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Friedrich Wehrung - One of the best experts on this subject based on the ideXlab platform.
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Coordinatization of lattices by regular rings without unit and banaschewski functions
Algebra Universalis, 2010Co-Authors: Friedrich WehrungAbstract: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.
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Coordinatization of lattices by regular rings without unit and Banaschewski functions
Algebra Universalis, 2010Co-Authors: Friedrich WehrungAbstract: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) countable 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. Jónsson 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.
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Coordinatization of lattices by regular rings without unit and banaschewski functions
arXiv: Rings and Algebras, 2009Co-Authors: Friedrich WehrungAbstract: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.
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von neumann Coordinatization is not first order
arXiv: General Mathematics, 2004Co-Authors: Friedrich WehrungAbstract:A lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. This forces L to be complemented modular. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jonsson, are first-order. Nevertheless, we prove that coordinatizability of lattices is not first-order, by finding a non-coordinatizable lattice K with a coordinatizable countable elementary extension L. This solves a 1960 problem of B. Jonsson. We also prove that there is no L\_{infinity, infinity} statement equivalent to coordinatizability. Furthermore, the class of coordinatizable lattices is not closed under countable directed unions; this solves another problem of B. Jonsson from 1962.
Christian Herrmann - One of the best experts on this subject based on the ideXlab platform.
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on the Coordinatization of primary arguesian lattices of low geometric dimension
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 2014Co-Authors: Christian HerrmannAbstract:Correcting claims made in Herrmann and Takach (Beitr Algebr Geom 46:215–239, 2005), we give lattice theoretic characterizations of lattices, \(L\), isomorphic to submodule lattices of finitely generated modules over commutative completely primary uniserial rings and of those isomorphic to subgroup lattices of finite abelian \(p\)-groups. Dealing with Coordinatization over arbitrary completely primary uniserial rings, we have to exclude the case that \(L\) has breadth \(\ge 3\) and all but \(2\) basis elements are atoms. Primary Arguesian lattices \(L\) of the latter type are shown to admit a cover preserving embedding into the subspace lattice of some vector space. The approach is that of Herrmann and Takach (Beitr Algebr Geom 46:215–239, 2005) but takes into account Monk’s construction of non-coordinatizable primary Arguesian lattices of the exceptional types.
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generators for complemented modular lattices and the von neumann jonsson Coordinatization theorems
Algebra Universalis, 2010Co-Authors: Christian HerrmannAbstract:Extending work of von Neumann, Jonsson has shown that each complemented modular lattice, L admitting a large partial n-frame with n ≥ 4, or with n ≥ 3 and L Arguesian, can be coordinatized as the lattice of all principal right ideals of some regular ring. His proof built on the embedding of L into the subgroup lattice of an abelian group which follows from Frink’s embedding of L into to a direct product of subspace lattices of irreducible projective spaces and Coordinatization of the latter. We offer a proof which, in addition to these results, employs only some elementary linear algebra. Luca Giudici’s thesis [6] is an important source for this approach.
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Generators for complemented modular lattices and the von Neumann–Jónsson Coordinatization theorems
Algebra universalis, 2010Co-Authors: Christian HerrmannAbstract:Extending work of von Neumann, Jónsson has shown that each complemented modular lattice, L admitting a large partial n -frame with n ≥ 4, or with n ≥ 3 and L Arguesian, can be coordinatized as the lattice of all principal right ideals of some regular ring. His proof built on the embedding of L into the subgroup lattice of an abelian group which follows from Frink’s embedding of L into to a direct product of subspace lattices of irreducible projective spaces and Coordinatization of the latter. We offer a proof which, in addition to these results, employs only some elementary linear algebra. Luca Giudici’s thesis [6] is an important source for this approach.
Mahmood Sohrabi - One of the best experts on this subject based on the ideXlab platform.
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elementary bilinearization and Coordinatization of finitely generated nilpotent groups
2013Co-Authors: Alexei Myasnikov, Mahmood SohrabiAbstract:This papers has two main parts. In the first part we develop an elementary Coordinatization for any nilpotent group taking exponents in a binomial PID A. In case that the additive group A of the A is finitely generated we prove using a classical result of Julia Robinson that one can obtain an elementary Z-Coordinatization of the group. In the second part we use a refinement of the Z-Coordinatization obtained above to give a new structural and algebraic criterion for elementary equivalence of finitely generated nilpotent groups.
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elementary Coordinatization of finitely generated nilpotent groups
arXiv: Group Theory, 2013Co-Authors: Alexei Myasnikov, Mahmood SohrabiAbstract:This paper has two main parts. In the first part we develop an elementary Coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$ is finitely generated we prove using a classical result of Julia Robinson that one can obtain a central series for $G$ where the action of the ring of integers $\Z$ on the quotients of each of the consecutive terms of the series except for one very specific gap, called the special gap, is interpretable in $G$. Then we use a refinement of this central series to give a criterion for elementary equivalence of finitely generated nilpotent groups in terms of the relationship between group extensions and the second cohomology group.
M. Greenberg - One of the best experts on this subject based on the ideXlab platform.
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Lattice tensor products. II
Acta Mathematica Hungarica, 2002Co-Authors: G. Grätzer, M. GreenbergAbstract:G. Grätzer and F. Wehrung has recently introduced the lattice tensor product, A ⊠ B , of the lattices A and B . In this note, for a finite lattice A and an arbitrary lattice B , we compute the ideal lattice of A ⊠ B , obtaining the isomorphism Id( A ⊠ B )≌ A ⊠Id B . This generalizes an earlier result of G. Grätzer and F. Wehrung proving this isomorphism for A = M _3 and B n -modular. We prove this isomorphism by utilizing the Coordinatization of A ⊠ B introduced in Part I of this paper.
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Lattice Tensor Products i. Coordinatization
Acta Mathematica Hungarica, 2002Co-Authors: G. Grätzer, M. GreenbergAbstract:G. Grätzer and F. Wehrung introduced the lattice tensor product, A ⊠ B , of the lattices A and B . One of the most important properties is that for a simple and bounded lattice A , the lattice A ⊠ B is a congruence-preserving extension of B . The lattice A ⊠ B is defined as the set of certain subsets of A ⊠ B ; there is no easy test when a subset belongs to A ⊠ B . A special case, M _3⊠ B , was earlier defined by G. Gräatzer and F. Wehrung as M _3, the it Boolean triple construct, defined as a subset of B ^3, with a simple criterion when a triple belongs. A~recent paper of G. Grätzer and E. T. Schmidt illustrates the importance of this Boolean triple arithmetic. In this paper we show that for any finite lattice A , we can ``coordinatize"" A ⊠ B , that is, represent A ⊠ B as a subset of B ^n (where n is the number of join-irreducible elements of A ), and provide an effective criteria to recognize the n -tuples of elements of B that occur in this representation. To show the utility of this Coordinatization, we reprove a special case of the above result: for a finite simple lattice A , the lattice A ⊠ B is a congruence-preserving extension of B .
Alexei Myasnikov - One of the best experts on this subject based on the ideXlab platform.
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elementary bilinearization and Coordinatization of finitely generated nilpotent groups
2013Co-Authors: Alexei Myasnikov, Mahmood SohrabiAbstract:This papers has two main parts. In the first part we develop an elementary Coordinatization for any nilpotent group taking exponents in a binomial PID A. In case that the additive group A of the A is finitely generated we prove using a classical result of Julia Robinson that one can obtain an elementary Z-Coordinatization of the group. In the second part we use a refinement of the Z-Coordinatization obtained above to give a new structural and algebraic criterion for elementary equivalence of finitely generated nilpotent groups.
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elementary Coordinatization of finitely generated nilpotent groups
arXiv: Group Theory, 2013Co-Authors: Alexei Myasnikov, Mahmood SohrabiAbstract:This paper has two main parts. In the first part we develop an elementary Coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$ is finitely generated we prove using a classical result of Julia Robinson that one can obtain a central series for $G$ where the action of the ring of integers $\Z$ on the quotients of each of the consecutive terms of the series except for one very specific gap, called the special gap, is interpretable in $G$. Then we use a refinement of this central series to give a criterion for elementary equivalence of finitely generated nilpotent groups in terms of the relationship between group extensions and the second cohomology group.