Sameness between based universal algebras

• Autorzy: Ricci G.
• Języki publikacji: EN

Abstrakty

EN

This is the continuation of the paper "Transformations between Menger systems". To define when two universal algebras with bases "are the same", here we propose a universal notion of transformation that comes from a triple characterization concerning three representation facets: the determinations of the Menger system, analytic monoid and endomorphism representation corresponding to a basis. Hence, this notion consists of three equivalent definitions. It characterizes another technical variant and also the universal version of the very semi-linear transformations that were coordinate-free. Universal transformations allow us to check the actual invariance of general algebraic constructions, contrary to the seeming invariance of representation-free thinking. They propose a new interpretation of free algebras as superpositions of "analytic spaces" and deny that our algebras differ from vector spaces at fundamental stages. Contrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.

Słowa kluczowe

EN

universal algebra; free algebra; universal matrix

PL

algebra uniwersalna; algebra wolna; macierz uniwersalna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2009
• Tom: Vol. 42, nr 1
• Strony: 3--22
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Ricci G.

• Universita de Parma Dipart. di Matematica I-43100 Parma, Italy,

Bibliografia

• [1] R. Baer, Linear Algebra and Projective Geometry, Academic Press Inc., New York, N. Y. 1952.
• [2] K. Głazek, Some old and new problems in the independence theory, Colloq. Math. 42 (1979),127-189.
• [3] K. Głazek, Morphisms of general algebras without fixed fundamental operations, Contemp. Math. 184 (1995), 117-137.
• [4] G. Grätzer, Universal Algebra, 2th ed., Springer-Verlag, New York-Heidelberg 1979.
• [5] B. Jónsson & A. Tarski, Two general theorems concerning free algebras, Bull. Amer. Math. Soc. 62 (1956), 554.
• [6] G. Ricci, Universal eigenvalue equations, Pure Math. Appl. Ser. B 3 (1992), 2-4, 231-288 (1993). (Most of the misprints appear in ERRATA to Universal eigenvalue equations, Pure Math. Appl. Ser. B 5 (1994), 2, 241-243. Anyway, the original version is in www.cs.unipr.it/?ricci/).
• [7] G. Ricci, Some analytic features of algebraic data, Discrete Appl. Math. 122 (2002), 1-3, 235-249.
• [8] G. Ricci, A semantic construction of two-ary integers, Discuss. Math. Gen. Algebra Appl. 25 (2005), 2, 165-219.
• [9] G. Ricci, Dilatations kill fields, Int. J. Math. Game Theory Algebra 16 (2007), 5-6, 13-34.
• [10] G. Ricci, Another characterization of vector spaces without fields, in G. Dorfer, G. Eigenthaler, H. Kautschitsch, W. More, W.B. Müller. (Hrsg.): Contributions to General Algebra 18. Klagenfurt: Verlag Heyn GmbH & Co KG, 31. February 2008, 139-150.
• [11] G. Ricci, Transformations between Menger systems, Demonstratio Math. 41 (2008), 743-762.
• [12] A. N. Whitehead, A Treatise on Universal Algebra with Applications, 1, Cambridge University Press, Cambridge, 1898.