The Trace of the Weak Topology and of the T-Topology of Lop Coincide on the Pseudo-Meet-Prime Elements of a Continuous Lattice L

Author(s):  
Rudolf-E. Hoffmann
2015 ◽  
Vol 48 (4) ◽  
Author(s):  
Y.T. Nai ◽  
D. Zhao

AbstractWe take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.


1980 ◽  
Vol 32 (2) ◽  
pp. 385-394 ◽  
Author(s):  
B. Banaschewski

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.


Author(s):  
Christopher Tomlins

As the linguistic/cultural turn of the last fifty years has begun to ebb, sociolegal and legal-humanist scholarship has seen an accelerating return to materiality. This chapter asks what relationship may be forthcoming between the “new materialisms” and “vibrant matter” of recent years, and the older materialisms—both historical and literary, both Marxist and non-Marxist—that held sway prior to post-structuralism. What impact might such a relationship have on the forms, notably “spatial justice,” that materiality is assuming in contemporary legal studies? To attempt answers, the chapter turns to two figures from more than half a century ago: Gaston Bachelard—once famous, now mostly forgotten; and Walter Benjamin—once largely forgotten, now famous. A prolific and much-admired writer between 1930 and 1960, Bachelard pursued two trajectories of inquiry: a dialectical and materialist and historical (but non-Marxist) philosophy of science; and a poetics of the material imagination based on inquiry into the literary reception and representation of the prime elements—earth, water, fire, and air. Between the late 1920s and 1940, meanwhile, Benjamin developed an idiosyncratic but potent form of historical materialism dedicated to “arousing [the world] from its dream of itself.” The chapter argues that by mobilizing Bachelard and Benjamin for scholarship at the intersection of law and the humanities, old and new materialisms can be brought into a satisfying conjunction that simultaneously offers a poetics for spatial justice and lays a foundation for a materialist legal historiography for the twenty-first century.


2020 ◽  
Vol 178 (3-4) ◽  
pp. 1125-1172
Author(s):  
Julio Backhoff-Veraguas ◽  
Daniel Bartl ◽  
Mathias Beiglböck ◽  
Manu Eder

Abstract A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.


1955 ◽  
Vol 28 (3) ◽  
pp. 125 ◽  
Author(s):  
John Wells Brace
Keyword(s):  

2011 ◽  
Vol 285 (2-3) ◽  
pp. 136-149 ◽  
Author(s):  
L. Agud ◽  
J. M. Calabuig ◽  
E. A. Sánchez Pérez

2011 ◽  
Vol 158 (9) ◽  
pp. 1131-1135
Author(s):  
Jerzy Ka̧kol ◽  
Wiesław Śliwa
Keyword(s):  

2017 ◽  
Vol 18 ◽  
pp. 48-57 ◽  
Author(s):  
Martin Eichenhofer ◽  
Joanna C.H. Wong ◽  
Paolo Ermanni

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