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Friedrich Wehrung - One of the best experts on this subject based on the ideXlab platform.

  • a non coordinatizable sectionally complemented Modular Lattice with a large jonsson four frame
    Advances in Applied Mathematics, 2011
    Co-Authors: Friedrich Wehrung
    Abstract:

    A sectionally complemented Modular Lattice L is coordinatizable if it is isomorphic to the Lattice L(R) of all principal right ideals of a von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame if it has a homogeneous sequence (a"0,a"1,a"2,a"3) such that the neutral ideal generated by a"0 is L. Jonsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable; whether the cofinal sequence assumption could be dispensed with was left open. We solve this problem by finding a non-coordinatizable sectionally complemented Modular Lattice L with a large 4-frame; it has cardinality @?"1. Furthermore, L is an ideal in a complemented Modular Lattice L^' with a spanning 5-frame (in particular, L^' is coordinatizable). Our proof uses Banaschewski functions. A Banaschewski function on a bounded Lattice L is an antitone self-map of L that picks a complement for each element of L. In an earlier paper, we proved that every countable complemented Modular Lattice has a Banaschewski function. We prove that there exists a unit-regular ring R of cardinality @?"1 and index of nilpotence 3 such that L(R) has no Banaschewski function.

  • coordinatization of Lattices by regular rings without unit and banaschewski functions
    Algebra Universalis, 2010
    Co-Authors: Friedrich Wehrung
    Abstract:

    A Banaschewski function on a bounded Lattice L is an antitone self-map of L that picks a complement for each element of L. We prove a set of results that include the following: Every countable complemented Modular Lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism. Every (not necessarily unital) countable von Neumann regular ring R has a map \({\varepsilon}\) from R to the idempotents of R such that \({x{R} = \varepsilon(x){R}}\) and \({\varepsilon(xy) = \varepsilon(x)\varepsilon(xy)\varepsilon(x)}\) for all \({x, y \in R}\). Every sectionally complemented Modular Lattice with a Banaschewski trace (a weakening of the notion of a Banaschewski function) embeds, as a neutral ideal and within the same quasivariety, into some complemented Modular Lattice. This applies, in particular, to any sectionally complemented Modular Lattice with a countable cofinal subset.

  • a non coordinatizable sectionally complemented Modular Lattice with a large j onsson four frame
    arXiv: Rings and Algebras, 2010
    Co-Authors: Friedrich Wehrung
    Abstract:

    A sectionally complemented Modular Lattice L is coordinatizable if it is isomorphic to the Lattice L(R) of all principal right ideals of some von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame if it has a homogeneous sequence (a_0,a_1,a_2,a_3) such that the neutral ideal generated by a_0 is L. Jonsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable; whether the cofinal sequence assumption could be dispensed with was left open. We solve this problem by finding a non-coordinatizable sectionally complemented Modular Lattice L with a large 4-frame; it has cardinality aleph one. Furthermore, L is an ideal in a (necessarily coordinatizable) complemented Modular Lattice with a spanning 5-frame. Our proof uses Banaschewski functions. A Banaschewski function on a bounded Lattice L is an antitone self-map of L that picks a complement for each element of L. In an earlier paper, we proved that every countable complemented Modular Lattice has a Banaschewski function. We prove that there exists a unit-regular ring R of cardinality aleph one and index of nilpotence 3 such that L(R) has no Banaschewski function.

  • coordinatization of Lattices by regular rings without unit and banaschewski functions
    arXiv: Rings and Algebras, 2009
    Co-Authors: Friedrich Wehrung
    Abstract:

    A Banaschewski function on a bounded Lattice L is an antitone self-map of L that picks a complement for each element of L. We prove a set of results that include the following: (1) Every countable complemented Modular Lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism. (2) Every (not necessarily unital) von Neumann regular ring R has a map e from R to the idempotents of R such that xR=e(x)R and e(xy)=e(x)e(xy)e(x) for all x,y in R. (3) Every sectionally complemented Modular Lattice with a ``Banaschewski trace'' (a weakening of the notion of a Banaschewski function) embeds, as a neutral ideal and within the same quasivariety, into some complemented Modular Lattice. This applies, in particular, to any sectionally complemented Modular Lattice with a countable cofinal subset. A sectionally complemented Modular Lattice L is coordinatizable, if it is isomorphic to the Lattice L(R) of all principal right ideals of a von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame, if it has a homogeneous sequence (a_0,a_1,a_2,a_3) such that the neutral ideal generated by a_0 is L. J\'onsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable. We prove that a sectionally complemented Modular Lattice with a large 4-frame is coordinatizable iff it has a Banaschewski trace.

M P Shushpanov - One of the best experts on this subject based on the ideXlab platform.

A G Gein - One of the best experts on this subject based on the ideXlab platform.

Patrick Solé - One of the best experts on this subject based on the ideXlab platform.

  • 2- and 3-Modular Lattice wiretap codes in small dimensions
    Applicable Algebra in Engineering Communication and Computing, 2015
    Co-Authors: Fuchun Lin, Frédérique Oggier, Patrick Solé
    Abstract:

    A recent line of work on Lattice codes for Gaussian wiretap channels introduced a new Lattice invariant called secrecy gain as a code design criterion which captures the confusion that Lattice coding produces at an eavesdropper. Following up the study of uniModular Lattice wiretap codes (Lin and Oggier in IEEE Trans Inf Theory 59(6):3295–3303, 2013 ), this paper investigates 2- and 3-Modular Lattices which can be constructed from linear codes and compares them with uniModular Lattices. Most even 2- and 3-Modular Lattices are found to have better performance (that is, a higher secrecy gain) than the best uniModular Lattices in dimension $$n,\ 2\le n\le 23$$ n , 2 ≤ n ≤ 23 . Odd 2-Modular Lattices are considered, too, and three Lattices are found to outperform the best uniModular Lattices.

  • 2 and 3 Modular Lattice wiretap codes in small dimensions
    arXiv: Number Theory, 2013
    Co-Authors: Fuchun Lin, Frédérique Oggier, Patrick Solé
    Abstract:

    A recent line of work on Lattice codes for Gaussian wiretap channels introduced a new Lattice invariant called secrecy gain as a code design criterion which captures the confusion that Lattice coding produces at an eavesdropper. Following up the study of uniModular Lattice wiretap codes [1], this paper investigates 2- and 3-Modular Lattices and compares them with uniModular Lattices. Most even 2- and 3-Modular Lattices are found to have better performance, that is, a higher secrecy gain than the best uniModular Lattices in dimension n, n is between 2 and 23. Odd 2-Modular Lattices are considered, too, and three Lattices are found to outperform the best uniModular Lattices.

Fuchun Lin - One of the best experts on this subject based on the ideXlab platform.

  • 2- and 3-Modular Lattice wiretap codes in small dimensions
    Applicable Algebra in Engineering Communication and Computing, 2015
    Co-Authors: Fuchun Lin, Frédérique Oggier, Patrick Solé
    Abstract:

    A recent line of work on Lattice codes for Gaussian wiretap channels introduced a new Lattice invariant called secrecy gain as a code design criterion which captures the confusion that Lattice coding produces at an eavesdropper. Following up the study of uniModular Lattice wiretap codes (Lin and Oggier in IEEE Trans Inf Theory 59(6):3295–3303, 2013 ), this paper investigates 2- and 3-Modular Lattices which can be constructed from linear codes and compares them with uniModular Lattices. Most even 2- and 3-Modular Lattices are found to have better performance (that is, a higher secrecy gain) than the best uniModular Lattices in dimension $$n,\ 2\le n\le 23$$ n , 2 ≤ n ≤ 23 . Odd 2-Modular Lattices are considered, too, and three Lattices are found to outperform the best uniModular Lattices.

  • 2 and 3 Modular Lattice wiretap codes in small dimensions
    arXiv: Number Theory, 2013
    Co-Authors: Fuchun Lin, Frédérique Oggier, Patrick Solé
    Abstract:

    A recent line of work on Lattice codes for Gaussian wiretap channels introduced a new Lattice invariant called secrecy gain as a code design criterion which captures the confusion that Lattice coding produces at an eavesdropper. Following up the study of uniModular Lattice wiretap codes [1], this paper investigates 2- and 3-Modular Lattices and compares them with uniModular Lattices. Most even 2- and 3-Modular Lattices are found to have better performance, that is, a higher secrecy gain than the best uniModular Lattices in dimension n, n is between 2 and 23. Odd 2-Modular Lattices are considered, too, and three Lattices are found to outperform the best uniModular Lattices.