Lattice Of Subsets Research Articles

Polynomial Lym Inequalities

For a Sperner family A ⊂ 2[n] let A i denote the family of all i-element sets in A. We sharpen the LYM inequality $$ {\sum\nolimits_i {{\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} \leqslant 1} } $$ by adding to the LHS all possible products of fractions $$ {\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} $$ , with suitable coefficients. A corresponding inequality is established also for the linear lattice and the lattice of subsets of a multiset (with all elements having the same multiplicity).

On the structure of lattices of subquasivarieties of congruence-noetherian quasivarieties

We study the structure of algebraic τ-closed subsets of an algebraic lattice L, where τ is some Browerian binary relation on L, in the special case when the lattice of such subsets is an atomistic lattice. This gives an approach to investigate the atomistic lattices of congruence-Noetherian quasivarieties.

Sublattices of Lattices of Convex Subsets of Vector Spaces

Let Co(V) be a lattice of convex subsets of a vector space V over a totally ordered division ring $$\mathbb{F}$$ . We state that every lattice L can be embedded into Co(V), for some space V over $$\mathbb{F}$$ . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional; in this case L also embeds into a finite lower bounded lattice of the form Co(V,Ω)={X⋂Ω | X ∈ Co(V)}, for some finite subset Ω of V. This result yields, in particular, a new universal class of finite lower bounded lattices.

Sublattices of lattices of order-convex sets, I. The main representation theorem

For a partially ordered set P, we denote by Co (P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co (P) iff L satisfies (S) , (U) , and (B) . Furthermore, if L has an embedding into some Co (P) , then it has such an embedding that preserves the existing bounds. If L is finite, then one can take P finite, with |P|⩽2 J(L) 2−5 J(L) +4, where J( L) denotes the set of all join-irreducible elements of L. On the other hand, the partially ordered set P can be chosen in such a way that there are no infinite bounded chains in P and the undirected graph of the predecessor relation of P is a tree.

Join congruence relations and closure relations

In this paper we show that, given a complete lattice $$ \mathcal{L} $$ , the following three lattices are the same: (1) the lattice of closure relations on $$ \mathcal{L} $$ , (2) the lattice of meet-closed subsets of $$ \mathcal{L} $$ , and (3) the lattice of complete join congruence relations on $$ \mathcal{L} $$ .

Exchange property and the natural preorder between simple modules over semi-Artinian rings

We prove that if R is a right semi-Artinian ring, then R is an exchange ring and every irredundant set Simp R of representatives of simple right R -modules carries a canonical structure of an Artinian poset, which is a Morita invariant. We investigate several basic features of this order structure and, for a wide class of right semi-Artinian rings, which we call nice , we establish a link between those (two-sided) ideals which are pure as left ideals and some upper subsets of Simp R . If R is nice and Simp R does not contain infinite antichains, then that link realizes an anti-isomorphism from the lattice of upper subsets of Simp R to the set of all ideals which are pure as left ideals. Further we show that every Artinian poset (possibly after adding a suitable maximal element if it is infinite) is order isomorphic to Simp R for some nice right semi-Artinian ring R .

Congruence relations on lattices of recursively enumerable sets

Let {We}e∈ω be a standard enumeration of the recursively enumerable (r. e.) subsets of ω = {0, 1, 2, …}. The lattice of recursively enumerable sets, is the structure ({We}e∈ω, ∪, ∩). is the sublattice of consisting of the recursive sets.Suppose is a lattice of subsets of ω. ≡ is said to be a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V ∈ , if U ≡ U′ and V ≡ V′ then U ∪ U′ ≡ V ∪ V′ and U ∩ U′ ≡ V ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V].The quotient lattices of (or of some sublattice ) correspond naturally with the congruence relations which give rise to them, and in turn the congruence relations of sublattices of can be characterized in part by their computational complexity. The aim of the present paper is to characterize congruence relations in some of the most important complexity classes.

Parallel frequent set counting

Computing the frequent subsets of large multi-attribute data is a key component of local pattern detection data mining algorithms. It is both computation- and data-intensive. The standard parallel algorithms require multiple passes through the data. The cost of data access may easily outweigh any performance gained by parallelizing the computational part. We address two opportunities for performance improvement: using a parallel approximate algorithm that requires only a single pass over the data; and using a probabilistic technique to avoid generating most of the lattice of subsets implied by each object's data. The computation required is only slightly greater than levelwise algorithms, but the amount of data access is much smaller.

Hausdorff compactifications and zero-one measures

It is well known that the Wallman-type compactifications of a Tychonoff space X can be obtained as spaces of all regular zero-one measures on suitable lattices of subsets of X (see [1, 2, 4, 12]). Using the technique developed in [5, 6], we find for any Tychonoff space X a Boolean algebra [Bscr ]X and a set [Lscr ]X of sublattices of [Bscr ]X having the following property: for any Hausdorff compactification cX of X there exists a (unique) LcX ∈ [Lscr ]X such that the maximal spectrum of LcX and the space of all u-regular zero-one measures on the Boolean subalgebra b(LcX) of [Bscr ]X, generated by LcX, are Hausdorff compactifications of X equivalent to cX. Let us give more details now.

Pseudo-finite dimensional representations of $ sl(2,k) $

Let k be an algebraically closed field of characteristic zero and L = sl(2,k) the Lie algebra of 2 × 2 traceless matrices over k. It is shown that there exists a von Neumann regular extension \( U(L) \subseteq U'(L) \) of the universal enveloping algebra, which is an epimorphism in the category of rings. The article is devoted to the study of the simple representations of U'(L), which may be topologized via the Ziegler topology on the set of injective indecomposable representations of U'(L) or via the Jacobson topology on the set of primitive ideals. These two topologies coincide and the finite dimensional simple representations of L form a dense, discrete and open subset. The field of fractions K(L) of the universal enveloping algebra is another simple representation of U'(L). If the point K(L) is removed from the Ziegler spectrum of U'(L), one obtains a compact totally disconnected topological space, which has the cardinality of the continuum. It is also shown that the lattice of ideals of U'(L) is isomorphic to the lattice of open subsets. The epimorphic ring extension \( U(L) \subseteq U'(L) \) is used to find an axiomatization of the finite dimensional representations of L in the language of left U(L)-modules. A representation V of L is called pseudo-finite dimensional if it satisfies these axioms. It is shown that a representation V of L is pseudo-finite dimensional if and only if for every central idempotent \( e \in U'(L) \) for which \( eK(L) \neq 0 \), whenever the subrepresentation eV is nonzero, then it has a nonzero highest weight space.

Normal characterizations of lattices

Let X be an arbitrary nonempty set and ℒ a lattice of subsets of X such that ∅, X ∈ ℒ. Let 𝒜(ℒ) denote the algebra generated by ℒ and I(ℒ) denote those nontrivial, zero‐one valued, finitely additive measures on 𝒜(ℒ). In this paper, we discuss some of the normal characterizations of lattices in terms of the associated lattice regular measures, filters and outer measures. We consider the interplay between normal lattices, regularity or σ‐smoothness properties of measures, lattice topological properties and filter correspondence. Finally, we start a study of slightly, mildly and strongly normal lattices and express then some of these results in terms of the generalized Wallman spaces.

On the invariant faces associated with a cone-preserving map

For an n×n nonnegative matrix P , an isomorphism is obtained between the lattice of initial subsets (of {1, · · · , n}) for P and the lattice of P -invariant faces of the nonnegative orthant R+. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum Index Theorem for a nonnegative matrix: If A leaves invariant a polyhedral cone K, then for each distinguished eigenvalue λ ofA forK, there is a chain of mλ distinct A-invariant join-irreducible faces of K, each containing in its relative interior a generalized eigenvector of A corresponding to λ (referred to as semi-distinguished A-invariant faces associated with λ), where mλ is the maximal order of distinguished generalized eigenvectors of A corresponding to λ, but there is no such chain with more than mλ members. We introduce the important new concepts of semi-distinguished A-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end. 1991 Mathematics Subject Classification. Primary 15A48; Secondary 47B65, 47A25, 46B42.

Characterization of lattices of convex subsets of posets

Characterization of lattices of convex subsets of posets

Connectedness in ditopological texture spaces

The notion of a texture space, under the name of fuzzy structure, was introduced by Brown (1993) as a means of representing a lattice of fuzzy sets as a lattice of crisp subsets of some base set. Corresponding to a fuzzy topology we then have the concept of a ditopology, and in this paper connectedness in ditopological texture spaces is defined, and some of its basic properties are obtained. In particular it is shown that connectedness is productive in the context of ditopological texture spaces. Zero-dimensionality and strong zero-dimensionality are defined, and it is shown that in stable zero-dimensional ditopological texture spaces the component of a point coincides with the quasi-component. Finally, it is shown that a fuzzy topological space is C 5 -connected if and only if the corresponding ditopological texture space is connected.

The Basis of a Knowledge Space and a Generalized Interval Order

Abstract Can one read off from the basis of a knowledge space in a simple way whether or not the space is ordinal? A positive answer can be derived from Birkhoffs Theorem: One regards the basis as an abstract partial order which is represented as a suborder of the Boolean lattice of subsets of the item set. If this set representation consists of the lower cones of the partial order then the knowledge space is ordinal (or quasi ordinal if slight modifications are observed). Some partial orders have the property that all their set representations which are parsimonious in a certain sense are equal to the representation by its lower cones. These orders allow only ordinal knowledge spaces. They are characterized by a special condition which generalizes the interval order condition.

Remarks on μ″‐measurbale sets: regularity, σ‐smootheness, and measurability

Let X be an arbitrary nonempty set and ℒ a lattice of subsets of X such that ϕ, X ∈ ℒ. 𝒜(ℒ) is the algebra generated by ℒ and ℳ(ℒ) denotes those nonnegative, finite, finitely additive measures μ on 𝒜(ℒ). I(ℒ) denotes the subset of ℳ(ℒ) of nontrivial zero‐one valued measures. Associated with μ ∈ I(ℒ) (or Iσ(ℒ)) are the outer measures μ′ and μ″ considered in detail. In addition, measurability conditions and regularity conditions are investigated and specific characteristics are given for , the set of μ″‐measurable sets. Notions of strongly σ‐smooth and vaguely regular measures are also discussed. Relationships between regularity, σ‐smoothness and measurability are investigated for zero‐one valued measures and certain results are extended to the case of a pair of lattices ℒ1, ℒ2 where ℒ1 ⊂ ℒ2.

Ideals in tensor powers of the enveloping algebra u(sl 2)

Let U = U(sl2)⊗n be the tensor power of n copies of the enveloping algebra U(sl 2) over an arbitrary field K of characteristic zero. In this paper we list the prime ideals of U by generators and classify them by height. If Z is the center of U and J is a prime ideal of Z, there are exactly 25 prime ideals I of U with I ∩ Z = J, where 0 ≤ s = s(J) ≤ n is an integer. Indeed, with respect to inclusion, they form a lattice isornorphic to the lattice of subsets of a set. When J is a maximal ideal of Z, there are only finitely many two-sided ideals of U containing J, They are presented by generators and their lattice is described, In particular, for each such J there exists a unique maximal ideal of U containing J and a unique ideal of U minimal with respect to the property that it properly contains JU. Similar results are given in the case when U is the tensor product of infinitely many copies of U(sl 2).

Ditopological texture spaces and intuitionistic sets

In this paper it is shown that the lattice of intuitionistic subsets of a set X in the sense of D. Çoker may be represented as a special type of texture space, called an intuitionistic texture on X, and various characterizations are given. It is established that intuitionistic topologies are mapped to ditopologies on the corresponding texture, and some notions of compactness and stability are considered.

Aq-Analog of Approximate Inclusion–Exclusion

We consider the lattice of subspaces of ann-dimensional vector spaceVnqover a finite fieldGF(q) and represent a family of such subspaces by elements of a setX. Theq-analog of the principle of inclusion–exclusion expresses the size of the union of elements ofXrepresenting subspaces ofVnqin terms of the sizes of subsets ofXwhose intersection contains a given subspace ofVnq. We study the problem of approximating the size of this union when intersection sizes are known only for some subspaces ofVnq. In particular, we consider the case where intersection sizes are given for subsets ofXwhose intersection contains a subspace ofVnqof dimension at mostk. We extend methods of Linial and Nisan (1990,Combinatorica10, No. 4, pp. 349–365), drawn from approximation theory, to show that the quality of approximation changes in a significant way around [equation]: if [equation], then any approximation may err by a factor of [equation]; while if [equation] , the size of the union may be approximated to within a multiplicative factor of [equation]. Our result, the firstq-analog of a computational property of the lattice of subsets of a finite set, answers in the affirmative a question posed by Linial and Nisan.

The foundations of enumeration theory for finite nilpotent groups

This paper is the first in a series of papers that lay the foundations of enumeration theory for finite groups including the classical inversion calculus on segments of the natural series and on lattices of subsets of finite sets. Since it became possible to calculate the Mobius function on all subgroups of finite nilpotent groups, the Mobius inversion on these groups began to play a decisive role. The efficiency of the inversion method as a regular technique suitable for solution of enumeration problems of group theory is illustrated with a number of concrete and very important enumerations. Bibliography: 13 titles.