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Ghislain Fourier - One of the best experts on this subject based on the ideXlab platform.
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marked Poset polytopes minkowski sums indecomposables and unimodular equivalence
Journal of Pure and Applied Algebra, 2016Co-Authors: Ghislain FourierAbstract:We analyze marked Poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked Poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked Poset polytope is the Minkowski sum of sets of lattice points for 0–1 polytopes. Moreover, we provide a decomposition of the marked Poset into indecomposable marked Posets, which respects this Minkowski sum decomposition for the marked chain polytopes.
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marked Poset polytopes minkowski sums indecomposables and unimodular equivalence
arXiv: Combinatorics, 2014Co-Authors: Ghislain FourierAbstract:We analyze marked Poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked Poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked Poset polytope is the Minkowski sum of sets of lattice points for 0-1 polytopes. Moreover, we provide a decomposition of the marked Poset into indecomposable marked Posets, which respects this Minkowski sum decomposition for the marked chain polytopes polytopes.
Harold N Gabow - One of the best experts on this subject based on the ideXlab platform.
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applications of a Poset representation to edge connectivity and graph rigidity
Foundations of Computer Science, 1991Co-Authors: Harold N GabowAbstract:A Poset representation for a family of sets defined by a labeling algorithm is investigated. Poset representations are given for the family of minimum cuts of a graph, and it is shown how to compute them quickly. The representations are the starting point for algorithms that increase the edge connectivity of a graph, from lambda to a given target tau = lambda + delta , adding the fewest edges possible. For undirected graphs the time bound is essentially the best-known bound to test tau -edge connectivity; for directed graphs the time bound is roughly a factor delta more. Also constructed are Poset representations for the family of rigid subgraphs of a graph, when graphs model structures constructed from rigid bars. The link between these problems is that they all deal with graphic matroids. >
Länger Helmut - One of the best experts on this subject based on the ideXlab platform.
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Kleene Posets and pseudo-Kleene Posets
2020Co-Authors: Chajda Ivan, Länger HelmutAbstract:The concept of a Kleene algebra (sometimes also called Kleene lattice) was already generalized by the first author for non-distributive lattices under the name pseudo-Kleene algebra. We extend these concepts to Posets and show how (pseudo-)Kleene Posets can be characterized by identities and implications of assigned commutative meet-directoids. Moreover, we prove that the Dedekind-MacNeille completion of a pseudo-Kleene Poset is a pseudo-Kleene algebra and that the Dedekind-MacNeille completion of a finite Kleene Poset is a Kleene algebra. Further, we introduce the concept of a strict (pseudo-)Kleene Poset and show that under an additional assumption a strict Kleene Poset can be organized into a residuated structure. Finally, we prove by using the so-called twist construction that every Poset can be embedded into a pseudo-Kleene Poset in some natural way
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The logic of orthomodular Posets of finite height
2020Co-Authors: Chajda Ivan, Länger HelmutAbstract:Orthomodular Posets form an algebraic formalization of the logic of quantum mechanics. The question is how to introduce the connective implication in such a logic. We show that this is possible when the orthomodular Poset in question is of finite height. The main point is that the corresponding algebra, called implication orthomodular Poset, i.e. a Poset equipped with a binary operator of implication, corresponds to the original orthomodular Poset and this operator is everywhere defined. We present here the complete list of axioms for implication orthomodular Posets. This enables us to derive an axiomatization in Gentzen style for the algebraizable logic of orthomodular Posets of finite height
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Extensions of Posets with an antitone involution to residuated structures
2020Co-Authors: Chajda Ivan, Kolařík Miroslav, Länger HelmutAbstract:We prove that every not necessarily bounded Poset P=(P,\le,') with an antitone involution can be extended to a residuated Poset E(P)=(E(P),\le,\odot,\rightarrow,1) where x'=x\rightarrow0 for all x\in P. If P is a lattice with an antitone involution then E(P) is a lattice, too. We show that a Poset can be extended to a residuated Poset by means of a finite chain and that a Boolean algebra (B,\vee,\wedge,',p,q) can be extended to a residuated lattice (Q,\vee,\wedge,\odot,\rightarrow,1) by means of a finite chain in such a way that x\odot y=x\wedge y and x\rightarrow y=x'\vee y for all x,y\in B
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Residuated operators in complemented Posets
2018Co-Authors: Chajda Ivan, Länger HelmutAbstract:Using the operators of taking upper and lower cones in a Poset with a unary operation, we define operators M(x,y) and R(x,y) in the sense of multiplication and residuation, respectively, and we show that by using these operators, a general modification of residuation can be introduced. A relatively pseudocomplemented Poset can be considered as a prototype of such an operator residuated Poset. As main results we prove that every Boolean Poset as well as every pseudo-orthomodular Poset can be organized into a (left) operator residuated structure. Some results on pseudo-orthomodular Posets are presented which show the analogy to orthomodular lattices and orthomodular Posets
Jong Yoon Hyun - One of the best experts on this subject based on the ideXlab platform.
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a subgroup of the full Poset isometry group
SIAM Journal on Discrete Mathematics, 2010Co-Authors: Jong Yoon HyunAbstract:Let $P$ be a Poset on $[n]$. We construct a subgroup $\mathcal{G}_P$ of the full Poset-isometry group $\mathrm{Iso}_P(F^n_q)$ and present the structure of $\mathcal{G}_P$ as well as its size. We also find Poset-metric spaces satisfying $\mathcal{G}_P=\mathrm{Iso}_P(F^n_q)$ by using a characterization of $\mathcal{G}_P$.
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Maximum distance separable Poset codes
Designs Codes and Cryptography, 2008Co-Authors: Jong Yoon HyunAbstract:We derive the Singleton bound for Poset codes and define the MDS Poset codes as linear codes which attain the Singleton bound. In this paper, we study the basic properties of MDS Poset codes. First, we introduce the concept of I -perfect codes and describe the MDS Poset codes in terms of I -perfect codes. Next, we study the weight distribution of an MDS Poset code and show that the weight distribution of an MDS Poset code is completely determined. Finally, we prove the duality theorem which states that a linear code C is an MDS $${\mathbb{P}}$$ -code if and only if $${C^\perp}$$ is an MDS $${\widetilde{\mathbb{P}}}$$ -code, where $${C^\perp}$$ is the dual code of C and $${\widetilde{\mathbb{P}}}$$ is the dual Poset of $${\mathbb{P}.}$$
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groups of linear isometries on Poset structures
Discrete Mathematics, 2008Co-Authors: Luciano Panek, Marcelo Firer, Hyun Kim, Jong Yoon HyunAbstract:Let V be an n-dimensional vector space over a finite field F"q and P={1,2,...,n} a Poset. We consider on V the Poset-metric d"P. In this paper, we give a complete description of groups of linear isometries of the metric space (V,d"P), for any Poset-metric d"P.
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the Poset structures admitting the extended binary hamming code to be a perfect code
Discrete Mathematics, 2004Co-Authors: Jong Yoon HyunAbstract:Brualdi et al. introduced the concept of Poset codes, and gave an example of Poset structure which admits the extended binary Hamming code to be a double-error-correcting perfect P-code. Our study is motivated by this example. In this paper we classify all Poset structures which admit the extended binary Hamming code to be a double or triple-error-correcting perfect P-code.
Einar Steingrımsson - One of the best experts on this subject based on the ideXlab platform.
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on the topology of the permutation pattern Poset
Journal of Combinatorial Theory Series A, 2015Co-Authors: Peter R W Mcnamara, Einar SteingrımssonAbstract:Abstract The set of all permutations, ordered by pattern containment, forms a Poset. This paper presents the first explicit major results on the topology of intervals in this Poset. We show that almost all (open) intervals in this Poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the Poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Mobius function of decomposable permutations given by Burstein et al. [9] .
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The topology of the permutation pattern Poset
Discrete Mathematics and Theoretical Computer Science, 2014Co-Authors: Peter Mcnamara, Einar SteingrımssonAbstract:The set of all permutations, ordered by pattern containment, forms a Poset. This extended abstract presents the first explicit major results on the topology of intervals in this Poset. We show that almost all (open) intervals in this Poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the Poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.
