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On the dualization in distributive lattices and related problems [article]

Oscar Defrain, Lhouari Nourine, Takeaki Uno
2020 arXiv   pre-print
In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem.  ...  This in particular allows us to study the complexity of the problem under various combined restrictions on graph classes and poset types, including bipartite, split, and co-bipartite graphs, and variants  ...  We show that the obtained problem is equivalent to the dualization in distributive lattices.  ... 
arXiv:1902.07004v2 fatcat:w7xlm72dqvbe3e7u3ihbuuoem4

Dualization in Lattices Given by Ordered Sets of Irreducibles [article]

Mikhail A. Babin, Sergei O. Kuznetsov
2015 arXiv   pre-print
However, if the lattice is distributive, dualization is shown to be possible in subexponential time.  ...  Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 to the set of its maximal 0 values.  ...  The second author was supported by the Basic Research Program of the National Research University Higher School of Economics (Moscow, Russia) and Russian Foundation for Basic Research.  ... 
arXiv:1504.01145v2 fatcat:n5zsomhlpnbdviuo4urk4ewve4

The Involutive Quantaloid of Completely Distributive Lattices [article]

Luigi Santocanale
2020 arXiv   pre-print
Let sLatt be the category of sup-lattices and join-continuous functions and let cdLatt be the full subcategory of sLatt whose objects are the completely distributive lattices.  ...  We prove the following result: Q(L) is an involutive (that is, non-commutative cyclic -autonomous) quantale if and only if L is a completely distributive lattice.  ...  The author is thankful to Srecko Brlek, Claudia Muresan, and André Joyal for the fruitful discussions these scientists shared with him on this topic during winter 2018.  ... 
arXiv:1911.01085v3 fatcat:rk6ewzmn3jc7rlnb3ua6sdk7ui

Dualization in lattices given by implicational bases [article]

Oscar Defrain, Lhouari Nourine
2020 arXiv   pre-print
Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independent-width and the size of the premises equal one.  ...  It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P=NP.  ...  Acknowledgement The first author would like to thank Simon Vilmin for extensive discussions on the topic of this paper.  ... 
arXiv:1901.07503v5 fatcat:wm7iya77hjdm5mrc4nwd36d4e4

Beyond Hypergraph Dualization [chapter]

Lhouari Nourine, Jean-Marc Petit
2016 Encyclopedia of Algorithms  
For distributive lattices class which contains boolean lattice and the product of chains, the dualization is open. 3.  ...  The Trans-Enum problem can also be formulated as a dualization problem in posets. Let (P, ≤) be a poset (i.e. ≤ is a reflexive, antisymmetric, and transitive relation on the set P ).  ... 
doi:10.1007/978-1-4939-2864-4_719 fatcat:uzler4wr6vbobozpcrtnid6ipa

Beyond Hypergraph Dualization [chapter]

Lhouari Nourine, Jean-Marc Petit
2014 Encyclopedia of Algorithms  
The Trans-Enum problem can also be formulated as a dualization problem in posets. Let (P, ≤) be a poset (i.e. ≤ is a reflexive, antisymmetric, and transitive relation on the set P ).  ...  For distributive lattices class which contains boolean lattice and the product of chains, the dualization is open. sch(R) and c ∈ D. Let Q r be the set of all possible conjunctive queries over R.  ... 
doi:10.1007/978-3-642-27848-8_719-1 fatcat:lu7ovfqp5rhnlcfehhp5xnwhnq

Translating between the representations of a ranked convex geometry [article]

Oscar Defrain and Lhouari Nourine and Simon Vilmin
2021 arXiv   pre-print
Quite surprisingly, we show that the problem in this context is already harder than the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem for which  ...  In Horn logic, these are respectively known as the Horn expressions and the characteristic models.  ...  In Section 3 we give hardness results in acyclic convex geometries in relation with the dualization problem in distributive lattices.  ... 
arXiv:1907.09433v2 fatcat:zshldvesr5d6hjoyvdf4dpxb7m

Dualizing Clones as Models of Lawvere Theories

Sebastian Kerkhoff
2014 Electronical Notes in Theoretical Computer Science  
be beneficial for the classical problem of studying the lattice of clones on a given set.  ...  In this paper, we argue that the possibility of applying duality theory might be such a reason, and we support this claim by outlining how treating and dualizing clones as models of Lawvere theories can  ...  Dualizing the bounded distributive lattice from Example 5.3 as a distributive lattice without the constants 0, 1 would for instance look as displayed in Figure 1 .  ... 
doi:10.1016/j.entcs.2014.02.005 fatcat:d7eyy7ksevckhjt6sfajempv3m

Page 83 of Mathematical Reviews Vol. , Issue 2002A [page]

2002 Mathematical Reviews  
Different cases of dualities are considered: For a cardinal « any «-dualizable algebra B via a cer- A. Priestley for bounded distributive lattices have been ex- NUMBER THEORY A.  ...  The digraph (G,—) corresponds to the groupoid whose underlying set is G and in which xy = x if and only if x A set G equipped with an equivalence relation p can be given the structure of a quasitrivial  ... 

Approaches to the sign problem in lattice field theory

Christof Gattringer, Kurt Langfeld
2016 International Journal of Modern Physics A  
We here review the sign problem in lattice field theories, focussing on two more recent methods: Dualization to world-line type of representations and the density-of-states approach.  ...  Various strategies to overcome this so-called Sign Problem or Complex Action Problem were proposed during the last thirty years.  ...  Sulejmanpasic, and A. Rago for discussions. KL is supported by the Leverhulme Trust (grant RPG-2014-118) and STFC (grant ST/L000350/1). CG is supported by DFG TRR55, and FWF grant I 1452-N27.  ... 
doi:10.1142/s0217751x16430077 fatcat:6f2zoufbwvg6fdhgalal4d74pe

A GENERAL DUALITY THEORY FOR CLONES

SEBASTIAN KERKHOFF
2013 International journal of algebra and computation  
Kearnes Inspired by work of Mašulović, we outline a general duality theory for clones that will allow us to dualize any given clone, together with its relational counterpart and the relationship between  ...  Afterwards, we put the approach to work and illustrate it by producing some specific results for concrete examples as well as some general results that come from studying the duals of clones in a rather  ...  Indeed, every step towards a richer duality theory for relational structures provides us with a new opportunity to obtain more results for clones over sets by using the machinery introduced in this paper  ... 
doi:10.1142/s0218196713500021 fatcat:5gqotzd6yzhg3jyg6gmx33as3q

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Irene A. Sharaya
2012 Reliable Computing  
The purpose of this paper is to inquire the connection between maximal inner interval estimates of the solution sets to interval linear system and solutions of the dualization equation in Kaucher interval  ...  The results of our work are as follows: 1) a criterion of inner interval estimate of the solution set, 2) a criterion for a solution of dualization equation to be a maximal inner interval estimate of the  ...  Lattice structure. "⊆", "≤" are partial order relations on IR defined by the order relation "≤" on R (see Figures 1 and 2 ): x ⊆ y deƒ ⇐⇒ (x ≥ y and x ≤ y), x ≤ y deƒ ⇐⇒ (x ≤ y and x ≤ y). "∨", "∧" are  ... 
doi:10.1023/a:1011428127620 fatcat:lsg2nhdv35aedetmaj6urmn44u

Page 4940 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
Algebraically, a BL-algebra is a bounded distributive lattice having also the structure of a commutative monoid and one additional binary implication, with all the basic operations connected by certain  ...  It turns out that the kernels of such mappings are congruence relations on the (absolutely free) term algebra of the considered type.  ... 

Open problems from NCS 2018

Jonathan Leech, João Pita Costa
2019 The Art of Discrete and Applied Mathematics  
Thus the reader is invited to refer back to the relevant article. It is hoped that in the months to follow some of these problems will be solved, or at least, considerable light will be shed on them.  ...  All participants in the workshop NCS 2018 were invited to submit a list of open problems, typically problems relevant to the subject matter of their talk.  ...  Background For an antilattice N , a congruence θ is an equivalence relation on N that is compatible with both ∧ and ∨. Congruence relations form a lattice, Con(N ). 9 Joao Pita Costa Problem 9.3.  ... 
doi:10.26493/2590-9770.1332.594 fatcat:4vmhj7dl5zhkxeyobknctef274

Page 5985 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews  
A fundamental problem in the theory of natural dualities is to determine which finite algebras M are dualizable, in the sense that there is a structure M = (M;G, H, R, 7 ) yielding a natural duality on  ...  For example, if M has a lattice reduct, it is sufficient to choose R to be S(M?). Herein lies the problem, for even when M is  ... 
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