Flocks in universal and Boolean algebras

• Autorzy: Gabriele Ricci
• Języki publikacji: EN

Abstrakty

EN

We propose the notion of flocks, which formerly were introduced only in based algebras, for any universal algebra. This generalization keeps the main properties we know from vector spaces, e.g. a closure system that extends the subalgebra one. It comes from the idempotent elementary functions, we call "interpolators", that in case of vector spaces merely are linear functions with normalized coefficients.
The main example, we consider outside vector spaces, concerns Boolean algebras, where flocks form "local" algebras with a sparseness similar to the one of vector spaces. We also outline the problem of generalizing the Segre transformations of based algebras, which used certain flocks, in order to approach a general transformation notion.

Słowa kluczowe

EN

combinators; elementary functions; closure system; interpolators; semi-affine lattice

Kategorie tematyczne

• 08B20: Free algebras
• 51D99: None of the above, but in this section

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2010
• Tom: 30
• Numer: 1
• Strony: 45-69

Daty

wydano

2010

otrzymano

2009-04-17

poprawiono

2009-12-20

Twórcy

autor

Gabriele Ricci

• Universitá di Parma, Dipart. di Matematica, I-43100 Parma, Italy

Bibliografia

• [1] R. Baer, Linear Algebra and Projective Geometry (Academic Press, New York 1952).
• [2] H. B. Curry and R. Feys, Combinatory Logic, Vol. I, (North-Holland, Amsterdam 1958).
• [3] K. Denecke and S.L. Wismath, Hyperidentities and Clones (Gordon and Breach Science Publishers, Amsterdam 2000).
• [4] K. Denecke, Menger algebras and clones of terms. East-West J. Math. 5 (2) (2003), 179-193.
• [5] K. Głazek, Algebras of Operations, in A.G. Pinus and K.N. Ponomaryov, Algebra and Model Theory 2 (Novosibirsk, 1999) 37-49.
• [6] J.D. Monk, Introduction to Set Theory (McGraw-Hill, New York 1969).
• [7] G. Ricci, P-algebras and combinatory notation, Riv. Mat. Univ. Parma 5(4) (1979), 577-589.
• [8] G. Ricci, Universal eigenvalue equations, Pure Math. and Appl. Ser. B, 3, 2-3-4 (1992), 231-288. (Most of the misprints appear in ERRATA to Universal eigenvalue equations, Pure Math. and Appl. Ser. B, 5, 2 (1994), 241-243. Anyway, the original version is in http://www.cs.unipr.it/~ricci/)
• [9] G. Ricci, Two isotropy properties of 'universal eigenspaces' (and a problem for DT0L rewriting systems), in G. Pilz, Contributions to General Algebra 9 (Verlag Hölder-Pichler-Tempsky, Wien 1995 - Verlag B.G. Teubner), 281-290.
• [10] G. Ricci, Some analytic features of algebraic data, Discrete Appl. Math. 122/1-3 (2002), 235-249. doi: 10.1016/S0166-218X(01)00323-7
• [11] G. Ricci, A semantic construction of two-ary integers, Discuss. Math. Gen. Algebra Appl. 25 (2005), 165-219. doi: 10.7151/dmgaa.1099
• [12] G. Ricci, Dilatations kill fields, Int. J. Math. Game Theory Algebra, 16 5/6 (2007), 13-34.
• [13] G. Ricci, All commutative based algebras have endowed dilatation monoids, (to appear on Houston J. of Math.).
• [14] 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.
• [15] G. Ricci, Transformations between Menger systems, Demonstratio Math. 41 (4) (2008), 743-762.
• [16] G. Ricci, Sameness between based universal algebras, Demonstratio Math. 42 (1) (2009), 1-20.
• [17] M. Steinby, On algebras as tree automata, Colloquia Mathematica Societatis János Bolyai, 17. Contributions to Universal Algebra, Szeged (1975), 441-455.
• [18] B. Vormbrock and R. Wille, Semiconcept and Protoconcept Algebras: The Basic Theorems, in B. Ganter, G. Stumme & R. Wille (eds.), Formal Concept Analysis: Foundations and Applications, (Springer-Verlag, Berlin 2005). doi: 10.1007/11528784₂

Identyfikatory

DOI

10.7151/dmgaa.1162