scholarly journals A topological representation of double Boolean lattices

2019 ◽  
Vol 64 (1) ◽  
pp. 11-23
Author(s):  
Brigitte E. Brecker ◽  
◽  
Christian Sacarea ◽  
Keyword(s):  
Topological Representation ◽  
Boolean Lattices

A topological duality for strong Boolean posets

2019 ◽  
Vol 69 (3) ◽  
pp. 497-506
Author(s):  
Zhenzhu Yuan ◽  
Qingguo Li
Keyword(s):  
Topological Representation ◽  
Topological Duality ◽  
New Class ◽  
Boolean Lattices

Abstract In this paper, we define a new class of posets which are complemented and ideal-distributive, we call these posets strong Boolean. This definition is a generalization of Boolean lattices on posets, and is different from Boolean posets. We give a topology on the set of all prime Frink ideals in order to obtain the Stone’s topological representation for strong Boolean posets. A discussion of a duality between the categories of strong Boolean posets and BP-spaces is also presented.

On Boolean Lattices of Module Classes

2018 ◽  
Vol 25 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Alejandro Alvarado-García ◽  
César Cejudo-Castilla ◽  
Hugo Alberto Rincón-Mejía ◽  
Ivan Fernando Vilchis-Montalvo ◽  
Manuel Gerardo Zorrilla-Noriega
Keyword(s):  
Torsion Theory ◽  
Boolean Lattices

Some properties of and relations between several (big) lattices of module classes are used in this paper to obtain information about the ring over which modules are taken. The authors reach characterizations of trivial rings, semisimple rings and certain rings over which every torsion theory is hereditary.

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices

Order ◽  
2017 ◽  
Vol 35 (2) ◽  
pp. 349-362
Author(s):  
Jun-Yi Guo ◽  
Fei-Huang Chang ◽  
Hong-Bin Chen ◽  
Wei-Tian Li
Keyword(s):  
Boolean Lattices ◽  
Families Of Subsets

scholarly journals Overlapping Representation of Basic Tastes in the Human Gustatory Cortex

2021 ◽  
Author(s):  
Du Zhang ◽  
Xiaoxiao Wang ◽  
Yanming Wang ◽  
Benedictor Alexander Nguchu ◽  
Zhoufang Jiang ◽  
...  
Keyword(s):  
Fundamental Property ◽  
Insular Cortex ◽  
Taste Preference ◽  
Topological Representation ◽  
Activation Patterns ◽  
Gustatory Cortex ◽  
Taste Response ◽  
Neural Activities ◽  
Sensory Cortices ◽  
Multivariate Pattern

The topological representation is a fundamental property of human primary sensory cortices. The human gustatory cortex (GC) responds to the five basic tastes: bitter, salty, sweet, umami, and sour. However, the topological representation of the human gustatory cortex remains controversial. Through functional magnetic resonance imaging(fMRI) measurements of human responses to the five basic tastes, the current study aimed to delineate the taste representations within the GC. During the scanning, the volunteers tasted solutions of the five basic tastes, then washed their mouths with the tasteless solution. The solutions were then sucked from the volunteers' mouths, eliminating the action of swallowing. The results showed that the bilateral mid-insula activated most during the taste task, and the active areas were mainly in the precentral and central insular sulcus. However, the regions responding to the five basic tastes are substantially overlapped, and the analysis of contrasts between each taste response and the averaged response to the remaining tastes does not report any significant results. Furthermore, in the gustatory insular cortex, the multivariate pattern analysis (MVPA) was unable to distinguish the activation patterns of the basic tastes, suggesting the possibility of weakly clustered distribution of the taste-preference neural activities in the human insular cortex. In conclusion, the presented results suggest overlapping representations of the basic tastes in the human gustatory insular cortex.

scholarly journals Characterization of logic program revision as an extension of propositional revision

2015 ◽  
Vol 16 (1) ◽  
pp. 111-138 ◽  
Author(s):  
NICOLAS SCHWIND ◽  
KATSUMI INOUE
Keyword(s):  
Logic Program ◽  
Nonmonotonic Reasoning ◽  
Knowledge Representation And Reasoning ◽  
Logic Programs ◽  
Characterization Result ◽  
International Conference ◽  
Complete Procedure ◽  
Boolean Lattices ◽  
Computational Properties

AbstractWe address the problem of belief revision of logic programs (LPs), i.e., how to incorporate to a LP P a new LP Q. Based on the structure of SE interpretations, Delgrande et al. (2008. Proc. of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR'08), 411–421; 2013b. Proc. of the 12th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'13), 264–276) adapted the well-known AGM framework (Alchourrón et al. 1985. Journal of Symbolic Logic 50, 2, 510–530) to LP revision. They identified the rational behavior of LP revision and introduced some specific operators. In this paper, a constructive characterization of all rational LP revision operators is given in terms of orderings over propositional interpretations with some further conditions specific to SE interpretations. It provides an intuitive, complete procedure for the construction of all rational LP revision operators and makes easier the comprehension of their semantic and computational properties. We give a particular consideration to LPs of very general form, i.e., the generalized logic programs (GLPs). We show that every rational GLP revision operator is derived from a propositional revision operator satisfying the original AGM postulates. Interestingly, the further conditions specific to GLP revision are independent from the propositional revision operator on which a GLP revision operator is based. Taking advantage of our characterization result, we embed the GLP revision operators into structures of Boolean lattices, that allow us to bring to light some potential weaknesses in the adapted AGM postulates. To illustrate our claim, we introduce and characterize axiomatically two specific classes of (rational) GLP revision operators which arguably have a drastic behavior. We additionally consider two more restricted forms of LPs, i.e., the disjunctive logic programs (DLPs) and the normal logic programs (NLPs) and adapt our characterization result to disjunctive logic program and normal logic program revision operators.

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