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HOW TO DERIVE FINITE SEMIMODULAR LATTICES FROM DISTRIBUTIVE LATTICES?
"... Abstract. It is proved that the class of finite semimodular lattices is the same as the class of coverpreserving joinhomomorphic images of direct products of finitely many finite chains. There is a trivial “representation theorem ” for finite lattices: each of them is a joinhomomorphic image of a ..."
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Abstract. It is proved that the class of finite semimodular lattices is the same as the class of coverpreserving joinhomomorphic images of direct products of finitely many finite chains. There is a trivial “representation theorem ” for finite lattices: each of them is a joinhomomorphic image of a finite distributive lattice. This follows from the fact that the finite free join semilattices (with zero) are the finite Boolean lattices. The goal of the present paper is to give two analogous but stronger representation theorems for finite semimodular (also called upper semimodular) lattices. Both theorems state that these lattices are very special joinhomomorphic images of appropriate finite distributive lattices. This way we generalize the main results of G. Grätzer and E. Knapp [4] and [5]. Since even the abovementioned trivial representation theorem was useful in [1], there is a hope that the new achievements will be of some interest in the future. To formulate our results we need the following notions. A sublattice {a1 ∧ a2,a1,a2,a1 ∨ a2} of a lattice is called a covering square if a1 ∧ a2 ≺ ai ≺ a1 ∨ a2 for i =1, 2. A planar lattice is called slim if every covering square is an interval. Now let L and K be finite lattices. A joinhomomorphism ϕ: L → K is said to be coverpreserving iff it preserves the relation �. Similarly, a joincongruence Φ of L is called coverpreserving if the natural joinhomomorphism L → L/Φ, x ↦ → [x]Φ is coverpreserving. As usual, J(L) stands for the poset of all nonzero joinirreducible elements of L. For a poset P,H(P) denotes the lattice of all hereditary subsets (order ideals) of P. The width w(P) of a (finite) poset P is defined to be max{n: P has an nelement antichain}. First we prove: Lemma. Let Φ be a joincongruence of a finite semimodular lattice M. Then Φ is coverpreserving if and only if for any covering square S = {a ∧ b, a, b, a ∨ b} if
Automated discovery of single axioms for ortholattices
 Algebra Universalis
, 2005
"... Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complemen ..."
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Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complementation. Proofs are omitted but are available in an associated technical report and on the Web. We used computers extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. 1.
Short equational bases for ortholattices
 Preprint ANL/MCSP10870903, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2004
"... Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. ..."
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Cited by 3 (3 self)
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Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. Proofs are omitted but are available in an associated technical report. Computers were used extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. The notion of computer proof is addressed. 1
A Note On Discrete Conduché Fibrations
, 1999
"... The class of functors known as discrete Conduché fibrations forms a common generalization of discrete fibrations and discrete opfibrations, and shares many of the formal properties of these two classes. F. Lamarche [7] conjectured that, for any small category B, the category DCF=B of discrete Conduc ..."
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Cited by 2 (0 self)
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The class of functors known as discrete Conduché fibrations forms a common generalization of discrete fibrations and discrete opfibrations, and shares many of the formal properties of these two classes. F. Lamarche [7] conjectured that, for any small category B, the category DCF=B of discrete Conduch'e fibrations over B should be a topos. In this note we show that, although for suitable categories B the discrete Conduché fibrations over B may be presented as the `sheaves' for a family of coverings on a category B tw constructed from B, they are in general very far from forming a topos.
Schmidt: Frankl’s conjecture for large semimodular and planar semimodular lattices
 Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica
"... Abstract. A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a joinirreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x. Frankl’s conjecture, also called as unionclosed sets conjecture, is wellknown in combinatorics, ..."
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Abstract. A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a joinirreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x. Frankl’s conjecture, also called as unionclosed sets conjecture, is wellknown in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let m denote the number nonzero joinirreducible elements of L. It is wellknown that L consists of at most 2 m elements. Let us say that L is large if it has more than 5 · 2 m−3 elements. It is shown that every large semimodular lattice satisfies Frankl’s conjecture. The second result states that every finite semimodular planar lattice L satisfies Frankl’s conjecture. If, in addition, L has at least four elements and its largest element is joinreducible then there are at least two choices for the abovementioned f. Given an melement finite set A = {a1,...,am}, m ≥ 3, a family (or, in other words, a set) F of at least two subsets of A, i.e. F⊆P (A), is called a unionclosed family (over A) ifX ∪ Y ∈Fwhenever X, Y ∈F. It was Peter Frankl in 1979
Disunification in ACI1 Theories
, 1999
"... Disunification is the problem of deciding satisfiability of a system of equations and disequations with respect to a given equational theory. In this paper we study the disunification problem in the context of ACI1 equational theories. This problem is of great importance, for instance, for the d ..."
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Disunification is the problem of deciding satisfiability of a system of equations and disequations with respect to a given equational theory. In this paper we study the disunification problem in the context of ACI1 equational theories. This problem is of great importance, for instance, for the development of constraint solvers over sets. Its solution opens new possibilities for developing automatic tools for static analysis and software verification. In this work we provide a characterization of the interpretation structures suitable to model the axioms in ACI1 theories. The satisfiability problem is solved using known techniques for the equality constraints and novel methodologies to transform disequation constraints to manageable solved forms. We propose four solved forms, each offering an increasingly more precise description of the set of solutions. For each solved form we provide the corresponding rewriting algorithm and we study the time complexity of the transformation. Remarkably, two of the solved forms can be computed and tested in polynomial time. All these solved forms are adequate to be used in the context of a Constraint Logic Programming systeme.g., they do not introduce universal quantifications, as instead happens in some of the existing solved forms for disunification problems.
A COVERPRESERVING EMBEDDING OF SEMIMODULAR LATTICES INTO GEOMETRIC LATTICES
"... Abstract. Extending former results by G. Grätzer and E. W. Kiss (1986) and M. Wild (1993) on finite (upper) semimodular lattices, we prove that each semimodular lattice L of finite length has a coverpreserving embedding into a geometric lattice G of the same length. The number of atoms of our G equ ..."
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Abstract. Extending former results by G. Grätzer and E. W. Kiss (1986) and M. Wild (1993) on finite (upper) semimodular lattices, we prove that each semimodular lattice L of finite length has a coverpreserving embedding into a geometric lattice G of the same length. The number of atoms of our G equals the number of joinirreducible elements of L. 1.
SUBLATTICES AND STANDARD CONGRUENCES
"... Abstract. In an earlier paper, the authors and H. Lakser proved that, for every lattice K and nontrivial congruence Φ of K, there is an extension L of K such that Φ is the restriction to K of a standard congruence on L. In this note, we give a very short proof of this result in a stronger form: the ..."
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Abstract. In an earlier paper, the authors and H. Lakser proved that, for every lattice K and nontrivial congruence Φ of K, there is an extension L of K such that Φ is the restriction to K of a standard congruence on L. In this note, we give a very short proof of this result in a stronger form: the L we construct is sectionally complemented and it has only one nontrivial congruence, the standard congruence. 1.