archimedean lattice
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2020 ◽  
Vol 101 (4) ◽  
Author(s):  
V. Peri ◽  
S. Ok ◽  
S. S. Tsirkin ◽  
T. Neupert ◽  
G. Baskaran ◽  
...  

2018 ◽  
Vol 30 (2) ◽  
pp. 513-526 ◽  
Author(s):  
Richard N. Ball ◽  
Vincenzo Marra ◽  
Daniel McNeill ◽  
Andrea Pedrini

AbstractWe use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered groupGwith a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriateG-indexed zero-dimensional compactification{w_{G}Z_{G}}of its space{Z_{G}}ofminimalprime ideals. The two further ingredients needed to establish this representation are the Yosida representation ofGon its space{X_{G}}ofmaximalideals, and the well-known continuous surjection of{Z_{G}}onto{X_{G}}. We then establish our main result by showing that the inclusion-minimal extension of this representation ofGthat separates the points of{Z_{G}}– namely, the sublattice subgroup of{\operatorname{C}(Z_{G})}generated by the image ofGalong with all characteristic functions of clopen (closed and open) subsets of{Z_{G}}which are determined by elements ofG– is precisely the classical projectable hull ofG. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.


2016 ◽  
Vol 45 (21) ◽  
pp. 8708-8711 ◽  
Author(s):  
Wenbin Guo ◽  
Yingying Tang ◽  
Suyun Zhang ◽  
Sihuai Chen ◽  
Hongping Xiang ◽  
...  

The last number of Archimedean lattice was realized for the first time in a layered phosphate, BaCo4(OH)2(H2PO4)(HPO4)2(PO4).


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