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Topological duality for distributive lattices, and applications [article]

Mai Gehrke, Sam van Gool
2022 arXiv   pre-print
This material will be published by Cambridge University Press as "Topological Duality for Distributive Lattices: Theory and Applications" by Mai Gehrke and Sam van Gool.  ...  This pre-publication is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works.  ...  as "Topological Duality for Distributive Lattices: Theory and Applications" by Mai Gehrke and Sam van Gool.  ... 
arXiv:2203.03286v3 fatcat:m7b27oubfffifgjylxacewe66e

Page 4763 of Mathematical Reviews Vol. , Issue 87i [page]

1987 Mathematical Reviews  
Continuous lattices and their applications (Bremen, 1982), 237-249, Lecture Notes in Pure and Appl. Math., 101, Dekker, New York, 1985.  ...  ., The Pontryagin duality of compact 0-dimensional semilattices and its applications, Lecture Notes in Math., 396, Springer, Berlin, 1974; MR 50 #7398; G.  ... 

Boolean Topological Distributive Lattices and Canonical Extensions

B. A. Davey, M. Haviar, H. A. Priestley
2007 Applied Categorical Structures  
Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets.  ...  By combining these dualities we obtain new insights into canonical extensions of distributive lattices.  ...  Acknowledgement The authors would like to thank the referee for pointing out several connections between our results and results in related areas.  ... 
doi:10.1007/s10485-007-9090-7 fatcat:l4sacjs5bzglzdy6edwsd7eb4y

Natural Dualities in Partnership

Brian A. Davey, Miroslav Haviar, Hilary A. Priestley
2011 Applied Categorical Structures  
A prototype for this procedure is provided by the passage from Priestley duality for bounded distributive lattices to Banaschewski duality for ordered sets.  ...  Given a duality, one may ask when the topology can be swapped to the other side to yield a partner duality (or, better, a dual equivalence) between a category of topological algebras and a category of  ...  Banaschewski's duality sets up a dual equivalence between the category D T of Boolean topological distributive lattices and the category P of ordered sets [1] .  ... 
doi:10.1007/s10485-011-9253-4 fatcat:dubia66vfnezvngfruzhdperiu

Page 3407 of Mathematical Reviews Vol. , Issue 86h [page]

1986 Mathematical Reviews  
She presents a broad spectrum of techniques and facts concerning distributive lattices and their order-topological dual counterparts.  ...  A. (4-0X) 86h:06028 Catalytic distributive lattices and compact zero-dimensional topological lattices. Algebra Universalis 19 (1984), no. 3, 322-329.  ... 

Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries

Hongyu Wang, Yingcheng Li, Yuting Hu, Yidun Wan
2020 Journal of High Energy Physics  
Aguado, Electric-magnetic duality of lattice systems with topological order , Nucl. Phys. B 876 (2013) 619 [arXiv:1006.5823] [INSPIRE]. [7] O. Buerschaper and M.  ...  Wan, Unified framework of topological phases with symmetry , Phys. Rev. B 90 (2014) 245125 [arXiv:1402.3356] [INSPIRE]. [20] F. Burnell, Anyon Condensation and Its Applications , Ann. Rev. Cond.  ... 
doi:10.1007/jhep02(2020)030 fatcat:yt33ap5glzdfvgkh5jzol4pugy

Page 2582 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews  
Further, he modi- fies the Stone duality to obtain a duality theory for distributive and bounded distributive lattices. This theory is used to derive the main results of the paper.  ...  This suggests a new way to deal with duality both in fuzzy topology and in the more general setting of topological molecular lattices.  ... 

Page 2395 of Mathematical Reviews Vol. , Issue 93e [page]

1993 Mathematical Reviews  
It encompasses the classical Stone dualities for Boolean lattices and for bounded distributive lattices [M. H. Stone, Casopis Pést. Mat. 67 (1937), 1-25; Zbl 18, 3], H. A.  ...  Priestley’s duality for bounded distributive lattices via compact to- tally order-disconnected spaces [Bull. London Math. Soc. 2 (1970), 186-190; MR 42 #153], and A.  ... 

2. Abstracts Invited Speakers

2013 Journal of Perinatal Medicine  
The first talk will take as its starting point Stone duality for Boolean algebras and Priestley duality for distributive lattices, and will present further dual equivalences which follow the same pattern  ...  There is a third duality-a hybrid between the algebraic and the topological dualities-that seems not to have been noticed before, even in the case of Boolean algebras, and that has important applications  ... 
doi:10.1515/jpm-2013-2001 fatcat:uzvmvs2t7rcbznk3yhwaaobvly

Viral Vectors I: Major Viral Vectors

2000 Molecular Therapy  
The first talk will take as its starting point Stone duality for Boolean algebras and Priestley duality for distributive lattices, and will present further dual equivalences which follow the same pattern  ...  There is a third duality-a hybrid between the algebraic and the topological dualities-that seems not to have been noticed before, even in the case of Boolean algebras, and that has important applications  ... 
doi:10.1006/mthe.2000.0143 fatcat:t7k2ymu7rjeirdh3zhe4cxuxju

Page 5308 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
A lattice L is called a topological lattice if it carries a Hausdorff topology for which the two lattice operations are continuous.  ...  A topological orthomodular lattice (TOML) is defined to be an or- thomodular lattice which is a topological lattice for which the orthocomplementation is also continuous.  ... 

Canonical Extensions and Discrete Dualities for Finitely Generated Varieties of Lattice-based Algebras

B. A. Davey, H. A. Priestley
2012 Studia Logica: An International Journal for Symbolic Logic  
In addition, we call on recent results from duality theory to show that topological and discrete dualities for A exist in partnership.  ...  The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed.  ...  To see this, take for example A = D, the variety of bounded distributive lattices.  ... 
doi:10.1007/s11225-012-9392-0 fatcat:mdjlbtzgenadxlalvyqutuazvy

Page 865 of Mathematical Reviews Vol. , Issue 85c [page]

1985 Mathematical Reviews  
to be a Boolean space), and Priestley’s duality for bounded distributive lattices (we give A = the two-element bounded distributive lattice the extra struc- ture of an ordered discrete space, and then  ...  The main result for later applications is the two-step duality theorem, which also gives a sufficient condition for duality of finite algebras B.  ... 

A view of canonical extension [article]

Mai Gehrke, Jacob Vosmaer
2010 arXiv   pre-print
In particular, they are doubly algebraic lattices and their interval topologies agree with their double Scott topologies and make them Priestley topological algebras.  ...  In addition some topological results about canonical extensions of lattices with additional operations in finitely generated varieties are given.  ...  Both authors would like to acknowledge the influence of discussions and work with H.A. Priestley on the content of this paper. In partic-  ... 
arXiv:1009.2803v1 fatcat:edbavypkenbpvi2puuoodj2m6e

A View of Canonical Extension [chapter]

Mai Gehrke, Jacob Vosmaer
2011 Lecture Notes in Computer Science  
In particular, they are doubly algebraic lattices and their interval topologies agree with their double Scott topologies and make them Priestley topological algebras.  ...  In addition some topological results about canonical extensions of lattices with additional operations in finitely generated varieties are given.  ...  Both authors would like to acknowledge the influence of discussions and work with H.A. Priestley on the content of this paper. In partic-  ... 
doi:10.1007/978-3-642-22303-7_6 fatcat:glfht7qipfgthkzre4xkucfm7m
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