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17
FORCING EXTENSIONS OF PARTIAL LATTICES
, 2005
"... Abstract. We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let ϕ: Conc K → D be a {∨,0}homomorphism, where Conc K denotes the {∨, 0}semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f: K → L, a ..."
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Cited by 4 (4 self)
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Abstract. We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let ϕ: Conc K → D be a {∨,0}homomorphism, where Conc K denotes the {∨, 0}semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f: K → L, and an isomorphism α: Conc L → D such that α ◦ Conc f = ϕ. Furthermore, L and f satisfy many additional properties, for example: (i) L is relatively complemented. (ii) L has definable principal congruences. (iii) If the range of ϕ is cofinal in D, then the convex sublattice of L generated by f[K] equals L. We mention the following corollaries, that extend many results obtained in the last decades in that area: — Every lattice K such that Conc K is a lattice admits a congruencepreserving extension into a relatively complemented lattice.
Lifting retracted diagrams with respect to projectable functors, Algebra Universalis 54
"... Abstract. 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 wi ..."
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Cited by 3 (3 self)
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Abstract. 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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Cited by 2 (2 self)
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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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Cited by 1 (0 self)
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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.
MULTISOLID VARIETIES
, 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〉. The set Hyp(τ) of all hypersubstitutions of type τ is a countable monoid. If M is a submonoid of Hyp(τ) 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. Precomplete congruences are satisfied in the pretrivial varieties. Studying the multihypersubstitutions we find out necessary and sufficient conditions a variety to be precomplete. Finally we give an automata realization of multihypersubstitutions and colored terms.
COMPOSITION OF TERMS AND ESSENTIAL POSITIONS IN DEDUCTION
, 802
"... Abstract. The paper deals with inductive, positional and Σ−compositions of terms in order to describe some varieties defined by such compositions. In algebra, equational logic, theoretical computer science, etc. the composition of terms is defined as replacements of some variables by external terms. ..."
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Abstract. The paper deals with inductive, positional and Σ−compositions of terms in order to describe some varieties defined by such compositions. In algebra, equational logic, theoretical computer science, etc. the composition of terms is defined as replacements of some variables by external terms. We consider the positional composition of terms as a tool, which allows us to extend the derivation rules in formal deduction of identities. Cases when closure operators of the sets of identities of a given type are fully invariant congruences are investigated. The concept of essential variables and essential positions of terms with respect to a set of identities is a key step in the simplification of the process of formal deduction. TSRclosure of the sets of identities is obtained by applying inductive and positional term composition in deduction of identities. Σ−composition of terms is defined as replacement between Σequal terms. This composition induces ΣR−deductively closed sets of identities. In analogy to balanced identities we introduce and investigate Σ−balanced identities for a given set Σ of identities.