10 found
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  1.  14
    L-algebras and three main non-classical logics.Wolfgang Rump - 2022 - Annals of Pure and Applied Logic 173 (7):103121.
  2.  14
    Non-commutative logical algebras and algebraic quantales.Wolfgang Rump & Yi Chuan Yang - 2014 - Annals of Pure and Applied Logic 165 (2):759-785.
    Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the way that every quantum B-algebra admits a (...)
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  3.  8
    Quantum B‐modules.Xia Zhang & Wolfgang Rump - 2022 - Mathematical Logic Quarterly 68 (2):159-170.
    Quantum B‐algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non‐commutative arithmetic, and quantum theory. A quantum B‐algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B‐module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi‐poset version of the completion of Dedekind and MacNeille.
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  4.  17
    L -effect Algebras.Wolfgang Rump & Xia Zhang - 2020 - Studia Logica 108 (4):725-750.
    L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a \-semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it is (...)
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  5.  15
    The Structure Group of a Generalized Orthomodular Lattice.Wolfgang Rump - 2018 - Studia Logica 106 (1):85-100.
    Orthomodular lattices with a two-valued Jauch–Piron state split into a generalized orthomodular lattice and its dual. GOMLs are characterized as a class of L-algebras, a quantum structure which arises in the theory of Garside groups, algebraic logic, and in connections with solutions of the quantum Yang–Baxter equation. It is proved that every GOML X embeds into a group G with a lattice structure such that the right multiplications in G are lattice automorphisms. Up to isomorphism, X is uniquely determined by (...)
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  6.  20
    Multi-posets in algebraic logic, group theory, and non-commutative topology.Wolfgang Rump - 2016 - Annals of Pure and Applied Logic 167 (11):1139-1160.
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  7.  41
    A Note on Bosbach’s Cone Algebras.Wolfgang Rump & Yichuan Yang - 2011 - Studia Logica 98 (3):375-386.
    In 2002, Dvurečenskij extended Mundici’s equivalence between unital abelian l -groups and MV-algebras to the non-commutative case. We analyse the relationship to Bosbach’s cone algebras and clarify the rôle of the corresponding pair of L -algebras. As a consequence, it follows that one of the two L -algebra axioms can be dropped.
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  8.  6
    Frobenius Quantales, Serre Quantales and the Riemann–Roch Theorem.Wolfgang Rump - 2021 - Studia Logica 110 (2):405-427.
    The Riemann–Roch theorem for algebraic curves is derived from a theorem for Girard quantales. Serre duality is shown to be a quantalic phenomenon. An example provides a Girard quantale satisfying the Riemann–Roch theorem, where the associated curve is non-connected and irreducible.
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  9.  6
    Linear L-Algebras and Prime Factorization.Wolfgang Rump - 2023 - Studia Logica 111 (1):57-82.
    A complete recursive description of noetherian linear _KL_-algebras is given. _L_-algebras form a quantum structure that occurs in algebraic logic, combinatorial group theory, measure theory, geometry, and in connection with solutions to the Yang-Baxter equation. It is proved that the self-similar closure of a noetherian linear _KL_-algebra is determined by its partially ordered set of primes, and that its elements admit a unique factorization by a decreasing sequence of prime elements.
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  10.  9
    Corrigendum and addendum to: “L-algebras and three main non-classical logics” [Ann. Pure Appl. Log. 173 (7) (2022) 103121]. [REVIEW]Wolfgang Rump - 2023 - Annals of Pure and Applied Logic 174 (3):103209.
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