM
Subvariety lattice (atoms)
′ ′ Subvariety lattice (joins) λb(a↔ b) ∧ λa(a↔ b) ≤ a ↔ b ≤ 1 Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #23 Lattice isomorphism
A A− A Title 1. The CNSs of , the CNMs of and the DF of form Outline lattices, denoted by CNS A , CNM A and Fil A , ( ) ( ) ( ) RL examples respectively. Congruences Congruences G, B 2. All the above lattices are isomorphic to the congruence Congruences R, M Con A A Congruences and sets lattice ( ) of via the maps defined above. Correspondence CNM to CNS 3. The composition of the above maps gives the CNS to congruence CNS to congruence corresponding map; e.g., Ms(Sc(θ)) = Mc(θ). Lattice isomorphism Compositions Generation Claim: Sc and Θs are inverse maps. Generation of CNM
Subvariety lattice (atoms) S = [1]Θs(S): a ∈ S implies a ↔ 1= a\1 ∧ a ∧ 1 ∈ S. Conversely, (a ↔ 1) ≤ a ≤ (a ↔ 1)\1. Subvariety lattice (joins)
Logic θ =Θs(Sc(θ)): If (a, b) ∈ Θs([1]θ), then a ↔ b ∈ [1]θ, so a ↔ b θ 1. Therefore, aθa(a ↔ b) ≤ a(a\b) ≤ b, so a ∨ b θ b. Representation - Frames Likewise, a ∨ bθa, so a θ b. Applications of frames Undecidability
Conversely, if a θ b, then References 1=(a\a ∧ b\b ∧ 1) θ (a\b ∧ b\a ∧ 1) = a ↔ b.
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #24 Compositions
Claim: S (F )= S (Θ (F )). (Sketch) Title f c f Outline
RL examples If a ∈ Sc(Θf (F )), then a Θf (F )1, so a\1, 1\a ∈ F . − Congruences Hence a, 1/a ∈ F . Since 1 ∈ F , we get x = a ∧ 1/a ∧ 1 ∈ F . Congruences G, B Congruences R, M Obviously, x ≤ a; also a ≤ (1/a)\1 ≤ x\1. Congruences and sets Thus, . Correspondence a ∈ Sf (F ) CNM to CNS − CNS to congruence Conversely, if a ∈ Sf (F ), then x ≤ a ≤ x\1, for some x ∈ F . CNS to congruence Lattice isomorphism So, a ∈ F and 1/(x\1) ≤ 1/a. Compositions Generation Since, x ≤ 1/(x\1), we have x ≤ 1/a and 1/a ∈ F . Generation of CNM
Thus both a/1 and 1/a are in F . Hence, a ∈ [1]Θf (F ). Subvariety lattice (atoms)
Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #25 Generation
− Title If X is a subset of A and Y is a subset of A, then Outline − 1. the CNM M(X) of A generated by X is equal to RL examples − Ξ ΠΓ(X). Congruences Congruences G, B 2. The CNS S(Y ) of A generated by Y is equal to ΞΠΓ∆(Y ). Congruences R, M Congruences and sets A Correspondence 3. The DF F (Y ) of generated by Y ⊆ A is equal to CNM to CNS . CNS to congruence ↑ ΠΓ(Y )= ↑ ΠΓ(Y ∧ 1) CNS to congruence A 2 Lattice isomorphism 4. The congruence Θ(P ) on generated by P ⊆ A is equal Compositions ′ ′ Generation to Θm(M(P )), where P = {a ↔ b|(a, b) ∈ P }. Generation of CNM
Subvariety lattice (atoms)
X ∧ 1= {x ∧ 1: x ∈ X} Subvariety lattice (joins)
∆(X)= {x ↔ 1: x ∈ X} Logic
Π(X)= {x1x2 ··· xn : n ≥ 1,xi ∈ X}∪{1} Representation - Frames Γ(X)= {γ(x): γ is an iterated conjugate } Applications of frames Ξ(X)= {a ∈ A : x ≤ a ≤ x\1, for some x ∈ X} Undecidability References Ξ−(X)= {a ∈ A : x ≤ a ≤ 1, for some x ∈ X} a ↔ b = a\b ∧ b\a ∧ 1
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #26 Generation of CNM
A− Title Clearly, if M is a CNM of that contains X, then it Outline contains , by normality, , since is closed under Γ(X) ΠΓ(X) M RL examples − product, and Ξ ΠΓ(X), since M is convex and contains 1. Congruences We will now show that Ξ−ΠΓ(X) itself is a CNM of A−; it Congruences G, B Congruences R, M obviously contains X. It is clearly convex and a submonoid of Congruences and sets − − Correspondence A . To show that it is convex, consider a ∈ Ξ ΠΓ(X) and CNM to CNS CNS to congruence u ∈ A. There are x1,...,xn ∈ X and iterated conjugates CNS to congruence Lattice isomorphism γ1,...,γn such that γ1(x1) ··· γn(xn) ≤ a ≤ 1. We have Compositions Generation Generation of CNM λ (γ (x )) ≤ λ ( γ (x )) ≤ λ (a) ≤ 1. Y u i i u Y i i u Subvariety lattice (atoms) Idea for n =2: Subvariety lattice (joins) Logic
λu(a1)λu(a2) =(u\a1u ∧ 1)(u\a2u ∧ 1) ≤ (u\a1u)(u\a2u) ∧ 1 Representation - Frames
Applications of frames
≤ u\a1u(u\a2u) ∧ 1 ≤ u\a1a2u ∧ 1= λu(a1a2). Undecidability Also, λu(γi(xi)) ∈ Γ(X) and Q λu(γi(xi)) ∈ ΠΓ(X), so References − − λu(a) ∈ Ξ ΠΓ(X). Likewise, we have ρu(a) ∈ Ξ ΠΓ(X).
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #27 Title Outline
RL examples
Congruences
Subvariety lattice (atoms) Subvariety lattice (atoms) Size BA and 2 BA: an atom Fin. gen. atoms Cancellative atoms Idempotent rep. atoms
Subvariety lattice (joins)
Logic
Representation - Frames
Applications of frames
Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #28 Size
We view RL as the subvariety of RL axiomatized by 0=1. Title p Outline The subvariety lattices of HA (Heyting algebras) and Br RL examples Congruences (Brouwerian algebras) are uncountable, hence so are Λ(RLp) Subvariety lattice (atoms) and Λ(RL). Size BA and 2 BA: an atom We will Fin. gen. atoms Cancellative atoms Idempotent rep. atoms ■ determine the size of the set of atoms in Λ(RL ). p Subvariety lattice (joins) ■ outline a method for finding axiomatizations of certain Logic varieties Representation - Frames
Applications of frames ■ Λ give a description of joins in (RLp). Undecidability
References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #29 BA and 2
The variety BA of Boolean algebras is generated by the Title Outline 2-element algebra 2. BA = HSP(2)= V(2). RL examples
H: homomorphic images Congruences S: subalgebras Subvariety lattice (atoms) Size P: direct products BA and 2 BA: an atom V = HSP Fin. gen. atoms Cancellative atoms Proof idea: Use the prime ideal-filter theorem for distributive Idempotent rep. atoms lattices to show that every Boolean algebra is a subdirect Subvariety lattice (joins) product of copies of 2. Logic Representation - Frames
Subdirect product: A subalgebra of a product such that all Applications of frames
projections are onto. Undecidability Clearly, 2 is subdirectly irreducible. References Subdirectly irreducible: non-trivial and ■ it cannot be written as a subdirect product of a family that does not contain it. ■ Alt. its congruence lattice is ∆ ∪↑ µ.
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #30 BA: an atom
The variety BA is an atom in the lattice of subvarieties of pRL. Title Outline
pRL is a congruence distributive variety (RL’s have lattice RL examples reducts) so Jònsson’s Lemma applies: Congruences
Subvariety lattice (atoms) Given a class K⊆ RLp, the subdirectly irreducible algebras Size V in the variety generated by a are in HSP . BA and 2 (K)SI K U(K) BA: an atom Fin. gen. atoms An ultraproduct A ∈ PU(K) is obtained by taking Cancellative atoms Idempotent rep. atoms ■ a product A of A ∈ K and then Qi∈I i i Subvariety lattice (joins)
■ ∼ Logic a quotient Qi∈I Ai/ =U by an ultrafilter U over I (maximal filter on P(U)): Representation - Frames ¯ ∼ ¯ Applications of frames for a,¯ b ∈ Qi∈I Ai, a¯ =U b iff {i ∈ I : ai = bi}∈ U. Undecidability First order formulas persist under ultraproducts. References
Now, HSPU(2)= {2, 1}, hence (V(2))SI = {2}.
Recall that V = V(VSI ).
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #31 Fin. gen. atoms
We define ⊤u = u⊤ = u. Title ⊤ Outline
T RL examples Note that n is strictly sim- 1 ple (has no non-trivial subal- Congruences ⊤\1 u Subvariety lattice (atoms) gebras or homomorphic im- Size ages). BA and 2 2 BA: an atom u Fin. gen. atoms Cancellative atoms So, V(Tn) is an atom of u3 Idempotent rep. atoms Λ(RL). . . Subvariety lattice (joins) n Moreover, all these atoms u Logic are distinct and Λ(RL) has Representation - Frames Applications of frames at least denumerably many Tn atoms. Undecidability References
Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #32 Cancellative atoms
Left cancellativity (ab = ac ⇒ b = c) can be written Title Outline equationally: x\(xy)= y. Right cancellativity is (yx)/x = y. RL examples CanRL denotes the variety of cancellative RL’s. Congruences − Prop. There are only 2 cancellative atoms: V(Z) and V(Z ). Subvariety lattice (atoms) Size BA and 2 Let L ∈ CanRL. For a ≤ 1, we have 1 ≤ 1/a. BA: an atom Fin. gen. atoms ∼ Z− Cancellative atoms Claim: If ∃a< 1 with 1/a =1, then Sg(a) = . Idempotent rep. atoms n+1 n Since a< 1, we get a Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #33 Idempotent rep. atoms For S ⊆ Z, we define Title . Outline a b = a , if i ∈ S and . i i i . RL examples a b = b , if i 6∈ S. i i i a−1 Congruences Subvariety lattice (atoms) Although, we may have a0 Size 2 ■ N ∼ N BA and S 6= T , but S = T a1 BA: an atom Fin. gen. atoms ■ NS =6∼ NT , but . Cancellative atoms . Idempotent rep. atoms N N . V( S) 6= V( T ) 1 . Subvariety lattice (joins) ■ V N is not an atom . ( S) Logic there are still continuum b1 Representation - Frames N many atoms V( S). Applications of frames b0 Undecidability b −1 References . . NS Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #34 Title Outline RL examples Congruences Subvariety lattice (joins) Subvariety lattice (atoms) Subvariety lattice (joins) Representable RL’s Joins Finite basis FSI PUF’s PUF and equations Axiomatization RRL Finite axiomatization Elementarity Applications Logic Representation - Frames Applications of frames Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #35 Representable RL’s A residuated lattice is called representable (or semi-linear) if Title Outline it is a subdirect product of totally ordered RL’s. RRL denotes RL examples the class of representable RL’s. Congruences Recall that a totally ordered RL satisfies the first-order Subvariety lattice (atoms) formula (∀x, y)(x ≤ y or y ≤ x) [(∀x, y)(1 ≤ x\y or 1 ≤ y\x)] Subvariety lattice (joins) Representable RL’s Joins Representable Heyting algebras form a variety axiomatized Finite basis FSI by 1=(x → y) ∨ (y → x). PUF’s PUF and equations Axiomatization Representable commutative RL’s form a variety axiomatized RRL by . Finite axiomatization 1 ≤ (x → y)∧1 ∨ (y → x)∧1 Elementarity Applications RRL is a variety axiomatized by 1 ≤ γ (x\y) ∨ γ (y\x). 1 2 Logic Goal: Given a class K of RL’s axiomatized by a set of Representation - Frames positive universal first-order formulas (PUF’s), provide an Applications of frames axiomatization for V(K). Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #36 Joins The meet of two varieties in Λ(RL ) is their intersection. Title p Outline Also, if V1 is axiomatized by E1 and V2 by E2, then V1 ∧V2 is RL examples axiomatized by E1 ∪ E2. Congruences On the other hand, the join of two varieties is the variety Subvariety lattice (atoms) Subvariety lattice (joins) generated by their union. Representable RL’s Joins Also, if V1 is axiomatized by E1 and V2 by E2, then V1 ∨V2 Finite basis FSI may not be axiomatized by E1 ∩ E2. PUF’s PUF and equations Axiomatization Goals RRL Finite axiomatization Elementarity ■ Applications Find an axiomatization of V1 ∨V2 in terms of E1 and E2. Logic ■ Find situations where: if E1 and E2 are finite, then V1 ∨V2 Representation - Frames is finitely axiomatized. Applications of frames Undecidability ■ Find V such that its finitely axiomatized subvarieties form a References lattice. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #37 Finite basis If V is a congruence distributive variety of finite type and Title Outline V is strictly elementary, then V is finitely axiomatized. F SI RL examples Strictly elementary: Axiomatized by a single FO-sentence. Congruences Finitely SI: ∆ is not the intersection of two non-trivial Subvariety lattice (atoms) congruences. Subvariety lattice (joins) Representable RL’s Joins Cor. For every variety V of RL’s, if VF SI is strictly elementary, Finite basis FSI then the finitely axiomatized subvarieties of V form a lattice. PUF’s PUF and equations Pf. For finitely axiomatized subvarieties V1, V2, Axiomatization RRL (V1 ∨V2)F SI =(V1 ∪V2)F SI is strictly elementary. Finite axiomatization Elementarity Applications Let V1, V2 be subvarieties of RL axiomatized by E1, E2, Logic respectively, where E1, E2 have no variables in common. Representation - Frames The class V1 ∪V2 is axiomatized by the universal closure of Applications of frames (AND E1) or (AND E2), over infinitary logic, which is equivalent Undecidability to the set {∀∀(ε1 or ε2) : ε1 ∈ E1, ε2 ∈ E2} of positive References universal first-order formulas (PUFs). Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #38 FSI In a RL, we say that 1 is weakly join irreducible, if for all Title Outline negative a, b, whenever 1= γ(a) ∨ γ′(b), for all all iterrated RL examples conjugates γ, γ′, then a =1 or b =1. Congruences Thm. A RL is FSI iff 1 is weakly join-irreducible. Subvariety lattice (atoms) Subvariety lattice (joins) − (⇐) Let F, G be CNS with F ∩ G = {1}. For all a ∈ F and Representable RL’s − ′ Joins b ∈ G , 1= γ(a) ∨ γ (b), for all iterated conjugates, because Finite basis ′ FSI if γ(a),γ (b) ≤ u, then u ∧ 1 ∈ F ∩ G = {1}, so 1 ≤ u. Since 1 PUF’s PUF and equations is weakly join-irreducible, a =1 or b =1. Axiomatization RRL Finite axiomatization (⇒) Let a, b be negative elements and assume that Elementarity u ∈ CNS−(a) ∩ CNS−(b). Then there exist products of Applications iterated conjugates p, q of a, b, resp., such that p, q ≤ u. If Logic 1= γ(a) ∨ γ′(b), for all iterated conjugates, then 1= p ∨ q. Representation - Frames Thus, u =1 and CNS−(a) ∩ CNS−(b)= {1}. Applications of frames Since A is FSI, CNS−(a)= {1} or CNS−(b)= {1}, hence Undecidability a =1 of b =1. References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #39 PUF’s Every PUF is equivalent to (the universal closure of) a Title Outline disjunction of conjunctions of equations. RL examples s = t iff (s ≤ t and t ≤ s) iff (1 ≤ s\t and 1 ≤ t\s). Congruences Subvariety lattice (atoms) Every conjunction of equations 1 ≤ p is equivalanent to the i Subvariety lattice (joins) equation 1 ≤ p1 ∧···∧ pn. Representable RL’s Joins Finite basis So, every PUF is equivalent to a formula of the form FSI PUF’s PUF and equations α = ∀ x (1 ≤ r1 or ··· or 1 ≤ rk) Axiomatization RRL Finite axiomatization Let α0 be (r1)∧1 ∨ ··· ∨ (rk)∧1 =1. Elementarity e Applications Also, for m> 0 and ℵ0 fresh variables Y , we define αm as Logic the set of all equations of the form e Representation - Frames Applications of frames γ1 ∨ ··· ∨ γk =1 Undecidability m where γi ∈ ΓY (ri) for each i ∈{1,...,k}. Set α = S n∈ω αn. References m e e Here ΓY (a)= {πy1 πy2 ··· πym (a∧1) | yi ∈ Y,πyi ∈{λyi , ρyi }}. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #40 PUF and equations Thm. For a PUF α and a FSI RL A, A |= α iff A |= α. Title Outline e RL examples Pf. (⇒) If a¯ are elements in A, then 1 ≤ ri(¯a) for some i. Congruences So, γ(ri(¯a)∧1)=1, for all γ; hence, Subvariety lattice (atoms) γ1(r1(¯a)∧1) ∨ ··· ∨ γk(rk(¯a)∧1)=1. Subvariety lattice (joins) Representable RL’s (⇐) We have 1= γ1(r1(¯a)∧1) ∨ ···∨ γk(rk(¯a)∧1), for all γi. Joins A Finite basis Since is FSI, 1 is weakly join irreducible, so ri(¯a)∧1 =1, for FSI PUF’s some i; i.e., ri(¯a) ≤ 1. PUF and equations Axiomatization RRL α = ∀ x (1 ≤ r1 or ··· or 1 ≤ rk) Finite axiomatization Elementarity Applications α = {γ1 ∨ ··· ∨ γk = 1 | γi ∈ ΓY (ri)} e Logic Representation - Frames Applications of frames Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #41 Axiomatization Thm. Let K be a class of RLs axiomatixed by a set Ψ of PUF. Title Outline Then V(K) is axiomatized, relative to RL, by Ψ. RL examples e Congruences Pf. Let A ∈ RLSI . By congruence distributivity and Jónsson’s Subvariety lattice (atoms) Lemma, A ∈ V(K) iff A ∈ HSPU(K). Furthermore, as PUFs Subvariety lattice (joins) are preserved under H, S and PU, A ∈ HSPU(K) iff A ∈ K. Representable RL’s Joins Finally, A ∈ K iff A |=Ψ iff A |= Ψ. Finite basis FSI e PUF’s Let V , V be subvarieties of RL axiomatized by E , E , PUF and equations 1 2 1 2 Axiomatization respectively, where E1, E2 have no variables in common. RRL Finite axiomatization The class V1 ∪V2 is axiomatized by the set of PUFs Elementarity Applications Ψ= {∀∀(1 ≤ r1 or 1 ≤ r2) | (1 ≤ r1) ∈ E1, (1 ≤ r2) ∈ E2}. Logic Thm. V1 ∨V2 is axiomatized by Representation - Frames Applications of frames Ψ= {γ1(r1)∨γ2(r2)=1 | (1 ≤ r1) ∈ E1, (1 ≤ r2) ∈ E2,γi ∈ Γ} Undecidability e References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #42 RRL Thm. The variety RRL generated by all totally ordered Title Outline residuated lattices is axiomatized by the 4-variable identity RL examples λz((x ∨ y)\x) ∨ ρw((x ∨ y)\y)=1. Congruences Pf. A RL is a chain iff it satisfies ∀x, y(x ≤ y or y ≤ x), or Subvariety lattice (atoms) Subvariety lattice (joins) ∀x, y(1 ≤ (x ∨ y)\x or 1 ≤ (x ∨ y)\y). Representable RL’s Joins Finite basis Thus, RRL is axiomatized by the identities FSI PUF’s PUF and equations 1= γ ((x ∨ y)\x) ∨ γ ((x ∨ y)\y); γ ,γ ∈ Γ (Γ) Axiomatization 1 2 1 2 RRL Finite axiomatization Elementarity So, RRL satisfies the identity Applications Logic λz((x ∨ y)\x) ∨ ρw((x ∨ y)\y)=1. (λ, ρ) Representation - Frames Conversely, the variety axiomatized by this identity satisfies Applications of frames Undecidability x∨y =1 ⇒ λz(x)∨y =1 x∨y =1 ⇒ x∨ρw(y)=1. (imp) References By repeated applications of (imp) on (λ, ρ), we get (Γ). Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #43 Finite axiomatization Let β = ∀x ∀x (1 ≤ x or 1 ≤ x ) and set B ⇒ B = Title 1 2 1 2 m m+1 Outline RL examples ∀x1 ∀x2 [( ∀ y ∀z AND βm ) =⇒ ( ∀ y ∀z AND βm+1 )] e e Congruences Subvariety lattice (atoms) Thm. Let V1 and V2 be two varieties of RLs that satisfy Subvariety lattice (joins) Bm ⇒ Bm+1. Then Representable RL’s Joins 1. V1 ∨V2 is axiomatized by Ψm + a finite set of equations. Finite basis e FSI 2. If and are finitely axiomatized then so is PUF’s V1 V2 V1 ∨V2 PUF and equations Axiomatization Pf. By congruence distributivity (V1 ∨V2)F SI ⊆V1 ∪V2, so RRL Finite axiomatization (V1 ∨V2)F SI satisfies Bm ⇒ Bm+1. V1 ∨V2 also satisfies Elementarity Applications Bm ⇒ Bm+1, because the latter is a special Horn sentence (Lyndon) and is preserved under subdirect products. Logic Representation - Frames By compactness of FOL, Bm ⇒ Bm+1 is a consequence of a Applications of frames finite set B of equations, valid in V1 ∨V2. Undecidability Note that V1 ∨V2 is axiomatized by Ψ and, using e References Bm ⇒ Bm+1, Ψm implies Ψn for all n>m. e e Hence, V1 ∨V2 is axiomatized by Ψm ∪ B. e Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #44 Elementarity Thm. For any variety V of RLs, V is an elementary class Title F SI Outline iff it satisfies B ⇒ B for some m. m m+1 RL examples Congruences Cor. For every variety V of RLs, if VF SI is elementary, then Subvariety lattice (atoms) the finitely axiomatized subvarieties of V form a lattice. Subvariety lattice (joins) Representable RL’s Joins Finite basis FSI PUF’s PUF and equations Axiomatization RRL Finite axiomatization Elementarity Applications Logic Representation - Frames Applications of frames Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #45 Applications RRLs satisfy B ⇒ B . Title 0 1 Outline 1 x ∨ y =1 ⇒ γ1(x) ∨ γ2(y)=1, for all γ1,γ2 ∈ ΓY . RL examples Congruences ℓ-groups satisfy B1 ⇒ B2. Subvariety lattice (atoms) Subvariety lattice (joins) −1 For a ≤ 1, we have λz(λw(a)) = λwz(a) and ρz(a)= λz (a). Representable RL’s Joins Finite basis Subcommutative RSs satisfy B0 ⇒ B1. FSI PUF’s k k PUF and equations k-subcommutative RSs are defined by (x ∧ 1) y = y(x ∧ 1) . Axiomatization RRL Finite axiomatization Elementarity Applications Logic Representation - Frames Applications of frames Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #46 Title Outline RL examples Congruences Logic Subvariety lattice (atoms) Subvariety lattice (joins) Logic A Hilbert system Substructural logics Algebraic semantics Substructural logics (examples) Substructural logics (examples) PLDT Applications to logic Representation - Frames Applications of frames Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #47 A Hilbert-style axiomatization (MP) {φ,φ → ψ} ⊢HL ψ Title e Outline (B) ⊢HLe (φ → ψ) → [(ψ → χ) → (φ → χ)] RL examples (C) ⊢HLe [φ → (ψ → χ)] → [ψ → (φ → χ)] Congruences Subvariety lattice (atoms) (I) ⊢HLe φ → φ Subvariety lattice (joins) (AD) {φ, ψ} ⊢HLe φ ∧ ψ Logic A Hilbert system (CLa) ⊢HLe (φ ∧ ψ) → φ Substructural logics HL Algebraic semantics (CLb) ⊢ e (φ ∧ ψ) → ψ Substructural logics (examples) Substructural logics (examples) (CR) ⊢HLe [(φ → ψ) ∧ (φ → χ)] → [φ → (ψ ∧ χ)] PLDT Applications to logic (DRa) ⊢HLe ψ → (φ ∨ ψ) Representation - Frames (DRb) ⊢HLe ψ → (φ ∨ ψ) Applications of frames (DL) ⊢HLe ((φ → χ) ∧ (ψ → χ)) → (φ ∨ ψ) → χ Undecidability References (PR) ⊢HLe φ → [ψ → (ψ · φ)] (PL) ⊢HLe [ψ → (φ → χ)] → [(φ · ψ) → χ] (U) ⊢HLe 1 (UP) ⊢HLe 1 → (φ → φ) Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #48 Substructural logics The system HL has the following inference rules: Title Outline φ φ\ψ φ ψ φ φ RL examples (mp) (adj) (pn) (pn) ψ φ ∧ ψ ψ\φψ ψφ/ψ Congruences Subvariety lattice (atoms) Subvariety lattice (joins) HL Logic We write Φ ⊢HL ψ, if the formula ψ is provable in from A Hilbert system Substructural logics the set of formulas Φ. Algebraic semantics Substructural logics (examples) We do not allow substitution instances of formulas in Φ. Substructural logics (examples) PLDT Applications to logic For example, p, p\q 6⊢HL r. Representation - Frames A set of formulas is called a substructural logic if it is closed Applications of frames Undecidability under ⊢HL and substitution. References Substructural logics form a lattice SL. In the following we identify (propositional) formulas over {∧, ∨, ·, \,/, 1} with terms over the same signature. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #49 Algebraic semantics For a set of equations E ∪{s = t}, we write Title Outline RL examples E |=RL s = t Congruences L if for every residuated lattice ∈ RL and for every Subvariety lattice (atoms) Fm L homomorphism f : → , Subvariety lattice (joins) f(u)= f(v), for all (u = v) ∈ E, implies f(s)= f(t). Logic A Hilbert system Substructural logics Theorem. The consequence relation ⊢HL is algebraizable, Algebraic semantics Substructural logics (examples) with RL as an equivalent algebraic semantics: Substructural logics (examples) PLDT 1. if Φ ∪{ψ} is a set of formulas, then Applications to logic Representation - Frames Φ ⊢HL ψ iff {1 ≤ φ|φ ∈ Φ}|=RL 1 ≤ ψ, and Applications of frames 2. if E ∪{t = s} is a set of equations, then Undecidability E |= t = s iff {u\v ∧ v\u|(u = v) ∈ E} ⊢HL t\s ∧ s\t. RL References 3. s = t =||=RL 1 ≤ t\s ∧ s\t 4. φ ⊣⊢HL 1\(1 ∧ φ) ∧ (φ ∧ 1)\1 Theorem. SL and Λ(RL) are dually isomorphic. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #50 Substructural logics (examples) Note that HL does not admit Title Outline (C) [x → (y → z)] → [y → (x → z)] (xy = yx) RL examples (K) y → (x → y) (x ≤ 1) Congruences 2 (W) [x → (x → y)] → (x → y) (x ≤ x ) Subvariety lattice (atoms) Subvariety lattice (joins) Examples of substructural logics include Logic A Hilbert system ■ classical: (C)+(K)+(W)+ ¬¬φ = φ (DN) Substructural logics Algebraic semantics ■ intuitionistic (Brouwer, Heyting): (C)+(K)+(W) Substructural logics (examples) Substructural logics (examples) ■ PLDT many-valued (Łukasiewicz): (C)+(K)+ Applications to logic (φ → ψ) → ψ = φ ∨ ψ Representation - Frames ■ basic (Hajek): (C)+(K)+ φ(φ → ψ)= φ ∧ ψ Applications of frames ■ MTL (Esteva, Godo): (C)+(K)+ (φ → ψ) ∨ (ψ → φ) Undecidability ■ relevance (Anderson, Belnap): (C)+(W)+ Distrib. (+ DN) References ■ (MA)linear logic (Girard): (C) Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #51 Substructural logics (examples) Relevance logic deals with relevance. Title Outline p → (q → q) is not a theorem. RL examples The algebraic models do not satisfy integrality x ≤ 1. Congruences p → (¬p → q) [or (p ·¬p) → q] is not a theorem, where Subvariety lattice (atoms) ¬p = p → 0. The algebraic models do not satisfy 0 ≤ x. Subvariety lattice (joins) Commutativity and distributivity are OK, so we get involutive Logic A Hilbert system CDRL (they satisfy ¬¬x = x). Substructural logics Algebraic semantics Substructural logics (examples) Substructural logics (examples) Intuitionistic logic deals with provability or constructibility. PLDT The algebraic models are Heyting algebras. Applications to logic Representation - Frames Many-valued logic allows different degrees of truth. Applications of frames [(p ∧ q) → r] ↔ [p → (q → r)] is not a theorem. Undecidability The algebraic models do not satisfy x ∧ y = x · y. References Linear logic is resourse sensitive. p → (p → p) [or (p · p) → p] and p → (p · p) are not theorems. The algebraic models do not satisfy contraction x ≤ x2. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #52 PLDT The deduction theorem for CPL states: Title Outline Σ, ψ ⊢CPL φ iff Σ ⊢CPL ψ → φ RL examples Congruences Theorem. Let Σ ∪ Ψ ∪{φ}⊆ FmL and L be a logic. Subvariety lattice (atoms) ■ L If is commutative, integral and contractive, then Subvariety lattice (joins) n Σ, Ψ ⊢L φ iff Σ ⊢L ( ψ ) → φ, Vi=1 i Logic A Hilbert system for some n ∈ ω,and ψi ∈ Ψ, i < n. Substructural logics ■ Algebraic semantics If L is commutative and integral, then Substructural logics (examples) n Substructural logics (examples) Σ, Ψ ⊢L φ iff Σ ⊢L ( ψ ) → φ, Qi=1 i PLDT Applications to logic for some n ∈ ω,and ψi ∈ Ψ, i < n. Representation - Frames ■ If L is commutative, then n Applications of frames Σ, Ψ ⊢L φ iff Σ ⊢L ( (ψi ∧ 1)) → φ, Qi=1 Undecidability for some n ∈ ω,and ψi ∈ Ψ, i < n. References ■ If L is any substructural logic, then n Σ, Ψ ⊢L φ iff Σ ⊢L (Qi=1 γi(ψi))\φ, for some n ∈ ω, iterated conjugates γi and ψi ∈ Ψ, i < n. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #53 Applications to logic ■ Hilbert systems (Algebraization) Title Outline ■ PLDT (Congruence generation for RL’s) RL examples ■ Maximal consistent logics (Atoms in Λ(RL)) Congruences ■ Axiomatizing intersections of logics (Joins in Λ(RL)) Subvariety lattice (atoms) Subvariety lattice (joins) ■ Translations (Glivenko, Kolmogorov) between logics, e.g., Logic ⊢CPL φ iff ⊢Int ¬¬φ (Structure of Λ(RL) and nuclei) A Hilbert system Substructural logics Algebraic semantics Algebra ↔ Logic Substructural logics (examples) Substructural logics (examples) PLDT congruence generation ↔ PLDT Applications to logic congruence extension ↔ localDT Representation - Frames EDPC ↔ deduction theorem Applications of frames subreduct axiomatization ↔ strong seperation (Hilbert) Undecidability decid. equational th. ↔ decid. provability (Gentzen) References finite generation ↔ cut elimination (+ fin. proof) amalgamation ↔ interpolation Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #54 Title Outline RL examples Congruences Representation - Frames Subvariety lattice (atoms) Subvariety lattice (joins) Logic Representation - Frames Lattice frames Residuated frames Formula hierarchy FL Basic substructural logics Examples of frames (FL) Examples of frames (FEP) GN Gentzen frames Proof Applications of frames Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #55 Lattice frames A lattice frame is a structure W ′ where and Title =(W, W , N) W Outline W ′ are sets and N is a binary relation from W to W ′. RL examples If L is a lattice, WL =(L,L, ≤) is a lattice frame. Congruences Subvariety lattice (atoms) For X ⊆ W and Y ⊆ W ′ we define Subvariety lattice (joins) ⊲ ′ X = {b ∈ W : xN b, for all x ∈ X} Logic ⊳ for all Y = {a ∈ W : aN y, y ∈ Y } Representation - Frames Lattice frames ⊲ ⊳ The maps ′ and ′ form a Residuated frames : P(W ) →P(W ) : P(W ) →P(W ) Formula hierarchy Galois connection. The map γN : P(W ) →P(W ), where FL ⊲⊳ Basic substructural logics γN (X)= X , is a closure operator. Examples of frames (FL) Examples of frames (FEP) GN Lemma. If L =(L, ∧, ∨) is a lattice and γ is a cl.op. on L, Gentzen frames Proof then (γ[L], ∧, ∨γ ) is a lattice. [x ∨γ y = γ(x ∨ y).] Applications of frames W Corollary. If is a lattice frame then the Galois algebra Undecidability W+ =(γN [P(W )], ∩, ∪γN ) is a complete lattice. References + If L is a lattice, WL is the Dedekind-MacNeille completion of L and x 7→ {x}⊳ is an embedding. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #56 Residuated frames A residuated frame is a structure W ′ where Title =(W, W ,N, ◦, 1) Outline W and W ′ are sets N ⊆ W × W ′, (W, ◦, 1) is a monoid and RL examples for all x, y ∈ W and w ∈ W ′ there exist subsets Congruences x w,w y ⊆ W ′ such that Subvariety lattice (atoms) (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y) Subvariety lattice (joins) Logic L W If is a RL, L =(L,L, ≤, ·, {1}) is a residuated frame. Representation - Frames Lattice frames A nucleus γ on a residuated lattice L is a closure operator on Residuated frames Formula hierarchy L such that γ(x)γ(y) ≤ γ(xy) (or γ(γ(x)γ(y)) = γ(xy)). FL Basic substructural logics L Examples of frames (FL) Theorem. Given a RL =(L, ∧, ∨, ·, \,/, 1) and a nucleus Examples of frames (FEP) L L GN on , the algebra γ =(Lγ , ∧, ∨γ , ·γ , \,/,γ(1)),isa Gentzen frames Proof residuated lattice, where x ·γ y = γ(x · y), x ∨γ y = γ(x ∨ y). Applications of frames Theorem. If W is a frame, then is a nucleus on γN Undecidability P(W, ◦, {1}). References Corollary. If W is a residuated frame then the Galois W+ algebra = P(W, ◦, 1)γN is a residuated lattice. Moreover, ⊳ for WL, x 7→ {x} is an embedding. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #57 Formula hierarchy ■ Polarity {∨, ·, 1}, {∧, \,/} Title Outline ■ The sets Pn, Nn of formulas are defined by: RL examples p p (0) P0 = N0 = the set of variables Congruences p6 p6 p p (P1) Nn ⊆Pn+1 Subvariety lattice (atoms) p p p p (P2) α,β ∈Pn+1 ⇒ α ∨ β,α · β, 1 ∈Pn+1 Subvariety lattice (joins) P N 3 3 (N1) Pn ⊆ Nn+1 Logic 6@I 6 (N2) α,β ∈ Nn+1 ⇒ α ∧ β ∈ Nn+1 Representation - Frames @ Lattice frames @ (N3) α ∈Pn+1,β ∈ Nn+1 ⇒ α\β,β/α ∈ Nn+1 Residuated frames Formula hierarchy P2 N2 ■ FL Pn+1 = hNniW,Q ; Nn+1 = hPniV,Pn+1\,/Pn+1 Basic substructural logics Examples of frames (FL) 6@I 6 ■ @ Pn ⊆Pn+1, Nn ⊆ Nn+1, S Pn = S Nn = Fm Examples of frames (FEP) @ GN ■ Gentzen frames P1-reduced: pi P1 N1 W Q Proof ■ N -reduced: (p p ··· p \r/q q ··· q ) Applications of frames 6@I@ 6 1 V 1 2 n 1 2 m @ Undecidability p p ··· p q q ··· q ≤ r References P0 N0 1 2 n 1 2 m ■ Sequent: a1,a2,...,an ⇒ a0 (x ⇒ a, a ∈ Fm, x ∈ Fm∗) Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #58 FL x⇒a y◦a◦z⇒c Title Outline y◦x◦z⇒c (cut) a⇒a (Id) RL examples y◦a◦z⇒c y◦b◦z⇒c x⇒a x⇒b Congruences (∧Lℓ) (∧Lr) (∧R) y◦a ∧ b◦z⇒c y◦a ∧ b◦z⇒c x⇒a ∧ b Subvariety lattice (atoms) y◦a◦z⇒c y◦b◦z⇒c x⇒a x⇒b Subvariety lattice (joins) (∨L) (∨Rℓ) (∨Rr) y◦a ∨ b◦z⇒c x⇒a ∨ b x⇒a ∨ b Logic Representation - Frames x⇒a y◦b◦z⇒c a ◦ x⇒b Lattice frames (\L) (\R) Residuated frames y◦x ◦ (a\b)◦z⇒c x⇒a\b Formula hierarchy FL Basic substructural logics x⇒a y◦b◦z⇒c x ◦ a⇒b Examples of frames (FL) (/L) (/R) Examples of frames (FEP) y◦(b/a) ◦ x◦z⇒c x⇒b/a GN Gentzen frames y◦a ◦ b◦z⇒c x⇒a y⇒b Proof (·L) (·R) Applications of frames y◦a · b◦z⇒c x ◦ y⇒a · b Undecidability y ◦ z⇒a (1L) (1R) References y◦1◦z⇒a ε⇒1 where a,b,c ∈ Fm, x,y,z ∈ Fm∗. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #59 FL Title x⇒a u[a]⇒c Outline (cut) (Id) u[x]⇒c a⇒a RL examples Congruences u[a]⇒c u[b]⇒c x⇒a x⇒b (∧Lℓ) (∧Lr) (∧R) Subvariety lattice (atoms) u[a ∧ b]⇒c u[a ∧ b]⇒c x⇒a ∧ b Subvariety lattice (joins) u[a]⇒c u[b]⇒c x⇒a x⇒b Logic (∨L) (∨Rℓ) (∨Rr) Representation - Frames u[a ∨ b]⇒c x⇒a ∨ b x⇒a ∨ b Lattice frames Residuated frames x⇒a u[b]⇒c a ◦ x⇒b Formula hierarchy (\L) (\R) FL u[x ◦ (a\b)]⇒c x⇒a\b Basic substructural logics Examples of frames (FL) Examples of frames (FEP) x⇒a u[b]⇒c x ◦ a⇒b GN (/L) (/R) Gentzen frames u[(b/a) ◦ x]⇒c x⇒b/a Proof Applications of frames u[a ◦ b]⇒c x⇒a y⇒b (·L) (·R) Undecidability u[a · b]⇒c x ◦ y⇒a · b References |u|⇒a (1L) (1R) u[1]⇒a ε⇒1 Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #59 Basic substructural logics If the sequent s is provable in FL from the set of sequents S, Title Outline we write S ⊢FL s. RL examples u[x ◦ y] ⇒ c Congruences (e) u[y ◦ x] ⇒ c (exchange) xy ≤ yx Subvariety lattice (atoms) Subvariety lattice (joins) u[x ◦ x] ⇒ c Logic Representation - Frames (c) 2 u[x] ⇒ c (contraction) x ≤ x Lattice frames Residuated frames Formula hierarchy FL |u|⇒ c Basic substructural logics Examples of frames (FL) (i) Examples of frames (FEP) u[x] ⇒ c (integrality) x ≤ 1 GN Gentzen frames We write FLec for FL + (e) + (c). Proof Applications of frames HL FL Theorem. The systems and are equivalent via the Undecidability maps s(ψ)=( ⇒ ψ) and References φ(a1,a2,...,an ⇒ a)= an\(... (a2\(a1\a)) ... ); Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #60 Examples of frames (FL) Consider the Gentzen system FL (full Lambek calculus). Title Outline We define the frame WFL, where RL examples Congruences ■ (W, ◦, ε) to be the free monoid over the set Fm of all formulas Subvariety lattice (atoms) Subvariety lattice (joins) ■ ′ W = SW × Fm, where SW is the set of all unary linear Logic polynomials u[x] = y◦x◦z of W , and Representation - Frames ■ Lattice frames x N (u, a) iff ⊢FL u[x] ⇒ a. Residuated frames Formula hierarchy For FL Basic substructural logics (u, a) x = {(u[_ ◦ x],a)} and x (u, a)= {(u[x ◦ _],a)}, Examples of frames (FL) Examples of frames (FEP) we have GN Gentzen frames x ◦ yN(u, a) iff ⊢FL u[x ◦ y] ⇒ a Proof Applications of frames iff ⊢FL u[x◦y] ⇒ a Undecidability iff xN(u[_ ◦ y],a) References iff yN(u[x ◦ _],a). Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #61 Examples of frames (FEP) Let A be a residuated lattice and B a partial subalgebra of A. Title Outline RL examples We define the frame WA,B, where Congruences ■ (W, ·, 1) to be the submonoid of A generated by B, Subvariety lattice (atoms) ■ ′ W = SB × B, where SW is the set of all unary linear Subvariety lattice (joins) polynomials u[x] = y◦x◦z of (W, ·, 1), and Logic ■ x N (u, b) by u[x] ≤A b. Representation - Frames Lattice frames Residuated frames For Formula hierarchy FL (u, a) x = {(u[_ · x],a)} and x (u, a)= {(u[x · _],a)}, Basic substructural logics Examples of frames (FL) we have Examples of frames (FEP) GN x · yN(u, a) iff u[x · y] ≤ a Gentzen frames iff xN(u[_ · y],a) Proof Applications of frames iff yN(u[x · _],a). Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #62 GN xNa aNz Title (CUT) (Id) Outline xNz aNa RL examples xNa bNz (\L) a ◦ xNb (\R) Congruences x ◦ (a\b)Nz xNa\b Subvariety lattice (atoms) Subvariety lattice (joins) xNa bNz (/L) x ◦ aNb (/R) (b/a) ◦ xNz xNb/a Logic Representation - Frames a ◦ bNz xNa yNb Lattice frames (·L) (·R) Residuated frames a · bNz x ◦ yNa · b Formula hierarchy FL aNz bNz xNa xNb Basic substructural logics ( L ) ( L ) ( R) Examples of frames (FL) ∧ ℓ ∧ r ∧ Examples of frames (FEP) a ∧ bNz a ∧ bNz xNa ∧ b GN aNz bNz xNa xNb Gentzen frames (∨L) (∨Rℓ) (∨Rr) Proof a ∨ bNz xNa ∨ b xNa ∨ b Applications of frames εNz (1L) (1R) Undecidability 1Nz εN1 References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #63 Gentzen frames The following properties hold for WL, WFL and WA B: Title , Outline 1. W is a residuated frame RL examples 2. B is a (partial) algebra of the same type, (B = L, Fm, B) Congruences Subvariety lattice (atoms) 3. B generates (W, ◦, ε) (as a monoid) Subvariety lattice (joins) ′ 4. W contains a copy of B (b ↔ (id, b)) Logic GN ′ Representation - Frames 5. N satisfies , for all a, b ∈ B, x, y ∈ W , z ∈ W . Lattice frames Residuated frames Formula hierarchy W B FL We call such pairs ( , ) Gentzen frames. Basic substructural logics Examples of frames (FL) A cut-free Gentzen frame is not assumed to satisfy the Examples of frames (FEP) GN (CUT)-rule. Gentzen frames Proof Theorem. Given a Gentzen frame (W, B), the map Applications of frames {}⊳ : B → W+, b 7→ {b}⊳ is a (partial) homomorphism. Undecidability (Namely, if a, b ∈ B and a • b ∈ B (• is a connective) then References ⊳ ⊳ ⊳ {a •B b} = {a} •W+ {b} ). Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #64 Proof Key Lemma. Let (W, B) be a Gentzen frame. For all Title Outline , W+ and for every connective , if , a, b ∈ B k, l ∈ • a • b ∈ B RL examples a ∈ X ⊆{a}⊳ and b ∈ Y ⊆{b}⊳, then Congruences ⊳ ⊳ + + 1. a •B b ∈ X •W Y ⊆{a •B b} (1B ∈ 1W ⊆{1B} ) Subvariety lattice (atoms) ⊳ ⊳ ⊳ 2. In particular, a •B b ∈{a} •W+ {b} ⊆{a •B b} . Subvariety lattice (joins) 3. Furthermore, because of (CUT), we have equality. Logic Representation - Frames ⊳ Lattice frames Proof Let • = ∨. If x ∈ X, then x ∈{a} ; so xNa and Residuated frames ⊳ ⊳ xNa ∨ b,by(∨Rℓ); hence x ∈{a ∨ b} and X ⊆{a ∨ b} . Formula hierarchy ⊳ ⊳ FL Likewise Y ⊆{a ∨ b} , so X ∪ Y ⊆{a ∨ b} and Basic substructural logics ⊳ Examples of frames (FL) X ∨ Y = γ(X ∪ Y ) ⊆{a ∨ b} . Examples of frames (FEP) GN ⊳ Gentzen frames On the other hand, let X ∨ Y ⊆{z} , for some z ∈ W . Then, Proof ⊳ a ∈ X ⊆ X ∨ Y ⊆{z} , so aNz. Similarly, bNz, so a ∨ bNz Applications of frames ⊳ by (∨L), hence a ∨ b ∈{z} . Thus, a ∨ b ∈ X ∨ Y . Undecidability We used that every closed set is an intersection of basic References closed sets {z}⊳, for z ∈ W . Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #65 Title Outline RL examples Congruences Applications of frames Subvariety lattice (atoms) Subvariety lattice (joins) Logic Representation - Frames Applications of frames DM-completion Completeness - Cut elimination FMP FEP Finiteness Equations 1 Equations 2 Structural rules Amalgamation-Interpolation Applications Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #66 DM-completion For a residuated lattice L, we associated the Gentzen frame Title Outline (WL, L). RL examples + The underlying poset of WL is the Dedekind-MacNeille Congruences completion of the underlying poset reduct of L. Subvariety lattice (atoms) Subvariety lattice (joins) ⊳ + Theorem. The map x 7→ x is an embedding of L into WL . Logic Representation - Frames Applications of frames DM-completion Completeness - Cut elimination FMP FEP Finiteness Equations 1 Equations 2 Structural rules Amalgamation-Interpolation Applications Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #67 Completeness - Cut elimination Fm B ¯ Fm W+ Title For every homomorphism f : → , let f : L → Outline ¯ ⊳ be the homomorphism that extends f(p)= {f(p)} (p: RL examples variable.) Congruences Corollary. If (W, B) is a cf Gentzen frame, for every Subvariety lattice (atoms) homomorphism f : Fm → B, we have f(a) ∈ f¯(a) ⊆↓ f(a). Subvariety lattice (joins) If we have (CUT), then f¯(a)=↓ f(a). Logic Representation - Frames We define WFL |= x ⇒ c by f(x) N f(c), for all f. Applications of frames + Theorem. If W · , then WFL . DM-completion FL |= x ≤ c |= x ⇒ c Completeness - Cut elimination ⊳ Idea: For Fm B, ¯ ¯ , so FMP f : → f(x) ∈ f(x) ⊆ f(c) ⊆{f(c)} FEP f(x) N f(c). Finiteness Equations 1 + Equations 2 Corollary. FL is complete with respect to WFL. Structural rules Amalgamation-Interpolation + Applications Corollary. The algebra WFL generates RL. Undecidability W f FL The frame FL corresponds to cut-free . References Corollary (CE). FL and FLf prove the same sequents. Corollary. FL and the equational theory of RL are decidable. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #68 Finite model property For WFL, given ′ (if , then Title (x,z) ∈ W × W z =(u, c) u(x) ⇒ c Outline ↑ is a sequent), we define (x,z) as the smallest subset of RL examples ′ W × W that contains (x,z) and is closed upwards with Congruences FLf ↑ respect to the rules of . Note that (x,z) is finite. Subvariety lattice (atoms) Subvariety lattice (joins) The new frame W′ associated with N ′ = N ∪ ((y,v)↑)c is Logic residuated and Gentzen. Representation - Frames Clearly, (N ′)c is finite, so it has a finite domain Dom((N ′)c) ′ c Applications of frames and codomain Cod((N ) ). DM-completion ′ c ⊳ ⊳ Completeness - Cut elimination For every z 6∈ Cod((N ) ), {z} = W . So, {{z} : z ∈ W } is FMP FEP ′ W′+ finite and a basis for γN . So, is finite. Finiteness FL Equations 1 Moreover, if u(x) ⇒ c is not provable in , then it is not valid Equations 2 W′+ Structural rules in . Amalgamation-Interpolation Applications Corollary. The system FL has the finite model property. Undecidability References Corollary. The variety of residuated lattices is generated by its finite members. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #69 FEP A class of algebras K has the finite embeddability property Title Outline (FEP) if for every A ∈ K, every finite partial subalgebra B of RL examples A can be (partially) embedded in a finite D ∈ K. Congruences Theorem. Every variety of integral RL’s axiomatized by Subvariety lattice (atoms) equartions over {∨, ·, 1} has the FEP. Subvariety lattice (joins) ■ B W+ ⊳ B W+ Logic embeds in A,B via {_} : → Representation - Frames + ■ W is finite Applications of frames A,B DM-completion Completeness - Cut elimination ■ W+ FMP A,B ∈V FEP Finiteness These varieties are generated as quasivarieties Equations 1 Corollary. Equations 2 by their finite members. Structural rules Amalgamation-Interpolation Corollary. The corresponding logics have the strong finite Applications model property: Undecidability if Φ 6⊢ ψ, for finite Φ, then there is a finite counter-model, References namely there is D ∈V and a homomorphism f : Fm → D, such that f(φ)=1, for all φ ∈ Φ, but f(ψ) 6=1. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #70 Finiteness W+ Title Idea: As every element in A,B is an intersection of basic Outline elements. So it suffices to prove that there are only finitely RL examples many such elements. Congruences W WM Subvariety lattice (atoms) Idea: Replace the frame A,B by one A,B, where it is easier to work. Subvariety lattice (joins) Logic Let M be the free monoid with unit over the set B and Representation - Frames f : M → W the extension of the identity map. Applications of frames DM-completion Completeness - Cut elimination N FMP f ′ M −→ W −− W FEP Finiteness Equations 1 . Equations 2 Structural rules Amalgamation-Interpolation Applications Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #71 Equations 1 Idea: Express equations over {∨, ·, 1} at the frame level. Title Outline For an equation ε over {∨, ·, 1} we distribute products over RL examples joins to get s1 ∨···∨ sm = t1 ∨···∨ tn. si, tj: monoid terms. Congruences Subvariety lattice (atoms) s ∨···∨ s ≤ t ∨···∨ t and t ∨···∨ t ≤ s ∨···∨ s . 1 m 1 n 1 n 1 m Subvariety lattice (joins) The first is equivalent to: &(sj ≤ t1 ∨···∨ tn). Logic Representation - Frames We proceed by example: x2y ≤ xy ∨ yx Applications of frames 2 DM-completion (x1 ∨ x2) y ≤ (x1 ∨ x2)y ∨ y(x1 ∨ x2) Completeness - Cut elimination FMP 2 2 FEP x1y ∨ x1x2y ∨ x2x1y ∨ x2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2 Finiteness Equations 1 Equations 2 x1x2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2 Structural rules Amalgamation-Interpolation Applications x1y ≤ v x2y ≤ v yx1 ≤ v yx2 ≤ v Undecidability x1x2y ≤ v References x ◦ yNz x ◦ yNz y ◦ x N z y ◦ x N z 1 2 1 2 R(ε) x1 ◦ x2 ◦ yNz Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #72 Equations 2 Theorem. If (W, B) is a Gentzen frame and ε an equation Title Outline over , then W B satisfies R iff W+ satisfies . {∨, ·, 1} ( , ) (ε) ε RL examples (The linearity of the denominator of R(ε) plays an important Congruences role in the proof.) Subvariety lattice (atoms) Subvariety lattice (joins) Corollary If an equation over {∨, ·, 1} is valid in A, then it is Logic W+ B A Representation - Frames also valid in A,B, for every partial subalgebra of . Applications of frames DM-completion W+ Completeness - Cut elimination Consequently, A,B ∈V. FMP FEP Finiteness Equations 1 Equations 2 Structural rules Amalgamation-Interpolation Applications Undecidability References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #73 Structural rules Given an equation ε of the form t ≤ t ∨···∨ t , where t Title 0 1 n i Outline are {·, 1}-terms we construct the rule R(ε) RL examples Congruences u[t1] ⇒ a ··· u[tn] ⇒ a (R(ε)) Subvariety lattice (atoms) u[t0] ⇒ a Subvariety lattice (joins) where the ti’s are evaluated in (W, ◦, ε). Such a rule is called Logic linear if all variables in t0 are distinct. Representation - Frames Applications of frames Theorem. Every system obtained from FL by adding linear DM-completion Completeness - Cut elimination rules has the cut elimination property. FMP FEP Finiteness A set of rules of the form R(ε) is called reducing if there is a Equations 1 Equations 2 complexity measure that decreases with upward applications Structural rules Amalgamation-Interpolation of the rules (and the rules of FL). Applications Undecidability Theorem. Every system obtained from FL by adding linear References reducing rules is decidable. The subvariety of residuated lattices axiomatized by the corresponding equations has decidable equational theory. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #74 Amalgamation-Interpolation Given algebras A, B, C, maps f : A → B and g : A → C and Title Outline Gentzen frames WB, WC, we define the frame W on B ∪ C, RL examples where N is specified by ΓB, ΓC N β iff there exists α ∈ A Congruences such that ΓC NC g(α) and ΓB,f(α) NB β. Subvariety lattice (atoms) ⊳ + Theorem. W is a Gentzen frame. Hence : B ∪ C → W Subvariety lattice (joins) is a quasihomomorhism. Logic Representation - Frames Let D = W+ and h, k the restrictions of ⊳ to B and C. Applications of frames DM-completion Corollary. The maps h : B → D and k : C → D are Completeness - Cut elimination FMP homomorphisms. Moreover, injections and surjections FEP Finiteness transfer: If f is injective (surjective), so is h. Equations 1 Equations 2 RL Structural rules Corollary. Commutative has the amalgamation property Amalgamation-Interpolation (f,g injective) and the congruence extension property (f Applications injective, g surjective). Undecidability References Corollary. FLe has the Craig interpolation propety and enjoys the Local Deduction Theorem. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #75 Applications ■ Cut-elimination (CE) and finite model property (FMP) for Title Outline FL, (cyclic) InFL. Generation by finite members for RL, RL examples InFL Congruences ■ The finite embeddability property (FEP) for integral RL with Subvariety lattice (atoms) {∨, ·, 1}-axioms. Subvariety lattice (joins) ■ The strong separation property for HL Logic Representation - Frames ■ The above extend to the non-associative case, as well as Applications of frames with the addition of suitable structural rules DM-completion Completeness - Cut elimination ■ FMP Amalgamation for commutative RL and interpolation for FEP commutative FL Finiteness Equations 1 ■ Equations 2 (Craig) Interpolation, Robinson Property, disjunction Structural rules Amalgamation-Interpolation property and Maximova variable separation property for Applications FL e Undecidability ■ Super-amalgamation, Transferable injections, Congruence References extension property for commutative RL Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #76 Title Outline RL examples Congruences Undecidability Subvariety lattice (atoms) Subvariety lattice (joins) Logic Representation - Frames Applications of frames Undecidability (Un)decidability Word problem (1) Word problem (2) Word problem (3) Word problem (4) Word problem (5) References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #77 (Un)decidability Theorem. The quasiequational theory of RL is undecidable. Title Outline (Because we can embed semigroups/monoids.) The same RL examples holds for commutative RL. Congruences Subvariety lattice (atoms) Theorem. The equational theory of modular RL is Subvariety lattice (joins) undecidable. (By transfering the corresponding result for Logic modular lattices). Representation - Frames Applications of frames Theorem. The equational theory of commutative, distributive Undecidability (Un)decidability RL is decidable. Word problem (1) Word problem (2) Word problem (3) Word problem (4) Word problem (5) References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #78 Word problem (1) A finitely presented algebra A =(X|R) (in a class K) has a Title Outline solvable word problem (WP) if there is an algorithm that, RL examples given any pair of words over X, decides if they are equal or Congruences not. Subvariety lattice (atoms) A class of algebras has solvable WP if all finitely presented Subvariety lattice (joins) algebras in it do. Logic Representation - Frames For example, the varieties of semigroups, groups, ℓ-groups, Applications of frames modular lattices have unsolvable WP. Undecidability (Un)decidability Word problem (1) Word problem (2) Main result: The variety CDRL of commutative, distributive Word problem (3) Word problem (4) residuated lattices has unsolvable WP. Word problem (5) References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #79 Word problem (2) Main idea: Embed semigroups, whose WP is unsolvable. Title Outline Residuated lattices have a semigroup operation ·, but RL examples commutative semigroups have a decidable WP. Congruences Subvariety lattice (atoms) Alternative approach: Come up with another term definable Subvariety lattice (joins) operation ⊙ in residuated lattices that is associative. Logic Representation - Frames Intuition: Coordinization in projective geometry and modular Applications of frames lattices. Undecidability (Un)decidability Word problem (1) (cki ∨ d) ∧ (ak ∨ aj) Word problem (2) Word problem (3) cjk Word problem (4) Word problem (5) ak References (b ∨ cjk) ∧ (ai ∨ ak) cik ai cij d bb ⊙ij d aj Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #80 Word problem (3) We define an n-frame in a residuated lattice consisting of Title Outline elements a , ··· ,a and c , for 1 ≤ i < j ≤ n and satisfying 1 n ij RL examples certain conditions (the a ’s are linearly independent, c is on i ij Congruences the line generated by a and a etc.). i j Subvariety lattice (atoms) We use the operations ∨ and ·. Subvariety lattice (joins) Logic Representation - Frames We define the ‘line’ Lij and the operation ⊙ij. Applications of frames Undecidability (Un)decidability Theorem Given an 4-frame in a residuated lattice the Word problem (1) Word problem (2) algebra (Lij, ⊙ij) is a semigroup. Word problem (3) Word problem (4) Word problem (5) References Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #81 Word problem (4) Given a finitely presented semigroup S and a variety V of Title Outline residuated lattices, we construct a finitely presented RL examples residuated lattice A(S, V) in V. Congruences Given a vector space W, its powerset forms a distributive Subvariety lattice (atoms) Subvariety lattice (joins) residuated lattice AW. Logic Theorem If Representation - Frames 1. V is a variery of distributive residuated lattices containing Applications of frames AW for some infinite-dimentional vector space W and Undecidability (Un)decidability S Word problem (1) 2. is a finitely presented semigroup with unsolvable WP, Word problem (2) A S Word problem (3) then the residuated lattice ( , V) in V has unsolvable WP. Word problem (4) Word problem (5) In the proof we show that for every pair of semigroup words References r, s, S satisfies r·(¯x)= s·(¯x) iff A(S, V) satisfies r⊙(¯x′)= s⊙(¯x′). Corollary The WP of CDRL is unsolvable. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #82 Word problem (5) A quasi-equation is a formula of the form Title Outline RL examples (s1 = t1 & s2 = t2 & ··· & sn = tn) ⇒ s = t Congruences The solvability/decidability of the WP states that given any Subvariety lattice (atoms) set of equations s1 = t1,s2 = t2,...sn = tn there is an Subvariety lattice (joins) algorithm that decides all quasi-equations of the above form. Logic Representation - Frames The solvability of the quasi-equational theory states that Applications of frames there is an algorithm that decides all quasi-equations of the Undecidability above form. (Un)decidability Word problem (1) Word problem (2) Word problem (3) Corollary The quasi-equational theory of CDRL is Word problem (4) undecidable. Word problem (5) References Corollary The equational theory of CDRL is decidable. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #83 Title Outline RL examples Congruences References Subvariety lattice (atoms) Subvariety lattice (joins) Logic Representation - Frames Applications of frames Undecidability References People Bibliography Bibliography Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #84 People Some people involved: Title Outline P. Bahls P. Jipsen RL examples Congruences F. Bernadinelli T. Kowalski Subvariety lattice (atoms) W. Blok H. Ono Subvariety lattice (joins) K. Blount L. Rafter Logic A. Ciabattoni J. Raftery Representation - Frames J. Cole K. Terui Applications of frames Undecidability R.P. Dilworth C. Tsinakis References N. Galatos C. van Alten People Bibliography J. Hart M. Ward Bibliography N. Galatos, P. Jipsen, T. Kowalski and H. Ono. Residuated Lattices: an algebraic glimpse at substructural logics, Studies in Logics and the Foundations of Mathematics, Elsevier, 2007. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #85 Bibliography P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis, Cancellative Title Outline Residuated Lattices, Algebra Universalis, 50(1) (2003), 83-106. RL examples F. Bernadinelli, H. Ono and P. Jipsen, Algebraic aspects of cut elimination, Congruences Studia Logics 68 (2001), 1-32. Subvariety lattice (atoms) W. Blok and C. van Alten, On the finite embeddability property for residuated Subvariety lattice (joins) lattices, pocrims and BCK-algebras, Algebra & substructural logics Logic (Tatsunokuchi, 1999). Rep. Math. Logic No. 34 (2000), 159–165. Representation - Frames K. Blount and C. Tsinakis, The structure of residuated lattices, Internat. J. Applications of frames Algebra Comput. 13 (2003), no. 4, 437–461. Undecidability A. Ciabattoni, N. Galatos and K. Terui. From Axioms to analytic rules in References nonclassical logics, LICS’08, 229 – 240. People Bibliography A. Ciabattoni, N. Galatos and K. Terui. The expressive power of structural Bibliography rules for FL, manuscript. N. Galatos. Equational bases for joins of residuated-lattice varieties, Studia Logica 76(2) (2004), 227-240. N. Galatos. Minimal varieties of residuated lattices, Algebra Universalis 52(2) (2005), 215-239. N. Galatos and P. Jipsen. Residuated frames and applications to decidability, manuscript. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #86 Bibliography N. Galatos, J. Olson and J. Raftery, Irreducible residuated semilattices and Title Outline finitely based varieties, Reports on Mathematical Logic 43 (2008), 85-108. RL examples N. Galatos and H. Ono. Algebraization, parameterized local deduction Congruences theorem and interpolation for substructural logics over FL, Studia Logica 83 Subvariety lattice (atoms) (2006), 279-308. Subvariety lattice (joins) N. Galatos and H. Ono. Glivenko theorems for substructural logics over FL, Logic Journal of Symbolic Logic 71(4) (2006), 1353-1384. Representation - Frames N. Galatos and H. Ono. Cut elimination and strong separation for Applications of frames non-associative substructural logics, manuscript. Undecidability J. Hart, L. Rafter and C. Tsinakis, The Structure of Commutative Residuated References Lattices, Internat. J. Algebra Comput. 12 (2002), no. 4, 509-524. People Bibliography H. Ono, Semantics for substructural logics, Substructural logics, 259–291, Bibliography Stud. Logic Comput., 2, Oxford Univ. Press, New York, 1993. K. Terui, Which Structural Rules Admit Cut Elimination? - An Algebraic Criterion, JSL 72(3) (2007), 738–754. C. van Alten and J. Raftery, Rule separation and embedding theorems for logics without weakening, Studia Logica 76(2) (2004), 241–274. M. Ward and R. P. Dilworth, Residuated Lattices, Transactions of the AMS 45 (1939), 335–354. Nikolaos Galatos, SSAOS, Treštˇ 2008 Residuated lattices - slide #87