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21
On the Minimum ManyValued Modal Logic over a Finite Residuated Lattice
, 2009
"... This article deals with manyvalued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum mod ..."
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Cited by 8 (5 self)
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This article deals with manyvalued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truthconstants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truthconstants in the language.
Forcing extensions of partial lattices
 CNRS, UMR 6139, Département de Mathématiques, BP 5186, Université de Caen, Campus 2, 14032 Caen
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Lifting retracted diagrams with respect to projectable functors
, 2008
"... We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean 〈∨,0〉semilattices with 〈∨, 0 ..."
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Cited by 3 (3 self)
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We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean 〈∨,0〉semilattices with 〈∨, 0〉embeddings, can be lifted with respect to the Conc functor on lattices, then so can every diagram, indexed by a lattice, of finite distributive 〈∨, 0〉semilattices with 〈∨, 0〉embeddings. If the premise of this statement held, this would solve in turn the (still open) problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lattice. We also outline potential applications of the method to other functors, such as the R ↦ → V (R) functor on von Neumann regular rings.
VON NEUMANN COORDINATIZATION IS NOT FIRSTORDER
, 2006
"... Abstract. A lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. This forces L to be complemented modular. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jónsson, are first ..."
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Cited by 2 (2 self)
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Abstract. A lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. This forces L to be complemented modular. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jónsson, are firstorder. Nevertheless, we prove that coordinatizability of lattices is not firstorder, by finding a noncoordinatizable lattice K with a coordinatizable countable elementary extension L. This solves a 1960 problem of B. Jónsson. We also prove that there is no L∞, ∞ statement equivalent to coordinatizability. Furthermore, the class of coordinatizable lattices is not closed under countable directed unions; this solves another problem of B. Jónsson from 1962. 1.
A NONCOORDINATIZABLE SECTIONALLY COMPLEMENTED MODULAR LATTICE WITH A LARGE JÓNSSON FOURFRAME
, 1003
"... Abstract. A sectionally complemented modular lattice L is coordinatizable if itisisomorphicto the lattice L(R)ofallprincipalrightidealsofavon Neumann regular (not necessarily unital) ring R. We say that L has a large 4frame if it has a homogeneous sequence (a0,a1,a2,a3) such that the neutral ideal ..."
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Abstract. A sectionally complemented modular lattice L is coordinatizable if itisisomorphicto the lattice L(R)ofallprincipalrightidealsofavon Neumann regular (not necessarily unital) ring R. We say that L has a large 4frame if it has a homogeneous sequence (a0,a1,a2,a3) such that the neutral ideal generated by a0 is L. Jónsson proved in 1962 that if L has a countable cofinal sequence and a large 4frame, then it is coordinatizable; whether the cofinal sequence assumption could be dispensed with was left open. We solve this problem by finding a noncoordinatizable sectionally complemented modular lattice L with a large 4frame; it has cardinality ℵ1. Furthermore, L is an ideal in a complemented modular lattice L ′ with a spanning 5frame (in particular, L ′ is coordinatizable). Our proof uses Banaschewski functions. A Banaschewski function on a bounded lattice L is an antitone selfmap of L that picks a complement for each element of L. In an earlier paper, we proved that every countable complemented modular lattice has a Banaschewski function. We prove that there exists a unitregular ring R of cardinality ℵ1 and index of nilpotence 3 such that L(R) has no Banaschewski function. 1.
THE COMPLEXITY OF VON NEUMANN COORDINATIZATION; 2DISTRIBUTIVE LATTICES
, 2004
"... Abstract. A complemented modular lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jónsson, are firstorder. Nevertheless, ..."
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Cited by 1 (1 self)
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Abstract. A complemented modular lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jónsson, are firstorder. Nevertheless, we prove that coordinatizability of complemented modular lattices is not firstorder, even for countable 2distributive lattices, thus solving a 1960 problem of B. Jónsson. This is established by expressing the coordinatizability of a countable 2distributive lattice L as a separation property of the set of all finite homomorphic images of L. This separation property is in Lω1,ω but not firstorder. In the uncountable case, it is no longer sufficient to characterize coordinatizability. In fact, we prove that there is no L∞, ∞ statement equivalent to coordinatizability. We also prove that the class of coordinatizable lattices is not closed under countable directed unions, thus solving another problem of B. Jónsson from 1962. 1.
A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY
"... Abstract. We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated w ..."
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Abstract. We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.
CalabiYau Jungles
, 2004
"... It was proposed that the CalabiYau geometry can be intrinsically connected with some new symmetries, some new algebras. In order to do this it has been analyzed the graphs constructed from K3fibre CYd (d ≥ 3) reflexive polyhedra. The graphs can be naturally get in the frames of Universal CalabiYa ..."
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It was proposed that the CalabiYau geometry can be intrinsically connected with some new symmetries, some new algebras. In order to do this it has been analyzed the graphs constructed from K3fibre CYd (d ≥ 3) reflexive polyhedra. The graphs can be naturally get in the frames of Universal CalabiYau algebra (UCYA) and may be decode by universal way with the changing of some restrictions on the generalized Cartan matrices associated with the Dynkin diagrams that characterize affine KacMoody algebras. We propose that these new Berger graphs can be directly connected with the generalizations of Lie and KacMoody algebras.
TREE AUTOMATA AND SEPARABLE SETS OF INPUT VARIABLES
, 2007
"... Abstract. We introduce the separable sets of variables for trees and tree automata. If a set Y of input variables is inseparable for a tree and an automaton then there a non empty family of distributive sets of Y. It is shown that if a tree t has ”many ” inseparable sets with respect to a tree autom ..."
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Abstract. We introduce the separable sets of variables for trees and tree automata. If a set Y of input variables is inseparable for a tree and an automaton then there a non empty family of distributive sets of Y. It is shown that if a tree t has ”many ” inseparable sets with respect to a tree automaton A then there is an effective way to reduce the complexity of A when running on t. 1.
MULTISOLID VARIETIES AND MHTRANSDUCERS
, 2008
"... Abstract. We consider the concepts of colored terms and multi hypersubstitutions. If t ∈ Wτ(X) is a term of type τ, then any mapping αt: Pos F (t) → IN of the nonvariable positions of a term into the set of natural numbers is called a coloration of t. The set W c τ (X) of colored terms consists o ..."
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Abstract. We consider the concepts of colored terms and multi hypersubstitutions. If t ∈ Wτ(X) is a term of type τ, then any mapping αt: Pos F (t) → IN of the nonvariable positions of a term into the set of natural numbers is called a coloration of t. The set W c τ (X) of colored terms consists of all pairs 〈t, αt〉. Hypersubstitutions are maps which assign to each operation symbol a term with the same arity. If M is a monoid of hypersubstitutions then any sequence ρ = (σ1, σ2,...) is a mapping ρ: IN → M, called a multihypersubstitution over M. An identity t ≈ s, satisfied in a variety V is an Mmultihyperidentity if its images ρ[t ≈ s] are also satisfied in V for all ρ ∈ M. A variety V is Mmultisolid, if all its identities are M−multihyperidentities. We prove a series of inclusions and equations concerning Mmultisolid varieties. Finally we give an automata realization of multihypersubstitutions and colored terms.