Embedding theorems for normal divisible residuated lattices Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics and Computer Science Siena, Italy June 3, 2008 (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 1 / Califo 27
Outline Background Poset sums From Heyting algebras to divisible residuated lattices Applications All normal divisible RL embed in poset sums of GMV-chains A Conrad-Harvey-Holland type embedding into algebras of R-valued functions Towards a representation theorem for n-potent divisible RL (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 2 / Califo 27
Definition A full Lambek algebra or FL-algebra is a system (L,,,, \, /,0, 1) where (L,, ) is a lattice (L,, 1) is a monoid \ and / are binary operations such that the residuation property holds: x y z iff y x\z iff x z/y Galatos, J., Kowalski, Ono, (2007) Residuated Lattices: An algebraic glimpse at substructural logics, Studies in Logic, Elsevier, xxi+509 pp. Definition A FL-algebra is divisible if x y implies x = y (y\x) = (x/y) y A residuated lattice is a FL-algebra that satisfies 0 = 1 (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 3 / Califo 27
Examples of divisible FL-algebras Boolean algebras x y = x y and x y = x/(y\x) and 0 x Heyting algebras x y = x y and 0 x MV-algebras xy = yx and x y = x/(y\x) and 0 x Pseudo-MV-algebras x y = x/(y\x) = (x/y)\x and 0 x Lattice ordered groups 0 = 1 and (1/x)x = 1 so x 1 = 1/x Generalized MV-algebras x y = x/((x y)\x) = (x/(x y))\x Basic logic algebras xy = yx and x\y y\x = 1 and 0 x Pseudo-BL-algebras x\y y\x = x/y y/x = 1 and 0 x Drl-monoids = duals of divisible residuated lattices Divisible residuated lattices = Generalized BL-algebras = GBL-algebras (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 4 / Califo 27
In any residuated lattice or FL-algebra distributes over all existing joins if exists then x = = x ( is usually omitted) in this case \ = (= / ) Moreover divisibility implies that the lattice reduct is distributive all idempotent elements 1, i.e. x 2 = x implies x 1 any lower bounded (e.g. finite) divisible RL is integral, i.e. 1 = Theorem (J., Montagna 2006) All finite divisible residuated lattices are commutative. (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 5 / Califo 27
Poset sums I Let P = (P, ) be a poset, and let A i (i P) be residuated lattices In addition we require that for nonmaximal i P each A i is integral, and for nonminimal i P each A i has a least element denoted by 0 i The poset sum is defined as i P A i = {a i P A i : for all j < k [ a j = 1 or a k = 0 i ]} The operations, and are defined pointwise (as in the direct product) If P is an antichain then the poset sum is the direct product If P is totally ordered then the poset sum is the ordinal sum of the A i (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 6 / Califo 27
Poset sums II A i = {a A i : for all j < k [ a j = 1 or a k = 0 i ]} i P i P Note that an element a is in the poset sum if and only if {i P : 0 i < a i < 1} is an antichain {i P : a i = 1} is downward closed {i P : a i = 0 i } is upward closed The constant function 1 is in the poset sum For the definition of the residuals, we have { ai \b (a\b) i = i if a j b j for all j < i 0 i otherwise and similarly for (a/b) i (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 7 / Califo 27
Theorem (J., Montagna 2008) The varieties of FL-algebras, residuated lattices, GBL-algebras and Heyting algebras are closed under poset sums for arbitrary posets P The varieties of (G)MV-algebras and (Generalized) Boolean algebras are not closed under poset sums When working with filters and congruences it is often convenient to consider dual poset sums defined as i P A i A (G)MV-chain, (G)BL-chain or residuated chain is a totally ordered (G)MV-algebra, (G)BL-algebra or residuated lattice respectively (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 8 / Califo 27
Two finite examples of dual poset sums 1 = a 0 a a 2. a n 1 1 2 0 = a n P MV n 0 i P MV 1 i P A i A 0 = MV 4, A 1 = MV 2 A 2 = MV 3 (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer JuneScience 3, 2008Orange, 9 / Califo 27
More examples of poset sums Recall that a lattice is complete if all joins exist J (L) = the set of completely join-irreducible elements of L A lattice is perfect if x = (J (L) x) for all x L Every complete perfect Heyting algebra A is the poset sum of 2-element Boolean algebras with P = J (A) Every complete atomic Boolean algebra is the poset sum of 2-element Boolean algebras with P = {atoms} Theorem (J., Montagna 2008) Every finite divisible residuated lattice A is (isomorphic to) a poset sum of MV-chains with P = {x A : x 2 = x} (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 10 / Califo 27
Aim: extend this result to infinite algebras Definition A subset F of a residuated lattice A is a filter if F is up-closed F is closed under the monoid operation and the meet operation 1 F A filter F is normal if it is closed under conjugation, i.e. x F and y A imply y\(xy), (yx)/y F The lattice of normal filters is isomorphic to the congruence lattice via θ {x : (x, 1) θ} and F {(x, y) : x\y, y\x F } (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 11 / Califo 27
A normal residuated lattice is one in which every filter is normal E.g. every commutative residuated lattice is normal Theorem (Galatos, Olson, Raftery 2008) A residuated (meet-semi-)lattice is normal iff x, y n [xy n yx and y n x xy] Definition A residuated lattice is n-potent if x n+1 = x n. Lemma (J., Montagna 2006) If x is idempotent in an integral divisible RL then xy = x y for all y. Proof. Integral divisible implies x y = x(x\y), xy x1 = x and xy 1y = y Now xx = x implies xy x y = x(x\y) = xx(x\y) = x(x y) xy (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 12 / Califo 27
n-potent x n idempotent x n y = yx n yx Therefore every n-potent divisible residuated lattice is normal Theorem (J., Montagna 2008) All n-potent divisible residuated lattices are commutative. Theorem (J., Montagna) Every n-potent (G)BL-algebra embeds into the poset sum of a family of finite and n-potent MV-chains This not a representation theorem, in the sense that not all n-potent GBL-algebras are isomorphic to a poset sum of n-potent MV-chains E.g. any poset sum of bounded residuated lattices has a bottom (the constantly zero function), whereas not all n-potent GBL-algebras are bounded (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 13 / Califo 27
Decomposition of divisible residuated lattices Theorem (Galatos, Tsinakis 2005) Every GBL-algebra (hence, every GMV-algebra) is a direct product of an l-group and an integral GBL-algebra (respectively GMV-algebra) Corollary Any GMV-chain is either an l-group, or a bounded integral GMV-algebra, or the negative cone of an l-group The latter case is covered by the middle one since any negative cone of an l-group can be embedded in a bounded integral GMV-algebra (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 14 / Califo 27
Theorem (Agliano, Montagna 2003) Every commutative integral GBL-chain C can be represented as an ordinal sum i I C i of commutative integral GMV-chains Theorem (Dvurečenskij 2007) Every integral GBL-chain C can be represented as an ordinal sum i I C i of integral GMV-chains Theorem (J., Montagna 2006) There exist noncommutative ordinal sum indecomposable integral GBL-algebras that are not GMV-algebras So we cannot replace GBL-chains with GBL-algebras in the above results However, poset sums and normal GBL-algebras allow a generalization (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 15 / Califo 27
Theorem (J., Montagna) Every normal GBL-algebra embeds into a (dual) poset sum of GMV-chains Corollary Every commutative GBL-algebra embeds into a (dual) poset sum of MV-chains and totally ordered abelian l-groups The proofs makes use of the following observations Lemma Every subdirectly irreducible and normal GMV-algebra is totally ordered. Lemma The class of normal GBL-algebras is closed under quotients, subalgebras and finite products. (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 16 / Califo 27
A FL-algebra is called representable if it can be embedded in a direct product of totally ordered FL-algebras A forest is a poset in which every principal downset x is a chain Theorem Let A be a GBL-algebra. The following are equivalent: (i) A is representable. (ii) A is embeddable in a poset sum i P A i such that each A i is a GMV-chain and the poset P is a forest Hence poset sums provide a way to characterize some classes of algebras The next result illustrates this approach (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 17 / Califo 27
Theorem Let B be a GBL-algebra B is a BL-algebra iff B = A i P A i such that each A i is an MV-chain, P is a forest and 0 = (0 i ) i P A B is an MV-algebra iff B = A i P A i such that each A i is an MV-chain, 0 A and P is an antichain B is an abelian l-group iff B = A i P A i such that each A i is a totally ordered abelian l-group and P is an antichain B is n-potent iff it is embeddable into a poset sum of totally ordered n-potent MV-algebras B is a Heyting algebra iff B = A i P A i where every A i is the two-element Boolean algebra and 0 A B is a Gödel algebra iff B = A i P A i where every A i is the two-element Boolean algebra, 0 A and P is a forest B is a Boolean algebra iff B = A i P A i where every A i is the two-element Boolean algebra, 0 A and P is an antichain (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 18 / Califo 27
Conrad-Harvey-Holland-style embedding theorems The Conrad-Harvey-Holland theorem says that every abelian l-group can be embedded into an l-group of functions from a root system (i.e. dual forest) into the set R of reals, with pointwise sum as group operation Aim to extend the result to commutative GBL-algebras using poset sums Definition Let = (, ) be a root system, i.e. δ is a chain for all δ For f : R, let Supp(f ) = {δ : f (δ) 0} Define a structure V(, R) as follows The universe of V(, R) is the set of all functions f from into R such that every non-empty subset of Supp(f ) has a maximal element The group operation is pointwise sum ( identity is 0) The positive cone of V(, R) consists of 0 together with all f V(, R) such that f (δ) > 0 for each maximal element δ Supp(f ). (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 19 / Califo 27
Constructing abelian l-groups using R and a poset E. g. if is a two element chain then V(, R) is the lexicographic product of two copies of R If is a two element antichain then V(, R) is the direct product of two copies of R The lattice operations are induced by the positive cone Theorem (Conrad, Harvey, Holland 1963, simplified version) The algebra V(, R) is an abelian l-group Every abelian l-group G embeds into an l-group of the form V(, R) for a suitable root system = (, ). (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 20 / Califo 27
Only need the version for totally ordered l-groups, first proved by Hahn Theorem (Hahn, 1907). If = (, ) is totally ordered, then V(, R) is a totally ordered abelian l-group. Every totally ordered abelian l-group G embeds into an l-group of the form V(, R) for a suitable totally ordered set. Note that the congruence lattice of an abelian l-group is isomorphic to the lattice of convex subgroups ordered by inclusion The proofs of both Hahn s theorem and of the Conrad-Harvey-Holland theorem take to be the set of completely join irreducible convex subgroups of the abelian l-group (called values in l-group theory) (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 21 / Califo 27
By our earlier corollary, every commutative GBL-algebra embeds into a (dual) poset sum i P d A i of algebras A i where each A i is an MV-chain or a totally ordered abelian l-group In the latter case, by Hahn s theorem, we can replace A i by an l-group of the form V( i, R) If A i is an MV-chain, then we use Mundici s [1986] Γ functor that shows each MV-algebra A is determined by an interval [0, u] in an abelian l-group Prefer to work with Γ (G, u) = [ u, 0] since then the l-group identity is the MV-algebra identity Theorem Every totally ordered MV-algebra embeds into an algebra of the form Γ (V(, R), u) for some totally ordered set (, ) with maximum element, and for some strong unit u of V(, R). Conclusion: any commutative GBL-algebra is determined by the additive group structure of R, together with posets P, i (i P) and u i : i R (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 22 / Califo 27
Commutative divisible RLs from (R, +) and posets Definition Let P = (P, ) be a poset that is a disjoint union of two posets P G, P MV Let Λ G = ( i : i P G ) and Λ MV = ( j, u j : j P MV ) be functions that label each element of i P G by a totally ordered set i and each j P MV by a totally ordered set j with maximum δ j and by a function u j V( j, R) such that u(δ j ) > 0 Then the real valued GBL-algebra A = GBL(P, P G, P MV, Λ G, Λ MV ) is defined as ( V( i, R)) Γ (V( j, R), u j ) i P G j P MV Theorem Every commutative GBL-algebra embeds into a real-valued GBL-algebra of the form GBL(P, P G, P MV, Λ G, Λ MV ) (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 23 / Califo 27
Theorem A commutative GBL-algebra B is integral iff it embeds into an algebra GBL(P, P G, P MV, Λ G, Λ MV ) in which P G = Λ G = an l-group iff it embeds into some GBL(P, P G, P MV, Λ G, Λ MV ) in which P MV = Λ MV = representable iff it embeds into some GBL(P, P G, P MV, Λ G, Λ MV ) in which P is a forest a BL-algebra iff B = A GBL(P, P G, P MV, Λ G, Λ MV ) in which P G = Λ G =, P is a forest and the function u = (u i ) is in A an MV-algebra iff B = A GBL(P, P G, P MV, Λ G, Λ MV ) with P G = Λ G =, u A and P is an antichain a Heyting algebra iff B = A GBL(P, P G, P MV, Λ G, Λ MV ) with P G = Λ G =, u A, and for all h A, i P MV, δ i, h(i)(δ) { u i, 0} a Gödel algebra iff in addition P is a forest a Boolean algebra iff in addition P is an antichain (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 24 / Califo 27
Towards a representation theorem for normal GBL-algebras Use the representation theory for bounded distributive residuated lattices (Galatos 2003 dissertation) Aim: generalize Esakia s representation for Heyting algebras to normal GBL-algebras Let (P,, τ) be an Esakia space For a GMV-chain A, define the (dual) poset sum over (P,, τ) by (P,,τ) A = {a A P : for all j < k [ a j = 0 or a k = 1 ] and for all b A {i P : a i = b} τ} This construction represents a large class of normal GBL-algebras, including all Heyting algebras (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 25 / Califo 27
However there are 2-potent MV-algebras that are not represented in this way Let A = (MV 2 ) N, where MV 2 = {0, m, 1} Consider the subalgebra B = {a A : a i = m for finitely many i} The poset sum defined above cannot represent this algebra with a single topology defined on N So the challenge is to find a good representation theorem for all n-potent (or normal) GBL-algebras (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 26 / Califo 27
References P. Aglianò and F. Montagna, Varieties of BL-algebras I: general properties, Journal of Pure and Applied Algebra 181, (2003), 105 129. A. Dvurečenskij, Aglianò-Montagna Type Decomposition of Pseudo Hoops and its Applications, Journal of Pure and Applied Algebra 211 (2007), 851 861. N. Galatos, J. S. Olson and J. G. Raftery, Irreducible residuated semilattices and finitely based varieties, Reports on Mathematical Logic, to appear. N. Galatos and C. Tsinakis, Generalized MV-algebras, Journal of Algebra 283 (2005), 254 291. P. Jipsen and F. Montagna, On the structure of generalized BL-algebras, Algebra Universalis 55 (2006), 226 237. P. Jipsen and F. Montagna, The Blok-Ferreirim for normal GBL-algebras and its applications, to appear in Algebra Universalis, 2008. D. Mundici, Interpretations of AFC*-algebras in Lukasiewicz sentential calculus, Journal of Functional Analysis 65 (1986), 15 63. (Chapman EmbeddingUniversity theorems Department for normal divisible of Mathematics RL and Computer June 3, Science 2008 Orange, 27 / Califo 27