Semantics: Residuated Frames

Transcription

1 Semantics: Residuated Frames Peter Jipsen, Chapman University, Orange, California, USA joint work with Nikolaos Galatos, University of Denver, Colorado, USA ICLA, January, 2009 P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

2 Outline Part I Universal Algebra Examples of residuated lattices Congruences and normal filters The lattice of subvarieties Varieties generated by positive universal classes Direct decompositions and poset products Part II Residuated Frames Decidability Poset products of residuated lattices P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

3 Part II: Frames of residuated lattices We now consider semantics for residuated lattices Kripke frames for modal logics are a useful tool for building counter models They also led to many interesting notions such as frame completeness, canonical frames, and correspondence results Aim: To present frames for arbitrary residuated lattices and connect them with the proof theory of substructural logic P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

4 Galois connections and closure operators For posets P, Q, maps f : P Q, g : Q P are a Galois connection if y f (x) x g(y), for all x P, y Q A map c : P P is a closure operator if x y implies c(x) c(y), x c(x) and c(c(x)) = c(x) Exercise: If f, g are a Galois connection then c(x) = g(f (x)) is a closure operator on P A lattice frame is a structure W = (W,W, N) where W and W are sets and N is a binary relation from W to W E.g. If L is a lattice, W L = (L, L, ) is a lattice frame Let J(L) be the set completely join irreducibles and M(L) be the set completely meet irreducibles of L. Then L + = (J(L), M(L), ) is a lattice frame. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

5 Lattice frames For X W and Y W we define the polarities X = {b W : x N b, for all x X } Y = {a W : a N y, for all y Y } Exercise: The maps : P(W) P(W ) and : P(W ) P(W) form a Galois connection If γ N (X) = X then γ N : P(W) P(W) is a closure operator Lemma. If L = (L,, ) is a lattice and γ is a closure operator on L, then (γ[l],, γ ) is a lattice where x γ y = γ(x y) Corollary. If W is a lattice frame then the Galois algebra W + = (γ N [P(W)],, γn ) is a complete lattice If L is a lattice, W + L is the Dedekind-MacNeille completion of L and x {x} is an embedding P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

6 Residuated frames A residuated frame is a structure W = (W,W, N,,, ) where W and W are sets, N W W, W 3, W W W and W W W such that for all x, y W, w W (x y) N w y N (x w) x N (w y) Here x y = {z (x, y, z) } and similarly for, We also use X N y to abbreviate x N y for all x X and likewise for x N Y A ternary relation structure W = (W, ) is said to be associative if it satisfies (x y) z = x (y z), i.e., if it satisfies the following equivalence u[(x, y, u) and (u, z, w) ] v[(x, v, w) and (y, z, v) ] It is said to have a unit E W if x E = {x} = E x, i.e., if e E[(x, e, y) ] x = y e E[(e, x, y) ] P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

7 Nuclei A nucleus γ on a residuated lattice L is a closure operator on L such that γ(x)γ(y) γ(xy) (or γ(γ(x)γ(y)) = γ(xy)). Theorem. Given a RL L = (L,,,, \, /,1) and a nucleus on L, the algebra L γ = (L γ,, γ, γ, \, /, γ(1)), is a residuated lattice, where x γ y = γ(x y), x γ y = γ(x y). Theorem. For a frame W, γ N is a nucleus on (P(W),,,, \, /, {1}) Corollary. If W is a residuated frame then the Galois algebra W + = (P(W),,,, \, /,1) γn is a complete residuated lattice. Moreover, for W L, x {x} is an embedding. If L is a RL, W L = (L, L,,, \, /) is a residuated frame. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

8 Frames of complete perfect lattices A lattice L is perfect if every element is a join of elements of J(L) and a meet of elements of M(L) E.g. a Boolean algebra is perfect if and only if it is atomic For a perfect residuated lattice A, let A + = (J(A), M(A),,,,, E) where x y = {z J(A) z xy} and E = {z J(A) z 1} Theorem A + is a residuated frame and if A is complete then (A + ) + = A In particular, any finite lattice is complete and perfect So for finite residuated lattices, residuated frames give a compact representation analogous to atom structures for relation algebras P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

9 All (dually-)nonisomorphic lattices with 7 elements K n,k = selfdual lattice of size n L n,k = nonselfdual lattices of size n K 1,1 K 2,1 K 3,1 K 4,1 K 4,2 L 5,1 K 5,1 K 5,2 K 5,3 L 6,1 L 6,2 L 6,3 L 6,4 K 6,1 K 6,2 K 6,3 K 6,4 K 6,5 K 6,6 K 6,7 L 7,1 L 7,2 L 7,3 L 7,4 L 7,5 L 7,6 L 7,7 L 7,8 L 7,9 L 7,10 L 7,11 L 7,12 L 7,13 L 7,14 L 7,15 L 7,16 L 7,17 L 7,18 L 7,19 L 7,20 K 7,1 K 7,2 K 7,3 K 7,4 K 7,5 K 7,6 K 7,7 K 7,8 K 7,9 K 7,10 K 7,11 K 7,12 K 7,13 P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

10 FL y a z c y a b z c ( Ll) y a z c y b z c x a y a z c y x z c (cut) a a (Id) ( L) y a b z c x a y b z c y x (a\b) z c (\L) y b z c y a b z c ( Lr) x a x b x a b x a y b z c y (b/a) x z c (/L) x a x a b ( Rl) a x b x a\b (\R) x a b x b/a (/R) y a b z c y a b z c ( L) x a y b x y a b ( R) y z a y 1 z a (1L) ε 1 (1R) where a, b, c Fm, x, y, z Fm. ( R) x b x a b ( Rr) P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

11 FL with context notation x a u[a] c (cut) u[x] c u[a] c u[a b] c ( Ll) u[a] c u[b] c ( L) u[a b] c x a u[b] c u[x (a\b)] c (\L) a a (Id) u[b] c u[a b] c ( Lr) x a x b x a b x a x a b ( Rl) a x b x a\b (\R) x a u[b] c u[(b/a) x] c (/L) x a b x b/a (/R) u[a b] c u[a b] c ( L) x a y b x y a b ( R) u[ε] a u[1] a (1L) ε 1 (1R) ( R) x b x a b ( Rr) P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

12 Basic substructural logics If the sequent s is provable in FL from the set of sequents S, we write S FL s. u[x y] c u[y x] c (e) (exchange) xy yx u[x x] c u[x] c (c) (contraction) x x 2 u[ε] c u[x] c (i) (integrality) x 1 We write FL ec for FL + (e) + (c). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

13 Examples of frames (FL) Consider the Gentzen system FL (full Lambek calculus). We define the frame W FL, where (W,, ε) to be the free monoid over the set Fm of all formulas W = S W Fm, where S W is the set of all unary linear polynomials u[x] = y x z of W, and x N (u, a) iff FL u[x] a. For (u, a) x = {(u[ x], a)} and x (u, a) = {(u[x ], a)}, we have x yn(u, a) iff FL u[x y] a iff FL u[x y] a iff xn(u[ y], a) iff yn(u[x ], a). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

14 Examples of frames (FEP) Let A be a residuated lattice and B a partial subalgebra of A. We define the frame W A,B, where (W,, 1) to be the submonoid of A generated by B, W = S B B, where S W is the set of all unary linear polynomials u[x] = y x z of (W,, 1), and x N (u, b) by u[x] A b. For (u, a) x = {(u[ x], a)} and x (u, a) = {(u[x ], a)}, we have x yn(u, a) iff u[x y] a iff xn(u[ y], a) iff yn(u[x ], a). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

15 GN anz a bnz ( Ll) anz bnz ( L) a bnz xna anz (CUT) xnz xna bnz x (a\b)nz (\L) xna bnz (b/a) xnz (/L) ana (Id) a xnb xna\b (\R) x anb xnb/a (/R) a bnz a bnz ( L) xna ynb x yna b ( R) bnz a bnz ( Lr) xna xnb xna b xna xna b ( Rl) εnz 1Nz (1L) εn1 (1R) ( R) xnb xna b ( Rr) P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

16 Gentzen frames The following properties hold for W L, W FL and W A,B : 1 W is a residuated frame 2 B is a (partial) algebra of the same type, (B = L,Fm,B) 3 B generates (W,, ε) (as a monoid) 4 W contains a copy of B (b (id, b)) 5 N satisfies GN, for all a, b B, x, y W, z W. We call such pairs (W,B) Gentzen frames. A cut-free Gentzen frame is not assumed to satisfy the (CUT)-rule. Theorem. Given a Gentzen frame (W,B), the map {} : B W +, b {b} is a (partial) homomorphism. (Namely, if a, b B and a b B ( is a connective) then {a B b} = {a} W + {b} ). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

17 Proof Key Lemma. Let (W,B) be a Gentzen frame. For all a, b B, X, Y W + and for every connective, if a b B, a X {a} and b Y {b}, then 1 a B b X W + Y {a B b} (1 B 1 W + {1 B } ) 2 In particular, a B b {a} W + {b} {a B b}. 3 Furthermore, because of (CUT), we have equality. Proof Let =. If x X, then x {a} ; so xna and xna b, by ( Rl); hence x {a b} and X {a b}. Likewise Y {a b}, so X Y {a b} and X Y = γ(x Y ) {a b}. On the other hand, let X Y {z}, for some z W. Then, a X X Y {z}, so anz. Similarly, bnz, so a bnz by ( L), hence a b {z}. Thus, a b X Y. We used that every closed set is an intersection of basic closed sets {z}, for z W. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

18 DM-completion For a residuated lattice L, we associated the Gentzen frame (W L,L). The underlying poset of W + L is the Dedekind-MacNeille completion of the underlying poset reduct of L. Theorem. The map x x is an embedding of L into W + L. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

19 Completeness - Cut elimination For every homomorphism f : Fm B, let f : Fm L W + be the homomorphism that extends f (p) = {f (p)} for any variable p Corollary. If (W,B) is a cf Gentzen frame then for every homomorphism f : Fm B, we have f (a) f (a) {f (a)}. With CUT f (a)={f (a)} We define W FL = x c if f (x) N f (c) for all f : Fm Fm Theorem. If W + FL = x c, then W FL = x c. Idea: For f : Fm B, f (x) f (x) f (c) {f (c)}, so f (x) N f (c). Corollary. FL is complete with respect to W + FL. Corollary. The algebra W + FL generates RL. The frame W FL f corresponds to cut-free FL. Corollary (CE). FL and FL f prove the same sequents. Corollary. FL and the equational theory of RL are decidable. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

20 Finite model property For W FL, given (x, z) W W (if z = (u, c), then u(x) c is a sequent), we define (x, z) as the smallest subset of W W that contains (x, z) and is closed upwards with respect to the rules of FL f. Note that (x, z) is finite. The new frame W associated with N = N ((y, v) ) c is residuated and Gentzen. (N ) c is finite, so has finite domain Dom((N ) c ) and codomain Cod((N ) c ) For every z Cod((N ) c ), {z} = W. So, {{z} : z W } is finite and a basis for γ N. So W + is finite. Moreover, if u(x) c is not provable in FL, then it is not valid in W +. Corollary. The system FL has the finite model property. Corollary. The variety RL is generated by its finite members. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

21 FEP 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 (partially) embedded in a finite D K. The corresponding logic has the strong finite model property: if Φ ψ, for finite Φ, then there is a finite counter-model, namely there is D K and a homomorphism f : Fm D, such that f (φ) = 1, for all φ Φ, but f (ψ) 1. For logics with finitely many axioms and rules, this implies that the deducibility relation is decidable For a finitely axiomatized variety, FEP implies the decidability of the universal theory P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

22 FEP for integral RLs with {,, 1}-equations Blok and van Alten 2002 proved FEP for integral RLs, and extended it to residuated groupoids (2005) Theorem. Every variety of integral RL s axiomatized by equations over {,, 1} has the FEP. B embeds in W + A,B via { } : B W + W + A,B is finite W + A,B V Corollary. These varieties are generated as quasivarieties by their finite members. Corollary. The corresponding logics have the strong finite model property P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

23 Finiteness Idea for finiteness: Every element in W + A,B is an intersection of basic elements. So it suffices to prove that there are only finitely many such elements. Replace the frame W A,B by one WA,B M, where it is easier to work. Let M be the free monoid with unit over the set B and f : M W the extension of the identity map.. M f W N W P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

24 Equations 1 Idea: Express equations over {,, 1} at the frame level. For an equation ε over {,, 1} we distribute products over joins to get s 1 s m = t 1 t n. s i, t j : monoid terms. s 1 s m t 1 t n and t 1 t n s 1 s m. The first is equivalent to: &(s j t 1 t n ). We proceed by example: x 2 y xy yx (x 1 x 2 ) 2 y (x 1 x 2 )y y(x 1 x 2 ) x 2 1 y x 1x 2 y x 2 x 1 y x 2 2 y x 1y x 2 y yx 1 yx 2 x 1 x 2 y x 1 y x 2 y yx 1 yx 2 x 1 y v x 2 y v yx 1 v yx 2 v x 1 x 2 y v x 1 y N z x 2 y N z y x 1 N z y x 2 N z R(ε) x 1 x 2 y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

25 Equations 2 Theorem. If (W,B) is a Gentzen frame and ε an equation over {,, 1}, then (W,B) satisfies R(ε) iff W + satisfies ε. (The linearity of the denominator of R(ε) plays an important role in the proof.) Corollary. If an equation over {,, 1} is valid in A, then it is also valid in W + A,B, for every partial subalgebra B of A. Consequently, W + A,B V. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

26 Structural rules Given an equation ε of the form t 0 t 1 t n, where t i are {, 1}-terms we construct the rule R(ε) u[t 1 ] a u[t n ] a u[t 0 ] a (R(ε)) where the t i s are evaluated in (W,, ε). Such a rule is called linear if all variables in t 0 are distinct. Theorem. Every system obtained from FL by adding linear rules has the cut elimination property. A set of rules of the form R(ε) is called reducing if there is a complexity measure that decreases with upward applications of the rules (and the rules of FL). Theorem. Every system obtained from FL by adding linear reducing rules is decidable. The subvariety of residuated lattices axiomatized by the corresponding equations has decidable equational theory. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

27 Applications Cut-elimination (CE) and finite model property (FMP) for FL and (cyclic) InFL. Generation by finite members for RL, InFL M. Kozak 2008 proved distributive FL has the FMP, and using our approach the same result holds for any extension of DFL with linear reducing structural rules The finite embeddability property (FEP) for integral RL with {,, 1}-axioms The above extend to the non-associative case, also with the addition of suitable structural rules P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

28 (Un)decidability Theorem. The quasiequational theory of RL is undecidable. (Because we can embed semigroups/monoids.) The same holds for commutative RL. A lattice is modular if x (y (x z)) = (x y) (x z) Theorem. The equational theory of modular RL is undecidable. (Using the corresponding result for modular lattices by Freese 1980). Theorem. The equational theory of commutative, distributive RL is decidable (Galatos Raftery 2004, from decidability of relevant logic RW by Brady 1990). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

29 Word problem A finitely presented algebra A = (X R) (in a class K) has a solvable word problem (WP) if there is an algorithm that, given any pair of words over X, decides if they are equal or not. A class of algebras has solvable WP if all finitely presented algebras in it do. For example, the varieties of semigroups, groups, l-groups, modular lattices have unsolvable WP. Theorem [Galatos 2002]: The variety CDRL of commutative, distributive residuated lattices has unsolvable WP. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

30 Word problem for CDRL is unsolvable Main idea: Embed semigroups, whose WP is unsolvable. Residuated lattices have a semigroup operation, but commutative semigroups have a decidable WP. Alternative approach: Come up with another term definable operation in commutative distributive residuated lattices that is associative and embeds all semigroups. Technique: Coordinization in projective geometry and modular lattices, developed by J. von Neumann for continuous geometries, and applied by R. Freese to modular lattices, A. Urquhart to relevance logics, H. Andreak, S. Givant, I. Nemeti to symmetric relation algebras, and N. Galatos to CDRL P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

31 Undecidability of quasiequational theory A quasi-equation is a formula of the form (s 1 = t 1 & s 2 = t 2 & & s n = t n ) s = t The decidability of the quasi-equational theory states that there is an algorithm that decides all quasi-equations of the above form. The equivalent logical notion is the decidability of the deducibility relation for formulas. Corollary The quasi-equational theory of CDRL is undecidable. Hence CDRL does not have the FEP, although we saw earlier that the equational theory is decidable. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

32 Further results on decidability C. Holland, S. H. McCleary 1979: l-groups have decidable equation theory A.M.W. Glass, Y. Gurevich 1983: l-groups have undecidable word problem N. G. Hisamiev 1966: abelian l-groups have decidable universal theory, by V. Weispfenning 1986, in fact co-np-complete, but by Y. Gurevich 1967, the first-order theory is hereditarily undecidable MV-algebras have FEP because of a connections to linear programming IGMV and GMV have decidable equational theory because of a connections to l-groups (N. Galatos, C. Tsinakis 2004), but no FMP P. Jipsen and F. Montagna 2006, 2008: GBL and IGBL do not have FMP but normal GBL-algebras have FEP P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

33 Poset products The poset product uses a partial order on the index set to define a subset of the direct product. Specifically, let X = (X, ) be a poset, and assume {A i i X } is a family of algebras that have two constant operations denoted 0, 1. The poset product of {A i i X } is A i = {f A i f (i) = 0 or f (j) = 1 for all i < j in X } i X X If X is an antichain then the poset product is the same as the direct product If X is a chain and the A i are ordered, then the poset product is the (amalgamated) ordinal sum of the A i P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

34 For an l-groupoid A define I A = {c A c x = cx = xc for all x A} Note that distributes over in I A, but I A need not be a subalgebra of A A GBL-algebra is a residuated lattice that satisfies x y x = (x/y)y = y(y\x) [J., Montagna 2006] prove that for bounded GBL-algebras, I A is a subalgebra, hence a Heyting algebra contained in A, and B(A) is the subalgebra of complemented elements of I A. For MV-algebras I A = B(A) Lemma Let A be a FL w -algebra and let a, b I A with a b. Then the interval [a, b] = {x A a x b} is a FL w -algebra, with 0 = a, 1 = b,,, inherited from A, and x\y = (x\ A y) b, x/y = (x/ A y) b. If A is a GBL-algebra, then so is [a, b]. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

35 We now generalize the poset sum decomposition result of [J., Montagna 2006] from finite GBL-algebras to certain FL w -algebras Theorem Consider a FL w -algebra A with a finite subalgebra C such that C I A, and let X be the dual of the partially ordered set of completely join irreducible elements of C. For c X, let c denote the unique lower cover of c in C. If Ac = c [c, c] for all c X then A = X [c, c]. The condition Ac = c [c, c] is actually satisfied for many GBL-algebras P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

36 A GBL-algebra is normal if every filter is a normal filter Theorem (J., Montagna) A Blok-Ferreirim decomposition for GBL-algebras: Every subdirectly irreducible normal integral GBL-algebra decomposes as the ordinal sum of an integral GBL-algebra and a linearly ordered integral GMV-algebra. A residuated lattice is n-potent if it satisfies x n+1 = x n [J., Montagna] prove that any n-potent GBL-algebra is commutative, hence normal, so e.g. any finite GBL-algebra is commutative Corollary Suppose A is an integral normal GBL-algebra such that I A is finite Then A is isomorphic to a poset product of linearly ordered IGMV-algebras P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

37 Open Problems Do cancellative residuated lattices have a decidable equational theory or a cut free Gentzen system? Do (I)GBL-algebras have a decidable equational theory or a cut free Gentzen system? Is provability in FL c decidable, i.e. does the variety FL c have a decidable equational theory? A cut-free Gentzen system is known Develop a structure theory for infinite IGBL-algebras Do commutative cancellative residuated lattices satisfy any lattice equations that do not hold in all lattices? Investigate the structure of free residuated lattices. Even the 1-generated case is not well understood Find practical decision procedures for deducibility in FL (e)w or for deciding quasiequations in (C)IRL P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

38 References I F. Belardinelli, H. Ono and P. Jipsen, Algebraic aspects of cut elimination, Studia Logics 68 (2001), 1-32 W. Blok and C. van Alten, On the finite embeddability property for residuated lattices, pocrims and BCK-algebras, Algebra Universalis, No. 48 (2002), W. Blok and C. van Alten, On the finite embeddability property for residuated ordered groupoids, Transactions of the AMS, 356(10) (2005), A. Ciabattoni, N. Galatos and K. Terui. From Axioms to analytic rules in nonclassical logics, LICS 08, A. Ciabattoni, N. Galatos and K. Terui. The expressive power of structural rules for FL, manuscript N. Galatos and P. Jipsen. Residuated frames and applications to decidability, preprint 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 N. Galatos and H. Ono. Algebraization, parameterized local deduction theorem and interpolation for substructural logics over FL, Studia Logica 83 (2006), N. Galatos and H. Ono. Cut elimination and strong separation for non-associative substructural logics, manuscript P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39

39 References II N. Galatos and J. Raftery, Adding involution to residuated structures, Studia Logica 77 (2004), P. Jipsen and C. Tsinakis. A survey of residuated lattices, in Ordered Algebraic Structures (J. Martinez, editor), Kluwer Academic Publishers, Dordrecht, 2002, P. Jipsen and F. Montagna, On the structure of generalized BL-algebras, Algebra Universalis, 55 (2006), P. Jipsen and F. Montagna, The Blok-Ferreirim theorem for normal GBL-algebras and its application, Algebra Universalis, 60, to appear P. Jipsen, Generalizations of Boolean products for lattice-ordered algebras, Annals of Pure and Applied Logic, to appear K. Terui, Which Structural Rules Admit Cut Elimination? - An Algebraic Criterion, JSL 72(3) (2007), P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, / 39