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Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law

Samuel Huang, Dov Tamari
1972 Journal of combinatorial theory. Series A  
The notions of right bracketing and its associated bracketing function or family of sets lead to a short proof of the latticeproperty of (T, , -+), the system of binary bracketings ordered by a semi-associative  ...  law. 1 "Bracket" is used synonymously for "pair of parentheses" and "openings" ("closings") for "opening (closing) parentheses". 7  ...  ., the jinite subsystems U',z , -), are lattices for all n. A proof of this theorem has been published in this journal [I] . Here a considerably simpler proof is offered.  ... 
doi:10.1016/0097-3165(72)90003-9 fatcat:kodv3gf4ovhxxkjdt5alt2jdzi

System Description: The Proof Transformation System CERES [chapter]

Tsvetan Dunchev, Alexander Leitsch, Tomer Libal, Daniel Weller, Bruno Woltzenlogel Paleo
2010 Lecture Notes in Computer Science  
The system CERES, an implementation of the CERES-method has been used successfully in analyzing nontrivial mathematical proofs (see [4] ).In this paper we describe the main features of the CERES system  ...  The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs.  ...  Summarizing, the CERES system has taken as input a proof in lattice theory which used the auxiliary notion of partial order.  ... 
doi:10.1007/978-3-642-14203-1_36 fatcat:lzusyqb645hn7ghxqhl4qfhgse

Lower homomorphisms on additive generalized algebraic lattices

Xueyou Chen, Zike Deng
2007 Applied General Topology  
generalized algebraic lattices form an additive generalized algebraic lattice, which yields the classical theorem: the function space between T0-topological spaces is also a T0-topological space with  ...  In this paper, with the additivity property ([8]), the generalized way-below relation ([15]) and the maximal system of subsets ([6]) as tools, we prove that all lower homomorphisms between two additive  ...  Acknowledgements The authors would like to thank the Editor Juanjo Font for his English revision of the paper, which has helped to improve the paper significantly.  ... 
doi:10.4995/agt.2007.1900 fatcat:jeuw5nngh5a2pnebfidowo7kau

Permutability of rules in lattice theory

Sara Negri, Jan von Plato
2002 Algebra Universalis  
The theory of linear lattices is presented as a system with multiple-conclusion rules.  ...  Decidability of derivability with the rules for linear lattices follows through the termination of proof-search.  ...  Introduction In [Negri and von Plato 2002] , lattice theory was presented as a system of rules of proof. The rules are used for the construction of formal proofs or derivations.  ... 
doi:10.1007/s000120200012 fatcat:fv7yngz5qrdxfmwciid2nt5ofq

Mechanizing Complemented Lattices Within Mizar Type System

Adam Grabowski
2015 Journal of automated reasoning  
The efficiency of this approach was tested e.g. on short axiom systems for Boolean algebras based on negation and disjunction. We also proved Nachbin theorem for spectra of distributive lattices.  ...  Recently some longstanding open lattice theory problems were solved with the help of automated theorem provers.  ...  , valuable hints and thoughts (including those of not fully scientific character) and making the MIZAR system moving forward.  ... 
doi:10.1007/s10817-015-9333-5 fatcat:o3gvggn7qzdb3mfvhde3wve3ly

Page 1256 of Mathematical Reviews Vol. , Issue 83c [page]

1983 Mathematical Reviews  
M. 83c:81079 dimensional lattice gauge theories. I. Z, pure gauge system. Nuclear Phys. B 170 (1980), no. 2,-FS 1, 211-227.  ...  Thus a no-go theorem for putting theories of weak interaction on a lattice is presented.  ... 

Page 1768 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
A key lemma in the proof states that the tensor product of two semistable systems of linear inequalities is again semistable.  ...  M. (1-IDA-CM; Princeton, NJ) On the involutions fixing the class of a lattice. (English summary) J. Number Theory 101 (2003), no. 1, 185-194. Let A Cc R" be an integral lattice of level /.  ... 

Stone compactification of additive generalized-algebraic lattices

Xueyou Chen, Quingguo Li, Zike Deng
2007 Applied General Topology  
In this paper, the notions of regular, completely regular, compact additive generalized algebraic lattices ([7]) are introduced, and Stone compactification is constructed.  ...  Theorem: An additive generalized algebraic lattice has a Stone compactification if and only if it is regular and completely regular. 2000 AMS Classification: 06B30, 06B35, 54D35, 54H10  ...  Acknowledgements We are grateful to the editor for his valuable comments and suggestions.  ... 
doi:10.4995/agt.2007.1901 fatcat:kwn4vqhahzbmlbhmyy3oyzw634

Positive definite lattices of rank at most 8

Robert L. Griess
2003 Journal of Number Theory  
We give a short uniqueness proof for the E 8 root lattice, and in fact for all positive definite unimodular lattices of rank up to 8.  ...  Our proof is done with elementary arguments, mainly these: (1) invariant theory for integer matrices; (2) an upper bound for the minimum of nonzero norms (either of the elementary bounds of Hermite or  ...  Let L be a lattice and M a finite index sublattice. Then detðMÞ ¼ detðLÞjL:Mj 2 : Proof. This follows from the theory of modules for a PID.  ... 
doi:10.1016/s0022-314x(03)00107-0 fatcat:yqvs35cgknbzjd2hy5bzb5uw5u

Non-Wellfounded Proof Theory For (Kleene+Action)(Algebras+Lattices)

Anupam Das, Damien Pous, Michael Wagner
2018 Annual Conference for Computer Science Logic  
We prove cut-elimination for a sequent-style proof system which is sound and complete for the equational theory of Kleene algebra, and where proofs are (potentially) non-wellfounded infinite trees.  ...  We recover the equational theory of all action lattices by restricting to regular proofs (with cut) -those proofs that are unfoldings of finite graphs.  ...  C S L 2 0 1 8 19:4 Non-Wellfounded Proof Theory For (Kleene+Action)(Algebras+Lattices) Palka proposed a sequent system for star continuous action lattices in [29] , for which she proved cut-elimination  ... 
doi:10.4230/lipics.csl.2018.19 dblp:conf/csl/DasP18 fatcat:5hg67ddwr5hu7fkiuryt4rl4ru

Page 4234 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews  
-theory for generalized Cauchy-Riemann systems, akin to that described by E. M. Stein and Weiss [Acta Math. 103 (1960), 25-62; MR 22 #12315).  ...  These existence theorems are vital for the proof of super-rigidity.  ... 

On the Classification of Type II Codes of Length 24 [article]

Noam D. Elkies, Scott D. Kominers
2009 arXiv   pre-print
We give a new, purely coding-theoretic proof of Koch's criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators.  ...  This approach is inspired by Venkov's approach to the classification of the root systems of Type II lattices in R^24, and gives a new instance of the analogy between lattices and codes.  ...  The authors would like to thank David Hansen for his helpful comments on an earlier draft of the manuscript.  ... 
arXiv:0902.1942v2 fatcat:wj7ryvqquvdkfivjkf7wspoala

Pawlak Algebra and Approximate Structure on Fuzzy Lattice

Ying Zhuang, Wenqi Liu, Chin-Chia Wu, Jinhai Li
2014 The Scientific World Journal  
The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering  ...  Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.  ...  These works would provide a new direction for the study of rough set theory and information systems.  ... 
doi:10.1155/2014/697107 pmid:25152922 pmcid:PMC4134832 fatcat:7giwgtktprax5mrt5o5jzbbu3e

Infinitary domain logic for finitary transition systems [chapter]

Marcello M. Bonsangue, Joost N. Kok
1997 Lecture Notes in Computer Science  
As a consequence soundness and completeness of the in nitary logic is obtained for the class of nitary transition systems.  ...  A corollary of this result is that the same holds for the in nitary Hennessy-Milner logic.  ...  We thank also the anonymous referees and Achim Jung for their constructive comments.  ... 
doi:10.1007/bfb0014553 fatcat:bmwkg5ofmvc5niog3lowwmzv6e

Theory of Completeness for Logical Spaces

Kensaku Gomi
2009 Logica Universalis  
Since È is identified with ¼ ½ and ¼ ½ is a typical lattice, a pair´ µ of a non-empty set and a subset of for a certain lattice is also called a -valued functional logical space.  ...  On the other hand, the abstract theory of logical spaces is a framework for practical logics, and as such could have been polished by sense of beauty and efficiency, and generalized by wish for a wider  ...  The author expresses his thanks to all who gave him support or inspiration for this work, particularly to M.S.s Yasuaki Mizumura, Hitoki Matsuda, Yohsuke Takaoka, Ken Sasaki, and Kazuhiro Takahashi for  ... 
doi:10.1007/s11787-009-0008-z fatcat:pd72tzz2ufb4hkedhmytgs3fqq
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