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17
Congruence amalgamation of lattices
 Acta Sci. Math. (Szeged
"... Abstract. J. T˚uma proved an interesting “congruence amalgamation ” result. We are generalizing and providing an alternate proof for it. We then provide applications of this result: (i) A.P. Huhn proved that every distributive algebraic lattice D with at most ℵ1 compact elements can be represented a ..."
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Cited by 15 (15 self)
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Abstract. J. T˚uma proved an interesting “congruence amalgamation ” result. We are generalizing and providing an alternate proof for it. We then provide applications of this result: (i) A.P. Huhn proved that every distributive algebraic lattice D with at most ℵ1 compact elements can be represented as the congruence lattice of a lattice L. We show that L can be constructed as a locally finite relatively complemented lattice with zero. (ii) We find a large class of lattices, the ωcongruencefinite lattices, that contains all locally finite countable lattices, in which every lattice has a relatively complemented congruencepreserving extension.
Characterization of desirable properties of general database decompositions
 Ann. Math. Art. Intell
, 1993
"... This paper appears in a special issue on database theory of the Annals of Mathematics ..."
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Cited by 10 (6 self)
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This paper appears in a special issue on database theory of the Annals of Mathematics
NonBoolean Descriptions for MindMatter Problems
"... A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmat ..."
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Cited by 6 (0 self)
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A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmatter distinction. Such a description in terms of a locally Boolean but globally nonBoolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely nonBoolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain. In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given apriori,wehavetoexpectcorrelations between these two domains. In physics it corresponds to Newton’s separation of universal laws of nature and contingent initial conditions. Both descriptions have a nonBoolean structure and can be encompassed into a single nonBoolean description. Tensed and tenseless time can be synchronized by holistic correlations. 1.
Decomposition of relational schemata into components defined by both projection and restriction
 ACM SIGACTSIGMOSSIGART Sym
, 1988
"... A generalized approach to the decomposition of relational schemata is developed in which the component views may be defined using both restriction and projection operators, thus admitting both horizontal and vertical decompositions. The realization of restrictions is enabled through the use of a Boo ..."
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Cited by 5 (2 self)
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A generalized approach to the decomposition of relational schemata is developed in which the component views may be defined using both restriction and projection operators, thus admitting both horizontal and vertical decompositions. The realization of restrictions is enabled through the use of a Boolean algebra of types, while true independence of projections is modelled by permitting null values in the base schema. The flavor of the approach is algebraic, with the the collection of all candidate views of a decomposition modelled within a latticelike framework, and the actual decompositions arising as Boolean subalgebras. Central to the framework is the notion of bidimensional join dependency, which generalizes the classical notion of join dependency by allowing the components of the join to be selected horizontally as well as vertically. Several properties of such dependencies are presented, including a generalization of many of the classical results known to be equivalent to schema acyclicity. Finally, a characterization of the nature of dependencies which participate in decompositions is presented. It is shown that there are two major types, the bidimensional join dependencies, which are tuple generating and allow tuple removal by implicit encoding of knowledge, and splitting dependencies, which simply partition the database
Geometric Identities in Lattice Theory
"... An Arguesian identity is an identity in GrassmannCayley algebras with certain multilinear properties of expressions in joins and meets of vectors and covectors. Many classical theorems of projective geometry and their generalizations to higher dimensions can be expressed as simple and elegant Argu ..."
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An Arguesian identity is an identity in GrassmannCayley algebras with certain multilinear properties of expressions in joins and meets of vectors and covectors. Many classical theorems of projective geometry and their generalizations to higher dimensions can be expressed as simple and elegant Arguesian identities. In a previous work we showed that an Arguesian identity can be unfolded with respect to a vector variable to get a lattice inequality, which holds in various lattices. In this paper, we extend this technique to an arbitrary variable. We prove that for any variable v of an Arguesian identity I , a lattice inequality can be obtained by unfolding I with respect to the variable v. This inequality and its dual are valid in the class of linear lattices if the identity is of order 2, and in the congruence variety of Abelian groups if the identity is of a higher order. Consequently, we obtain a family of lattice identities which are selfdual over the class of linear lattices. In p...
Directions in Lattice Theory
, 1994
"... This paper is about three problems raised by Bjarni Jonsson and their influence on the development and direction of lattice theory ..."
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This paper is about three problems raised by Bjarni Jonsson and their influence on the development and direction of lattice theory
On the Structure of QuasiOrdering Lattices
"... We investigate the structure of the I) of all quasiorderings on a set I. We describe a natural set of the so called fundamental inequalities defining all minimal inequalities in \Omega\Gamma I) and develop an axiomatic characterization of \Omega\Gamma I). We further describe the automorphism gr ..."
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We investigate the structure of the I) of all quasiorderings on a set I. We describe a natural set of the so called fundamental inequalities defining all minimal inequalities in \Omega\Gamma I) and develop an axiomatic characterization of \Omega\Gamma I). We further describe the automorphism group of \Omega\Gamma I).
4. Representation by Equivalence Relations
"... No taxation without representation! So far we have no analogue for lattices of the Cayley theorem for groups, that every group is isomorphic to a group of permutations. The corresponding representation theorem for lattices, that every lattice is isomorphic to a lattice of equivalence relations, turn ..."
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No taxation without representation! So far we have no analogue for lattices of the Cayley theorem for groups, that every group is isomorphic to a group of permutations. The corresponding representation theorem for lattices, that every lattice is isomorphic to a lattice of equivalence relations, turns out to be considerably deeper. Its proof uses a recursive construction technique that has become a standard tool of lattice theory and universal algebra. An equivalence relation on a set X is a binary relation E satisfying, for all x,y,z ∈ X, (1) x E x, (2) x E y implies y E x, (3) if x E y and y E z, then x E z. We think of an equivalence relation as partitioning the set X into blocks of Erelated elements, called equivalence classes. Conversely, any partition of X into a disjoint union of blocks induces an equivalence relation on X: x E y iff x and y are in the same block. As usual with relations, we write x E y and (x,y) ∈ E interchangeably. The most important equivalence relations are those induced by maps. If Y is another set, and f: X → Y is any function, then ker f = {(x,y) ∈ X 2: f(x) = f(y)} is an equivalence relation, called the kernel of f. If X and Y are algebras and f: X → Y is a homomorphism, then ker f is a congruence relation. Thinking of binary relations as subsets of X 2, the axioms (1)–(3) for an equivalence relation are finitary closure rules. Thus the collection of all equivalence relations on X forms an algebraic lattice Eq X. The order on Eq X is given by set containment, i.e., R ≤ S iff R ⊆ S in P(X 2) iff (x,y) ∈ R = ⇒ (x,y) ∈ S. The greatest element of Eq X is the universal relation X 2, and its least element is the equality relation =. The meet operation in Eq X is of course set intersection, which means that (x,y) ∈ ∧ i∈I Ei if and only if x Ei y for all i ∈ I. The join