Generators for complemented modular lattices and the von Neumann–Jónsson coordinatization theorems

Herrmann, Christian

doi:10.1007/s00012-010-0064-5

Cited by 5 publications

(7 citation statements)

References 19 publications

(28 reference statements)

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“…This has been extended to 'large partial orthogonal d-frames' by Jónsson [Jóns60], including the case d = 3 under the stronger Arguesian Law (cf. [Her10b]) which is valid in all MOL of projections. In particular, this applies to all simple Arguesian MOLs of height ≥ 3.…”

Section: Towards the Infinite-dimensional Casementioning

confidence: 92%

Computational Complexity of Quantum Satisfiability

Herrmann

Ziegler

2011

2011 IEEE 26th Annual Symposium on Logic in Computer Science

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Quantum logic generalizes, and in dimension one coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic formulas; and show both 퓝 퓟-complete in dimension two as well. For higher-dimensional spaces ℝ and ℂ with ≥ 3 fixed, on the other hand, we show the problem to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory; and adds (a perhaps more natural and combinatorially flavoured) one to the still sparse list of 퓝 퓟ℝ-complete problems, mostly pertaining to real algebraic geometry. Our proofs rely on (a careful examination of) works by John von Neumann as well as contributions by Hagge et.al (2005,2007,2009). We finally investigate the problem over indefinite finite dimensions and relate it to noncommutative semialgebraic geometry.

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Section: Towards the Infinite-dimensional Casementioning

confidence: 92%

Computational Complexity of Quantum Satisfiability

Herrmann

Ziegler

2011

2011 IEEE 26th Annual Symposium on Logic in Computer Science

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“…We refer to Section 2 for the definition of a large n-frame. Jónsson's result extends von Neumann's classical Coordinatization Theorem; his proof has been recently substantially simplified by Christian Herrmann [9]. On another track, the author proved that there is no first-order axiomatization for the class of all coordinatizable lattices with unit [18].…”

Section: Introductionmentioning

confidence: 94%

“…As the element a is large in L, it follows easily from [10, Lemma 1.4] that a is large in each L ↓ e i 0 as well. Now it is observed in [11,Theorem 10.4] that every complemented modular lattice that admits a large 4-frame, or which is Arguesian and that admits a large 3-frame, is uniquely coordinatizable; the conclusion is strengthened to "uniquely rigidly coordinatizable" in [13,Corollary 4.12], see also [9,Theorem 18]. In particular, all the lattices L ↓ e i 0 , for i ∈ Λ, are uniquely rigidly coordinatizable.…”

Section: Banaschewski Traces and Coordinatizabilitymentioning

confidence: 99%

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Coordinatization of lattices by regular rings without unit and Banaschewski functions

Wehrung

2010

Algebra Univers.

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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 ε from R to the idempotents of R such that xR = ε(x)R and ε(xy) = ε(x)ε(xy)ε(x) for all x, y ∈ 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 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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“…For the most recent developments in coordinatization theory the reader can see Herrmann [10], Wehrung [17] and [18], and their bibliography.…”

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confidence: 99%

The ring of an outer von Neumann frame in modular lattices

Czédli

Skublics

2010

Algebra Univers.

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We prove the following theorem. Let (a 1 , . . . , am, c 12 , . . . , c 1m ) be a spanning von Neumann m-frame of a modular lattice L, and let (u 1 , . . . , un, v 12 , . . . , v 1n ) be a spanning von Neumann n-frame of the interval [0, a 1 ]. Assume that either m ≥ 4, or L is Arguesian and m ≥ 3. Let R * denote the coordinate ring of (a 1 , . . . , am, c 12 , . . . , c 1m ). If n ≥ 2, then there is a ring S * such that R * is isomorphic to the ring of all n × n matrices over S * . If n ≥ 4 or L is Arguesian and n ≥ 3, then we can choose S * as the coordinate ring of (u 1 , . . . , un, v 12 , . . . , v 1n ).

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Generators for complemented modular lattices and the von Neumann–Jónsson coordinatization theorems

Cited by 5 publications

References 19 publications

Computational Complexity of Quantum Satisfiability

Computational Complexity of Quantum Satisfiability

Coordinatization of lattices by regular rings without unit and Banaschewski functions

The ring of an outer von Neumann frame in modular lattices

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