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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1162,
author = {Gabriele Ricci},
title = {Flocks in universal and Boolean algebras},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {30},
year = {2010},
pages = {45-69},
zbl = {1245.08002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1162}
}
Gabriele Ricci. Flocks in universal and Boolean algebras. Discussiones Mathematicae - General Algebra and Applications, Tome 30 (2010) pp. 45-69. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1162/
[000] [1] R. Baer, Linear Algebra and Projective Geometry (Academic Press, New York 1952).
[001] [2] H. B. Curry and R. Feys, Combinatory Logic, Vol. I, (North-Holland, Amsterdam 1958). | Zbl 0081.24104
[002] [3] K. Denecke and S.L. Wismath, Hyperidentities and Clones (Gordon and Breach Science Publishers, Amsterdam 2000). | Zbl 0960.08001
[003] [4] K. Denecke, Menger algebras and clones of terms. East-West J. Math. 5 (2) (2003), 179-193. | Zbl 1083.08004
[004] [5] K. Głazek, Algebras of Operations, in A.G. Pinus and K.N. Ponomaryov, Algebra and Model Theory 2 (Novosibirsk, 1999) 37-49. | Zbl 0953.08002
[005] [6] J.D. Monk, Introduction to Set Theory (McGraw-Hill, New York 1969).
[006] [7] G. Ricci, P-algebras and combinatory notation, Riv. Mat. Univ. Parma 5(4) (1979), 577-589.
[007] [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/)
[008] [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. | Zbl 0884.08001
[009] [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 | Zbl 1002.68031
[010] [11] G. Ricci, A semantic construction of two-ary integers, Discuss. Math. Gen. Algebra Appl. 25 (2005), 165-219. doi: 10.7151/dmgaa.1099 | Zbl 1098.08004
[011] [12] G. Ricci, Dilatations kill fields, Int. J. Math. Game Theory Algebra, 16 5/6 (2007), 13-34.
[012] [13] G. Ricci, All commutative based algebras have endowed dilatation monoids, (to appear on Houston J. of Math.).
[013] [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.
[014] [15] G. Ricci, Transformations between Menger systems, Demonstratio Math. 41 (4) (2008), 743-762. | Zbl 1165.08003
[015] [16] G. Ricci, Sameness between based universal algebras, Demonstratio Math. 42 (1) (2009), 1-20. | Zbl 1172.08005
[016] [17] M. Steinby, On algebras as tree automata, Colloquia Mathematica Societatis János Bolyai, 17. Contributions to Universal Algebra, Szeged (1975), 441-455.
[017] [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