Geodesic
Boolean matrices ... neither Boolean nor matrices
Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 1, pp. 141-151

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: universal matrix, functional application, generalized matrix, analytic monoid 
Ricci, Gabriele. Boolean matrices ... neither Boolean nor matrices. Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 1, pp. 141-151. http://geodesic.mathdoc.fr/item/DMGAA_2000_20_1_a11/
@article{DMGAA_2000_20_1_a11,
     author = {Ricci, Gabriele},
     title = {Boolean matrices ... neither {Boolean} nor matrices},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {141--151},
     year = {2000},
     volume = {20},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2000_20_1_a11/}
}
TY  - JOUR
AU  - Ricci, Gabriele
TI  - Boolean matrices ... neither Boolean nor matrices
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2000
SP  - 141
EP  - 151
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2000_20_1_a11/
LA  - en
ID  - DMGAA_2000_20_1_a11
ER  - 
%0 Journal Article
%A Ricci, Gabriele
%T Boolean matrices ... neither Boolean nor matrices
%J Discussiones Mathematicae. General Algebra and Applications
%D 2000
%P 141-151
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/DMGAA_2000_20_1_a11/
%G en
%F DMGAA_2000_20_1_a11

[1] J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, John Wiley Sons, New York 1990.

[2] S.L. Bloom and Z. Ésik, Matrix and iteration theories, I and II, J. Comput. System Sci. 46 (1993), 381-408 and 409-439.

[3] S.L. Bloom and Z. Ésik, Iteration Theories, The Equational Logic of Iterative Processes, Springer-Verlag, Berlin 1993.

[4] C.C. Elgot, Matricial Theories, J. Algebra 42 (1976), 391-421.

[5] K. Głazek, Some old and new problems in the independence theory, Colloq. Math. 42 (1979), 127-189.

[6] J.R. Hindley and J.P. Seldin, Introduction to Combinators and λ-Calculus, Cambridge University Press, London 1986.

[7] K.-H. Kim, Boolean Matrix Theory and Applications, M. Dekker, New York 1982.

[8] E.G. Manes, Algebraic Theories, Springer-Verlag, Berlin 1976.

[9] J.D. Monk, Introduction to Set Theory, McGraw-Hill, New York 1969.

[10] G. Ricci, Universal eigenvalue equations, Pure Math. Appl., Ser. B, 3 (1992), 231-288.

[11] G. Ricci, ERRATA to Universal eigenvalue equations, ibidem, 5 (1994), 241-243.

[12] G. Ricci, A Whitehead Generator, Quaderni del Dipartimento di Matematica 86, Universitá di Parma, Parma, 1993.

[13] G. Ricci, Two isotropy properties of 'universal eigenspaces' (and a problem for DT0L rewriting systems), Contributions to General Algebra 9 (1995), 281-290.

[14] G. Ricci, New characterizations of universal matrices show that neural networks cannot be made algebraic, Contributions to General Algebra 10 (1998), 268-291.

[15] G. Ricci, Analytic monoids, to appear in the proceedings: 'Atti Convegno Strutture Geometriche, Combinatoria e loro applicazioni (Caserta Febr. 25-27, 1999)'.

[16] J.H.M. Wedderburn, Boolean linear associative algebra, Ann. of Math. 35 (1934), 185-194.

[17] A.N. Whitehead, A Treatise on Universal Algebra with Applications, 1, Cambridge University Press, Cambridge 1898.