scholarly journals On colorings of the Boolean lattice avoiding a rainbow copy of a poset

2020 ◽  
Vol 276 ◽  
pp. 108-114
Author(s):  
Balázs Patkós
Keyword(s):  
Boolean Lattice

Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size

2003 ◽  
Vol 24 (2) ◽  
pp. 219-228 ◽  
Author(s):  
Tim Hsu ◽  
Mark J. Logan ◽  
Shahriar Shahriari ◽  
Christopher Towse
Keyword(s):  
Boolean Lattice ◽  
Uniform Size

Flowers and Steps in the Boolean Lattice of Hexagrams

2004 ◽  
Vol 31 (4) ◽  
pp. 489-504
Author(s):  
Andreas Schöter
Keyword(s):  
Boolean Lattice

A FCA-Based Cognitive Diagnosis Model for CAT

2011 ◽  
pp. 151-170
Author(s):  
Yang Shuqun ◽  
Ding Shuliang
Keyword(s):  
Large Scale ◽  
Matrix Theory ◽  
Estimation Method ◽  
Boolean Lattice ◽  
Cognitive Diagnosis ◽  
Remedial Measure ◽  
Concept Lattices ◽  
Knowledge States ◽  
Attribute Hierarchy ◽  
Developing Area

There is little room for doubt about that cognitive diagnosis has received much attention recently. Computerized adaptive testing (CAT) is adaptive, fair, and efficient, which is suitable to large-scale examination. Traditional cognitive diagnostic test needs quite large number of items, the efficient and tailored CAT could be a remedy for it, so the CAT with cognitive diagnosis (CD-CAT) is prospective. It is more beneficial to the students who live in the developing area without rich source of teaching, and distance education is adopted there. CD is still in its infancy (Leighton at el.2007), and some flaws exist, one of which is that the rows/columns could form a Boolean lattice in Tatsuoka’s Q-matrix theory. Formal Concept Analysis (FCA) is proved to be a useful tool for cognitive science. Based on Rule Space Model (RSM) and the Attribute Hierarchy Method (AHM), FCA is applied into CD-CAT and concept lattices are served as the models of CD. The algorithms of constructing Qr matrice and concept lattices for CAT, and the theory and methods of diagnosing examinees and offering the best remedial measure to examinees are discussed in detail. The technology of item bank construction, item selection strategies in CD-CAT and estimation method are considered to design a systemic CD-CAT, which diagnoses examinees on-line and offers remedial measure for examinees in time. The result of Monte Carlo study shows that examinees’ knowledge states are well diagnosed and the precision in examinees’ abilities estimation is satisfied.

scholarly journals On blockers in bounded posets

2001 ◽  
Vol 26 (10) ◽  
pp. 581-588 ◽  
Author(s):  
Andrey O. Matveev
Keyword(s):  
Boolean Lattice ◽  
Bounded Poset ◽  
Finite Set ◽  
Bounded Posets

Antichains of a finite bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers are discussed.

scholarly journals Chains in generalized Boolean lattices

1976 ◽  
Vol 21 (2) ◽  
pp. 234-240
Author(s):  
Richard D. Byrd ◽  
Roberto A. Mena
Keyword(s):  
Distributive Lattice ◽  
Boolean Lattice ◽  
Boolean Lattices ◽  
A Chain

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

scholarly journals Entropy Inequalities for Lattices

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 784 ◽  
Author(s):  
Peter Harremoës
Keyword(s):  
Group Theory ◽  
Conditional Independence ◽  
Lattice Theory ◽  
Boolean Lattice ◽  
Functional Dependencies ◽  
Modular Lattices ◽  
Special Lattice ◽  
Entropy Inequalities ◽  
Lattice Of Subgroups

We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice.

scholarly journals REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

2012 ◽  
Vol 15 (03) ◽  
pp. 1250019
Author(s):  
PHILIP FEINSILVER
Keyword(s):  
Special Functions ◽  
Regular Representation ◽  
Boolean Lattice ◽  
Irreducible Representations ◽  
Leibniz Rule ◽  
Krawtchouk Polynomials ◽  
Lattice Representation ◽  
Boolean Poset ◽  
Group Law

Starting with the zero-square "zeon algebra," the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard–Sylvester matrices are shown as well.

ON SOME LATTICES OF MODULE CLASSES

2006 ◽  
Vol 05 (01) ◽  
pp. 105-117 ◽  
Author(s):  
ALEJANDRO ALVARADO GARCÍA ◽  
HUGO ALBERTO RINCÓN-MEJÍA ◽  
JOSÉ RÍOS MONTES
Keyword(s):  
Principal Ideal ◽  
Boolean Lattice

In this work we continue the discussion of the conatural classes introduced in [1]. We prove that the collection R-conat of all conatural classes of left modules over a ring R is a set, and it is a boolean lattice. Afterwards we study relationships between some lattices of module classes: R-conat and R-tors, R-her and R-quot. As a consequence of the developed theory we obtain characterizations of left MAX rings and artinian principal ideal rings.

On an algebra of lattice-valued logic

2005 ◽  
Vol 70 (1) ◽  
pp. 282-318
Author(s):  
Lars Hansen
Keyword(s):  
Boolean Algebra ◽  
Complete Lattice ◽  
Algebraic Theory ◽  
Propositional Calculus ◽  
Boolean Lattice ◽  
Finite Chain ◽  
Axiomatic Theory ◽  
Truth Values ◽  
Algebraic Generalization

AbstractThe purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.

Sign in / Sign up

Export Citation Format

Share Document