The direct sum of $q$-matroids

Ceria, Michela; Jurrius, Relinde

doi:10.48550/arxiv.2109.13637

Cited by 3 publications

(24 citation statements)

References 9 publications

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“…Our Proposition 4.5 illustrates this fact for a very simple class of representable q-matroids. Just recently, a first attempt of defining a direct sum of q-matroids has been put forward in [3]. We will show that it plays a prominent role in our framework.…”

Section: Introductionmentioning

confidence: 93%

“…In this section we show that none of the first 5 categories has a coproduct, and in the next section we establish the existence of a coproduct in q-Mat l-w . It is in fact the direct sum as introduced recently in [3]. The non-existence of a coproduct in q-Mat s and q-Mat l-s stands in contrast to the case of classical matroids, where the direct sum (see [9,Sec.…”

Section: Non-existence Of Coproducts In Categories Of Q-matroidsmentioning

confidence: 99%

“…It turns out that the only category that admits a coproduct is the one where the morphisms are linear weak maps. In that case, the direct sum introduced in [3] is the coproduct. The non-existence of a coproduct in the category with strong maps (or linear strong maps) stands in contrast to the classical case, where the direct sum is indeed the coproduct.…”

Section: Introductionmentioning

confidence: 99%

“…4.9]. However, the smallest non-representable q-matroid is the following one, found just recently in [3]. (or any other partial spread of size 4).…”

Section: Introductionmentioning

confidence: 99%

“…4.7] that this does indeed define a q-matroid M = (F 4 2 , ρ). In [3,Sec. 3.3] it has been shown that M is a non-representable q-matroid.…”

Section: Introductionmentioning

confidence: 99%

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Coproducts in Categories of q-Matroids

Gluesing-Luerssen¹,

Jany²

2021

Preprint

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q-Matroids form the q-analogue of classical matroids. In this paper we introduce various types of maps between q-matroids. These maps are not necessarily linear, but they must map subspaces to subspaces and respect the q-matroid structure in certain ways. The various types of maps give rise to different categories of q-matroids. We show that only one of these categories possesses a coproduct. This is the category where the morphisms are linear weak maps, that is, the rank of the image of any subspace is not larger than the rank of the subspace itself. The coproduct in this category is the very recently introduced direct sum of q-matroids.

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Section: Introductionmentioning

confidence: 93%

Section: Non-existence Of Coproducts In Categories Of Q-matroidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…4.9]. However, the smallest non-representable q-matroid is the following one, found just recently in [3]. (or any other partial spread of size 4).…”

Section: Introductionmentioning

confidence: 99%

“…4.7] that this does indeed define a q-matroid M = (F 4 2 , ρ). In [3,Sec. 3.3] it has been shown that M is a non-representable q-matroid.…”

Section: Introductionmentioning

confidence: 99%

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Coproducts in Categories of q-Matroids

Gluesing-Luerssen¹,

Jany²

2021

Preprint

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The Cyclic Flats of a $q$-Matroid

Alfarano¹,

Byrne²

2022

Preprint

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In this paper we develop the theory of cyclic flats of q-matroids. We show that the lattice of cyclic flats, together with their ranks, uniquely determines a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of F q m -independence of an F qsubspace of F n q and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.

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Decompositions of q-Matroids Using Cyclic Flats

Gluesing-Luerssen¹,

Jany²

2023

Preprint

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We study the direct sum of q-matroids by way of their cyclic flats. Using that the rank function of a q-matroid is fully determined by the cyclic flats and their ranks, we show that the cyclic flats of the direct sum of two q-matroids are exactly all the direct sums of the cyclic flats of the two summands. This simplifies the rank function of the direct sum significantly. A qmatroid is called irreducible if it cannot be written as a (non-trivial) direct sum. We provide a characterization of irreducibility in terms of the cyclic flats and show that every q-matroid can be decomposed into a direct sum of irreducible q-matroids, which are unique up to equivalence.

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The direct sum of $q$-matroids

Cited by 3 publications

References 9 publications

Coproducts in Categories of q-Matroids

Coproducts in Categories of q-Matroids

The Cyclic Flats of a $q$-Matroid

Decompositions of q-Matroids Using Cyclic Flats

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