Co-Stone Residuated Lattices Claudia MURES ¸AN c.muresan@yahoo.com, cmuresan11@gmail.com University of Bucharest Faculty of Mathematics and Computer Science Bucharest December 12, 2009 C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 1 / 37
Contents Preliminaries 1 Co-Stone Algebras 2 The Strongly Co-Stone Hull of a Residuated Lattice 3 C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 2 / 37
Preliminaries 1 Co-Stone Algebras 2 The Strongly Co-Stone Hull of a Residuated Lattice 3 C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 3 / 37
Definition Residuated lattice : ( A , ∨ , ∧ , ⊙ , → , 0 , 1), with: ( A , ∨ , ∧ , 0 , 1) bounded lattice; ( A , ⊙ , 1) commutative monoid; the law of residuation : for all a , b , c ∈ A , a ≤ b → c iff a ⊙ b ≤ c . monoidal logic A residuated lattice a , b ∈ A n ∈ N ∗ a ↔ b = ( a → b ) ∧ ( b → a ) ¬ a = a → 0 a n = a ⊙ . . . ⊙ a ; a 0 = 1 � �� � n of a C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 4 / 37
A lattice (residuated lattice) F , G filters of A F ∨ G = < F ∪ G > More generally: { F t | t ∈ T } family of filters of A � � F t = < F t > t ∈ T t ∈ T ( F ( A ) , ∨ , ∩ , { 1 } , A ) = the lattice of filters of A C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 5 / 37
A bounded lattice or residuated lattice B ( A ) = the Boolean center of A for A a bounded distributive lattice or a residuated lattice, B ( A ) is a Boolean algebra with the operations induced by those of A C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 6 / 37
A lattice (residuated lattice) Spec ( A ), Max ( A ) topological spaces with the Stone topologies RL = the category of residuated lattices D 01 = the category of bounded distributive lattices C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 7 / 37
A bounded distributive lattice or residuated lattice ∅ � = X ⊆ A x ∈ A the co-annihilator of X : X ⊤ = { a ∈ A | ( ∀ y ∈ X ) a ∨ y = 1 } the co-annihilator of x : x ⊤ = { x } ⊤ = { a ∈ A | a ∨ x = 1 } X ⊤⊤ = ( X ⊤ ) ⊤ x ⊤⊤ = ( x ⊤ ) ⊤ the co-annihilators are filters of A Definition Let A be a bounded distributive lattice or a residuated lattice. Then A is said to be co-Stone (respectively strongly co-Stone ) iff, for all x ∈ A (respectively all ∅ � = X ⊆ A ), there exists an element e ∈ B ( A ) such that x ⊤ = < e > (respectively X ⊤ = < e > ). C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 8 / 37
A bounded distributive lattice or residuated lattice CoAnn ( A ) = { X ⊤ |∅ � = X ⊆ A } F , G ∈ CoAnn ( A ) F ∨ ⊤ G = ( F ⊤ ∩ G ⊤ ) ⊤ More generally: { F t | t ∈ T } family of filters of A � � � ⊤ � ⊤ F t = F ⊤ t t ∈ T t ∈ T Proposition Let A be a bounded distributive lattice or a residuated lattice. Then ( CoAnn ( A ) , ∨ ⊤ , ∩ , ⊤ , { 1 } , A ) is a complete Boolean algebra. C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 9 / 37
Definition Let m be an infinite cardinal. An m -complete lattice is a lattice L with the property that any subset X of L with | X | ≤ m has an infimum and a supremum in L . Theorem Let L be a bounded distributive lattice and m an infinite cardinal. Then the following are equivalent: (i) for each nonempty subset X of L with | X | ≤ m, there exists an element e ∈ B ( L ) such that X ⊤ = < e > ; (ii) L is a co-Stone lattice and B ( L ) is an m-complete Boolean algebra; (iii) L ⊤⊤ = { l ⊤⊤ | l ∈ L } is an m-complete Boolean sublattice of F ( L ) ; (iv) for all l , p ∈ L, ( l ∨ p ) ⊤ = l ⊤ ∨ p ⊤ and, for each nonempty subset X of L with | X | ≤ m, there exists an element x ∈ L such that X ⊤⊤ = x ⊤ ; (v) for each nonempty subset X of L with | X | ≤ m, X ⊤ ∨ X ⊤⊤ = L. A bounded distributive lattice will be called an m-co-Stone lattice iff the conditions of the previous theorem hold for it. C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 10 / 37
Definition Let A be a bounded lattice (residuated lattice) and B a subalgebra of A . We say that B is co-dense in A iff, for all a ∈ A \ { 1 } , there exists b ∈ B such that a ≤ b < 1 (that is a ≤ b ≤ 1 and b � = 1). C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 11 / 37
Previous definitions of the reticulation: H. Simmons: commutative rings (1980, [21]) L. P. Belluce: MV-algebras (1986, [2]) L. P. Belluce: non-commutative rings (1991, [3]) G. Georgescu: quantales (1995, [9]) L. Leu¸ stean: BL-algebras (2003, [15], [16]) In each of the papers cited above, although it is not explicitely defined this way, the reticulation of an algebra A is a pair ( L ( A ) , λ ) consisting of a bounded distributive lattice L ( A ) and a surjection λ : A → L ( A ) such that the function given by the inverse image of λ induces (by restriction) a homeomorphism between the prime spectrum of L ( A ) and that of A . This construction allows many properties to be transferred between L ( A ) and A , and this transfer of properties between the category of bounded distributive lattices and another category (in our case that of residuated lattices) is the very purpose of the reticulation. C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 12 / 37
An axiomatic purely algebraic definition of the reticulation: innovation in the study of the reticulation, as in previous work the reticulation of an algebra was defined by its construction very useful in practice Definition [17] Let A be a residuated lattice. A reticulation of A is a pair ( L , λ ), where L is a bounded distributive lattice and λ : A → L is a function that satisfies conditions 1)-5) below: 1) for all a , b ∈ A , λ ( a ⊙ b ) = λ ( a ) ∧ λ ( b ); 2) for all a , b ∈ A , λ ( a ∨ b ) = λ ( a ) ∨ λ ( b ); 3) λ (0) = 0; λ (1) = 1; 4) λ is surjective; 5) for all a , b ∈ A , λ ( a ) ≤ λ ( b ) iff ( ∃ n ∈ N ∗ ) a n ≤ b . C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 13 / 37
In [17] and [18] we proved that this definition is in accordance with the general notion of reticulation applied to residuated lattices, more precisely that, given a residuated lattice A and a pair ( L , λ ) consisting of a bounded distributive lattice L and a function λ : A → L , we have: if λ satisfies conditions 1)-5) above, then its inverse image induces (by restriction) a homeomorphism between the prime spectrum of L and that of A ; and conversely: if the function given by the inverse image of λ takes prime filters of L to prime filters of A and its restriction to the prime spectrum of L is a homeomorphism between the prime spectrum of L and that of A , then λ satisfies conditions 1)-5) from the definition above. C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 14 / 37
The following theorem states the existence and uniqueness of the reticulation for any residuated lattice. usefulness of the axiomatic purely algebraic definition of the reticulation: simple algebraic proof for the uniqueness of the reticulation in previous work the argument for the uniqueness of the reticulation was of topological nature and consisted of the fact that there is at most one bounded distributive lattice whose prime spectrum is homeomorphic to a given topological space Theorem [17] Let A be a residuated lattice. Then there exists a reticulation of A. Let ( L 1 , λ 1 ) , ( L 2 , λ 2 ) be two reticulations of A. Then there exists an isomorphism of bounded lattices f : L 1 → L 2 such that f ◦ λ 1 = λ 2 . C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 15 / 37
Definition of the reticulation functor L : RL → D 01: A residuated lattice ( L ( A ) , λ A ) the reticulation of A L ( A ) = L ( A ) B residuated lattice ( L ( B ) , λ B ) the reticulation of B f : A → B a morphism of residuated lattices L ( f ) : L ( A ) = L ( A ) → L ( B ) = L ( B ) for all a ∈ A , L ( f )( λ A ( a )) = λ B ( f ( a )) f ✲ A B λ A λ B ❄ ❄ ✲ L ( A ) L ( B ) L ( f ) This definition makes L a covariant functor from RL to D 01. C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 16 / 37
Preliminaries 1 Co-Stone Algebras 2 The Strongly Co-Stone Hull of a Residuated Lattice 3 C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 17 / 37
structure = bounded distributive lattice or residuated lattice any strongly co-Stone structure is co-Stone and any complete co-Stone structure is strongly co-Stone the trivial structure is strongly co-Stone any chain is strongly co-Stone, because a chain A clearly has all co-annihilators equal to { 1 } = < 1 > , except for 1 ⊤ , which is equal to A = < 0 > C. Mure¸ san (University of Bucharest) Co-Stone Residuated Lattices December 12, 2009 18 / 37
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