Boolean matrices ... neither Boolean nor matrices

Gabriele Ricci

Discussiones Mathematicae - General Algebra and Applications (2000)

  • Volume: 20, Issue: 1, page 141-151
  • ISSN: 1509-9415

Abstract

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Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.

How to cite

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Gabriele Ricci. "Boolean matrices ... neither Boolean nor matrices." Discussiones Mathematicae - General Algebra and Applications 20.1 (2000): 141-151. <http://eudml.org/doc/287705>.

@article{GabrieleRicci2000,
abstract = {Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.},
author = {Gabriele Ricci},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {universal matrix; functional application; generalized matrix; analytic monoid; Boolean matrices; incidence matrices of a graph},
language = {eng},
number = {1},
pages = {141-151},
title = {Boolean matrices ... neither Boolean nor matrices},
url = {http://eudml.org/doc/287705},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Gabriele Ricci
TI - Boolean matrices ... neither Boolean nor matrices
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2000
VL - 20
IS - 1
SP - 141
EP - 151
AB - Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.
LA - eng
KW - universal matrix; functional application; generalized matrix; analytic monoid; Boolean matrices; incidence matrices of a graph
UR - http://eudml.org/doc/287705
ER -

References

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  11. [11] G. Ricci, ERRATA to Universal eigenvalue equations, ibidem, 5 (1994), 241-243. Zbl0818.15009
  12. [12] G. Ricci, A Whitehead Generator, Quaderni del Dipartimento di Matematica 86, Universitá di Parma, Parma, 1993. 
  13. [13] G. Ricci, Two isotropy properties of 'universal eigenspaces' (and a problem for DT0L rewriting systems), Contributions to General Algebra 9 (1995), 281-290. Zbl0884.08001
  14. [14] G. Ricci, New characterizations of universal matrices show that neural networks cannot be made algebraic, Contributions to General Algebra 10 (1998), 268-291. Zbl0907.08003
  15. [15] G. Ricci, Analytic monoids, to appear in the proceedings: 'Atti Convegno Strutture Geometriche, Combinatoria e loro applicazioni (Caserta Febr. 25-27, 1999)'. 
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  17. [17] A.N. Whitehead, A Treatise on Universal Algebra with Applications, 1, Cambridge University Press, Cambridge 1898. Zbl29.0066.03

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