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Quantum B-algebras

Wolfgang Rump — 2013

Open Mathematics

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,...

Representation theory of two-dimensionalbrauer graph rings

Wolfgang Rump — 2000

Colloquium Mathematicae

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 [ 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)...

Differentiation and splitting for lattices over orders

Wolfgang Rump — 2001

Colloquium Mathematicae

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 - l a t / [ ] δ u Λ - l a t / [ 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...

Stable short exact sequences and the maximal exact structure of an additive category

Wolfgang Rump — 2015

Fundamenta Mathematicae

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.

The essential cover and the absolute cover of a schematic space

Wolfgang RumpYi Chuan Yang — 2009

Colloquium Mathematicae

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,...

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