Properties of semi-elementary imsets as sums of elemen- tary imsets

Kashimura, Takuya; Sei, Tomonari; Takemura, Akimichi; Tanaka, Kentaro

doi:10.18409/jas.v2i1.8

Cited by 1 publication

(4 citation statements)

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“…We denote this face by F A,B | C . In Kashimura et al 7 we have shown some remarkable facts on F A,B | C . Here we establish more basic facts on F A,B | C .…”

Section: Structure Of Faces Of Semi-elementary Imsetsmentioning

confidence: 66%

“…Our final result of this section concerns a product of skeletal functions for disjoint sets. These functions played an important role in Kashimura et al 7 Proposition 4.3. Let AB = N , A ∩ B = ∅ and consider f of the form…”

Section: Results On Extreme Rays Of the Supermodular Conementioning

confidence: 97%

“…The fact that dim(F A,B | C ) does not depend on C can also be seen from the one-to-one linear correspondence between Lin(E A,B | C ) and Lin(E A,B | ∅ ) given by Remark 5.3. In our previous manuscript Kashimura et al 7 we studied how the semi-elementary imset u A,B | C is expressed as a non-negative integer combination of elements of E A,B | C . By Theorem 5.1, if u A,B | C is expressed as a non-negative integer combination of all elementary imsets, then the coefficients of u ∈ E A,B | C have to be zero.…”

Section: In This Section We Establish the Following Basic Facts On Fmentioning

confidence: 99%

“…In Kashimura et al 7 we gave a detailed study of the set of all possible non-negative integer combinations of elementary imsets which are equal to a semi-elementary imset. We consider R 2 |N | as equipped with the standard inner product •, • .…”

Section: Basic Facts On Supermodular Functions and Imsetsmentioning

confidence: 99%

See 3 more Smart Citations

Cones of Elementary Imsets and Supermodular Functions: A Review and Some New Results

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et al. 2012

Harmony of Gröbner Bases and the Modern Industrial Society - The Second CREST-CSBM International Conference

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In this paper we give a review of the method of imsets introduced by Studený 1 from a geometric point of view. Elementary imsets span a polyhedral cone and its dual cone is the cone of supermodular functions. We review basic facts on the structure of these cones. Then we derive some new results on the following topics: i) extreme rays of the cone of standardized supermodular functions, ii) faces of the cones, iii) small relations among elementary imsets, and iv) some computational results on Markov basis for the toric ideal defined by elementary imsets.

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“…We denote this face by F A,B | C . In Kashimura et al 7 we have shown some remarkable facts on F A,B | C . Here we establish more basic facts on F A,B | C .…”

Section: Structure Of Faces Of Semi-elementary Imsetsmentioning

confidence: 66%

Section: Results On Extreme Rays Of the Supermodular Conementioning

confidence: 97%

Section: In This Section We Establish the Following Basic Facts On Fmentioning

confidence: 99%

Section: Basic Facts On Supermodular Functions and Imsetsmentioning

confidence: 99%

See 2 more Smart Citations