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1
Content available remote Unimodular brackets and related structures
100%
Rump W.
Acta Arithmetica
|
2005
|
tom 120
|
nr 4
379-394
2
Content available remote Generalized radical rings, unknotted biquandles, and quantum groups
100%
Rump W.
Colloquium Mathematicum
|
2007
|
tom 109
|
nr 1
85-100
EN
Generalized radical rings (braces) were introduced for the study of set-theoretical solutions of the quantum Yang-Baxter equation. We discuss their relationship to groups of I-type, virtual knot theory, and quantum groups.
3
Content available remote Stable short exact sequences and the maximal exact structure of an additive category
100%
Rump W.
Fundamenta Mathematicae
|
2015
|
tom 228
|
nr 1
87-96
EN
It was recently proved that every additive category has a unique maximal exact structure, while it remained open whether the distinguished short exact sequences of this canonical exact structure coincide with the stable short exact sequences. The question is answered by a counterexample which shows that none of the steps to construct the maximal exact structure can be dropped.
4
Content available remote Differentiation and splitting for lattices over orders
100%
Rump W.
Colloquium Mathematicum
|
2001
|
tom 89
|
nr 1
7-42
EN
We extend our module-theoretic approach to Zavadskiĭ's differentiation techniques in representation theory. Let R be a complete discrete valuation domain with quotient field K, and Λ an R-order in a finite-dimensional K-algebra. For a hereditary monomorphism u: P ↪ I of Λ-lattices we have an equivalence of quotient categories $∂̃_{u}:Λ-lat/[ℋ ] ⭇ δ_{u}Λ-lat/[B]$ which generalizes Zavadskiĭ's algorithms for posets and tiled orders, and Simson's reduction algorithm for vector space categories. In this article we replace u by a more general type of monomorphism, and the derived order $δ_{u}Λ$ by some over-order $∂_{u}Λ ⊃ δ_{u}Λ$. Then $∂̃_{u}$ remains an equivalence if $δ_{u}Λ-lat$ is replaced by a certain subcategory of $∂_{u}Λ-lat$. The extended differentiation comprises a splitting theorem that implies Simson's splitting theorem for vector space categories.
5
Content available remote Quantum B-algebras
100%
Rump W.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1881-1899
EN
The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.
6
Content available remote Representation theory of two-dimensionalbrauer graph rings
100%
Rump W.
Colloquium Mathematicum
|
2000
|
tom 86
|
nr 2
239-251
EN
We consider a class of two-dimensional non-commutative Cohen-Macaulay rings to which a Brauer graph, that is, a finite graph endowed with a cyclic ordering of edges at any vertex, can be associated in a natural way. Some orders Λ over a two-dimensional regular local ring are of this type. They arise, e.g., as certain blocks of Hecke algebras over the completion of $ℤ[q,q^{-1}]$ at (p,q-1) for some rational prime $p$. For such orders Λ, a class of indecomposable maximal Cohen-Macaulay modules (see introduction) has been determined by K. W. Roggenkamp. We prove that this list of indecomposables of Λ is complete.
7
Content available remote On the maximal exact structure of an additive category
100%
Rump W.
Fundamenta Mathematicae
|
2011
|
tom 214
|
nr 1
77-87
EN
We prove that every additive category has a unique maximal exact structure in the sense of Quillen.
8
Content available remote Addendum to "Generalized radical rings, unknotted biquandles, and quantum groups" (Colloq. Math. 109 (2007), 85-100)
100%
Rump W.
Colloquium Mathematicum
|
2009
|
tom 117
|
nr 2
295-298
9
Content available remote Green walks in a hypergraph
100%
Rump W.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 1
133-147
10
Content available remote The essential cover and the absolute cover of a schematic space
63%
Rump W. , Yang Y.
Colloquium Mathematicum
|
2009
|
tom 114
|
nr 1
53-75
EN
A theorem of Gleason states that every compact space admits a projective cover. More generally, in the category of topological spaces with continuous maps, covers exist with respect to the full subcategory of extremally disconnected spaces. Such a cover of a space is called its absolute. We prove that the absolute exists within the category of schematic spaces, i.e. the spaces underlying a scheme. For a schematic space, we use the absolute to generalize Bourbaki's concept of irreducible component, so that embedded and multiple components may arise. We introduce the essential cover of a schematic space, and show that it parametrizes the generalized components.
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