Lattice Of Subsets Research Articles

Robustness, Scott continuity, and computability

AbstractRobustness is a property of system analyses, namely monotonic maps from the complete lattice of subsets of a (system’s state) space to the two-point lattice. The definition of robustness requires the space to be a metric space. Robust analyses cannot discriminate between a subset of the metric space and its closure; therefore, one can restrict to the complete lattice of closed subsets. When the metric space is compact, the complete lattice of closed subsets ordered by reverse inclusion is $\omega$ -continuous, and robust analyses are exactly the Scott-continuous maps. Thus, one can also ask whether a robust analysis is computable (with respect to a countable base). The main result of this paper establishes a relation between robustness and Scott continuity when the metric space is not compact. The key idea is to replace the metric space with a compact Hausdorff space, and relate robustness and Scott continuity by an adjunction between the complete lattice of closed subsets of the metric space and the $\omega$ -continuous lattice of closed subsets of the compact Hausdorff space. We demonstrate the applicability of this result with several examples involving Banach spaces.

Profunctors Between Posets and Alexander Duality

We consider profunctors between posets and introduce their graph and ascent. The profunctors text {Pro}(P,Q) form themselves a poset, and we consider a partition mathcal {I}sqcup mathcal {F} of this into a down-set mathcal {I} and up-set mathcal {F}, called a cut. To elements of mathcal {F} we associate their graphs, and to elements of mathcal {I} we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of Q times P. Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study text {Pro}({mathbb N}, {mathbb N}). Such profunctors identify as order preserving maps f: {mathbb N}rightarrow {mathbb N}cup {infty }. For our applications when P and Q are infinite, we also introduce a topology on text {Pro}(P,Q), in particular on profunctors text {Pro}({mathbb N},{mathbb N}).

Maranda’s theorem for pure-injective modules and duality

AbstractLet R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . Let $\Lambda $ be an R-order such that $Q\Lambda $ is a separable Q-algebra. Maranda showed that there exists $k\in \mathbb {N}$ such that for all $\Lambda $ -lattices L and M, if $L/L\pi ^k\simeq M/M\pi ^k$ , then $L\simeq M$ . Moreover, if R is complete and L is an indecomposable $\Lambda $ -lattice, then $L/L\pi ^k$ is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective $\Lambda $ -modules.As an application of this extension, we show that if $\Lambda $ is an order over a Dedekind domain R with field of fractions Q such that $Q\Lambda $ is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of $\Lambda $ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of $\Lambda $ .Furthermore, with k as in Maranda’s theorem, we show that if M is R-torsion-free and $H(M)$ is the pure-injective hull of M, then $H(M)/H(M)\pi ^k$ is the pure-injective hull of $M/M\pi ^k$ . We use this result to give a characterization of R-torsion-free pure-injective $\Lambda $ -modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $ -modules.

Subsystems and Fuzzy Relation Equations of Fuzzy Automata Based on Generalized Residuated Lattice

The objective of this paper is to establish the concepts of subsystems, reverse subsystems and double subsystems of a fuzzy automata based on generalized residuated lattices where the fuzzy sets are equipped with generalized residuated lattice in which the commutative property does not hold and characterizes them in terms of fuzzy relation inequalities and equations. Also, the opening and closure systems in the lattices of fuzzy subsets of [Formula: see text] can be discussed for these six subsystems.

Which Distributive Lattices are Lattices of Open Sets of P-Spaces?

Representing lattices by topologies has been studied to a great extent. In this paper, we prove that a complete lattice L is isomorphic to the lattice of open subsets of a P-space iff L is order generated by its countably prime elements. We establish the dual equivalence between the category of complete lattices order generated by their countably prime elements with morphisms preserving arbitrary sups and countable infs, and the category of countably sober P-spaces with morphisms of continuous maps. Finally, we show that the category of countably sober P-spaces is reflective in the category of P-spaces.

English

Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their f-vectors and checking the validity of the following five conjectures: Barany, unimodality, 3d, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension d, an isocanted alcoved polytope has 2d+1 − 2 vertices, its face lattice is the lattice of proper subsets of [d + 1] and its diameter is d + 1. They are realizations of d-elementary cubical polytopes. The f-vector of a d-dimensional isocanted alcoved polytope attains its maximum at the integer ⌊d/3⌋.

Definable Operators on Stable Set Lattices

A fundamental result from Boolean modal logic states that a first-order definable class of Kripke frames defines a logic that is validated by all of its canonical frames. We generalise this to the level of non-distributive logics that have a relational semantics provided by structures based on polarities. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose stable set lattices have additional operators that are first-order definable in the underlying structure. We show that such classes generate varieties of algebras that are closed under canonical extensions. The proof makes use of a relationship between canonical extensions and MacNeille completions.

A lattice isomorphism theorem for cluster groups of mutation-Dynkin type An

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type An case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, each parabolic subgroup has a presentation given by restricting the presentation of the whole group.

Erdős–Ko–Rado theorems on the weak Bruhat lattice

Let ℒ=(X,⪯) be a lattice. For P⊆X we say that P is t-intersecting if rank(x∧y)≥t for all x,y∈P. The seminal theorem of Erdős, Ko and Rado describes the maximum intersecting P in the lattice of subsets of a finite set with the additional condition that P is contained within a level of the lattice. The Erdős–Ko–Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak Bruhat lattice. In this setting, we prove analogs of certain extremal results on intersecting set systems. In particular we give a characterization of the maximum intersecting families of permutations in the Bruhat lattice. We also characterize the maximum intersecting families of permutations within the rth level of the Bruhat lattice of permutations of size n, provided that n is large relative to r.

Canonical extensions and ultraproducts of polarities

J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames.

Lattices Related to Extensions of Presentations of Transversal Matroids

For a presentation $\mathcal{A}$ of a transversal matroid $M$, we study the ordered set $T_{\mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $\mathcal{A}$; extensions are ordered by the weak order. We show that $T_{\mathcal{A}}$ is a distributive lattice, and that each finite distributive lattice is isomorphic to $T_{\mathcal{A}}$ for some presentation $\mathcal{A}$ of some transversal matroid $M$. We show that $T_{\mathcal{A}}\cap T_{\mathcal{B}}$, for any two presentations $\mathcal{A}$ and $\mathcal{B}$ of $M$, is a sublattice of both $T_{\mathcal{A}}$ and $T_{\mathcal{B}}$. We prove sharp upper bounds on $|T_{\mathcal{A}}|$ for presentations $\mathcal{A}$ of rank less than $r(M)$ in the order on presentations; we also give a sharp upper bound on $|T_{\mathcal{A}}\cap T_{\mathcal{B}}|$. The main tool we introduce to study $T_{\mathcal{A}}$ is the lattice $L_{\mathcal{A}}$ of closed sets of a certain closure operator on the lattice of subsets of $\{1,2,\ldots,r(M)\}$.

HEREDITARY TORSION THEORIES OF A LOCALLY NOETHERIAN GROTHENDIECK CATEGORY

Let${\mathcal{A}}$be a locally noetherian Grothendieck category. We construct closure operators on the lattice of subcategories of${\mathcal{A}}$and the lattice of subsets of$\text{ASpec}\,{\mathcal{A}}$in terms of associated atoms. This establishes a one-to-one correspondence between hereditary torsion theories of${\mathcal{A}}$and closed subsets of$\text{ASpec}\,{\mathcal{A}}$. If${\mathcal{A}}$is locally stable, then the hereditary torsion theories can be studied locally. In this case, we show that the topological space$\text{ASpec}\,{\mathcal{A}}$is Alexandroff.

Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set

Abstract We give necessary and sufficient conditions for a family M ${\cal M}$ of subsets of a set X which completely separates X to be a sublattice of the lattice of C ( X , ≤ ) $\mathscr{C}(X,\leq)$ of all convex subsets of X, with respect a suitable linear ordering ≤ of X. As an application we give a characterization of Hausdorff topological spaces which are orderable or suborderable.

On the Hierarchy of d-structures

In this paper we identify and study several lattice structures in the context of directed topology. The set of d-structures on a topological space is a Heyting algebra. The implication is constructed explicitely. There is a Galois connection between the lattice of subsets of the space and the lattice of d-structures which clarifies the idea of removing a subset of the space which has only constant dipaths. Hence, variation of d-structures and variation of the “forbidden area” may be considered in one structure. Moreover, the lattice of d-structures gives rise to a lattice of directed paths between a fixed pair of points. That lattice permits us to discuss a perspective on generalised persistence. Furthermore, we consider a lattice structure in the hierarchy of structures on the n-cube.

Fast Zeta Transforms for Lattices with Few Irreducibles

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Möbius algebras of finite lattices. We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O ( vn ) for computing the zeta transform and its inverse, thus enabling fast multiplication in the Möbius algebra. Furthermore, the circuit construction in fact gives optimal (up to constants) monotone circuits for several lattices of combinatorial and algebraic relevance, such as the lattice of subsets of a finite set, the lattice of set partitions of a finite set, the lattice of vector subspaces of a finite vector space, and the lattice of positive divisors of a positive integer.

Lattices of Convex Subsets

We characterize lattices that are isomorphic to the lattices of convex subsets of partially ordered sets of finite length. We also describe the class of lattices that are isomorphic to the lattices of convex subsets of n-ary trees for any integer n > 1. Bibliography: 10 titles.

Quasicontinuous Domains and the Smyth Powerdomain

In Domain Theory quasicontinuous domains pop up from time to time generalizing slightly the powerful notion of a continuous domain. It is the aim of this paper to show that quasicontinuous domains occur in a natural way in relation to the powerdomains of finitely generated and compact saturated subsets. Properties of quasicontinuous domains seem to be best understood from that point of view. This is in contrast to the previous approaches where the properties of a quasicontinuous domain were compared primarily with the properties of the lattice of Scott-open subsets. We present a characterization of those domains that occur as domains of nonempty compact saturated subsets of a quasicontinuous domain.A set theoretical lemma due to M. E. Rudin has played a crucial role in the development of quasicontinuous domains. We present a topological variant of Rudinʼs Lemma where irreducible sets replace directed sets. The notion of irreducibility here is that of a nonempty set that cannot be covered by two closed sets except if already one of the sets is covering it. Since directed sets are the irreducible sets for the Alexandroff topology on a partially ordered set, this is a natural generalization. It allows a remarkable characterization of sober spaces.For this we denote by QX the space of nonempty compact saturated subsets (with the upper Vietoris topology) of a topological space X. The following properties are equivalent: (1) X is sober, (2) QX is sober, (3) X is strongly well-filtered in the following sense: Whenever A is an irreducible subset of QX and U an open subset of X such that ⋂A⊆U, then K⊆U for some K∈A. This result fills a gap in the existing literature.

Exponential Kleisli Monoids as Eilenberg–Moore Algebras

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

Generalization of convolution and incidence algebras

The paper extends semigroup rings and incidence algebras into a single algebra called general convolution algebra. It is based on three ingredients: (1) a partial hypergroupoid on a set B, (2) an algebra on a set A whose addition is a commutative monoid and whose neutral element 0 is a both-sided absorbing element of the multiplication and (3) an ideal of the lattice of subsets of B. The elements of the structure are the maps from B into A whose supports (with respect to 0) belong to the ideal and satisfy certain finiteness restriction with respect to the partial hypergroupoid. The sum is pointwise and the product, called a general convolution, extends the product in semigroup-rings and incidence algebras. We characterize general convolution algebras with some common properties, like commutativity, associativity and idempotence.

The Connection between Concepts in Formal Concept Analysis and Concepts in a Dual Description Logic

Formal concept analysis (FCA) and description logics (DLs) are two major formalisms around concepts. However, concepts in FCA and concepts in DLs are of different nature. The former are tuples/properties pairs in a given relation, the later are just subsets of domains in terminological interpretations. To represent the logical difference of tuples and attribute values in relations, a dual description logic (DDL) is introduced, in which constants are classified into two classes: tuple constants and value constants, and they are interpreted into two disjoint parts in domain, respectively. Given a model M for DDL, we can obtain a normal relation RM. This paper demonstrates how to connect FCA-concepts in RM and concepts in DDL. To do so, we introduce logical operations negation, disjunction and conjunction on values and on attribute-value pairs, and then build Φ-statements from attribute-value pairs inductively with these operations, and further define a kind of extended FCA-concepts, called Γ-concepts. Each Γ-concept consists of two parts: the extent (tuples the concept covers) and the intent (Φ-statements describing the concept), and there are the following results: the union of any number of extents is always an extent; the complement of an extent is always an extent. Consequently, we obtain two isomorphic lattices: one is LM=(Θ, ∪,∩,–), which is the lattice of subsets of Θ, and the other is concept lattice L(↑RM), where Θ consists of the interpretations of DDL-concepts, and ↑RM is an extend relation from the normal relation RM.