Residuated lattices Nikolaos Galatos University of Denver [email protected] 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.