The paper establishes a duality between a category of free subreducts of affine spaces and a corresponding category of generalized hypercubes with constants. This duality yields many others, in particular a duality between the category of (finitely generated) free barycentric algebras (simplices of real affine spaces) and a corresponding category of hypercubes with constants.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1063,
author = {Krzysztof J. Pszczo\l a and Anna B. Romanowska and Jonathan D.H. Smith},
title = {Duality for some free modes},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {23},
year = {2003},
pages = {45-61},
zbl = {1060.08009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1063}
}
Krzysztof J. Pszczoła; Anna B. Romanowska; Jonathan D.H. Smith. Duality for some free modes. Discussiones Mathematicae - General Algebra and Applications, Tome 23 (2003) pp. 45-61. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1063/
[000] [1] R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, MO, 1974.
[001] [2] D.M. Clark and B.A. Davey, Natural Dualities for the Working Algebraists, Cambridge University Press, Cambridge 1998. | Zbl 0910.08001
[002] [3] B. Csákány, Varieties of affine modules, Acta Sci Math. 37 (1975), 3-10.
[003] [4] B.A. Davey, Duality theory on ten dollars a day, 'Algebras and Orders', Kluwer Acad. Publ. 1993, 71-111. | Zbl 0793.08001
[004] [5] B.A. Davey and R.W. Quackenbush, Bookkeeping duality for paraprimal algebras, Contributions to General Algebra 9 (1995), 19-26. | Zbl 0884.08009
[005] [6] B.A. Davey and H. Werner, Dualities and equivalences for varieties of algebras, Colloq. Math. Soc. J. Bolyai 33 (1980), 101-275. | Zbl 0503.08011
[006] [7] G. Grätzer, Universal Algebra, Springer-Verlag, Berlin 1979.
[007] [8] K.H. Hofmann, M. Mislove and A. Stralka, The Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications, Springer-Verlag,Berlin 1974. | Zbl 0281.06004
[008] [9] J. Jezek and T. Kepka, Free commutative idempotent abelian groupoids and quasigroups, Acta Univ. Carolin. Math. Phys. 17 (1976), 13-19. | Zbl 0394.20058
[009] [10] J. Jezek and T. Kepka, Medial Groupoids, Academia, Praha 1983.
[010] [11] S. MacLane, Categories for the Working Mathematician, Springer-Verlag, Berlin 1971.
[011] [12] A. I. Mal'cev, Algebraic Systems, Springer-Verlag, New York 1973.
[012] [13] W.D. Neumann, On the quasivariety of convex subsets of affine spaces, Arch. Math. 21 (1970), 11-16. | Zbl 0194.01502
[013] [14] K. Pszczoła, Duality for affine spaces over finite fields, Contributions to General Algebra 13 (2001), 285-293. | Zbl 0993.08008
[014] [15] K.J. Pszczoła, A.B. Romanowska and J.D.H. Smith, Duality for quadrilaterals, Contribution to General Algebra, to appear. | Zbl 1049.08001
[015] [16] A.B. Romanowska, Barycentric algebras, 'General Algebra and Applications', Shaker Verlag, Aachen 2000, 167-181.
[016] [17] A.B. Romanowska and J.D.H. Smith, Modal Theory, Heldermann-Verlag, Berlin 1985.
[017] [18] A.B. Romanowska and J.D.H. Smith, Semilattice-based dualities, Studia Logica 56 (1996), 225-261. | Zbl 0854.08007
[018] [19] A.B. Romanowska and J.D.H. Smith, Duality for semilattice representations, J. Pure Appl. Algebra 115 (1997), 289-308. | Zbl 0872.18001
[019] [20] A.B. Romanowska and J.D.H. Smith, Embedding sums of cancellatice modes into functorial sums of affine spaces, 'Unsolved Problems on Mathematics for the 21st Century, a Tribute to Kiyoshi Iseki's 80th Birthday', IOS Press, Amsterdam 2001, 127-139.
[020] [21] A.B. Romanowska and J.D.H. Smith, Modes, World Scientific, Singapore 2002.
[021] [22] A.B. Romanowska and J.D.H. Smith, Poset extensions, convex sets, and semilattice presentations, preprint, 2002. | Zbl 1112.06002
[022] [23] J.D.H. Smith and A. B. Romanowska, Post-Modern Algebra, Wiley, New York, NY, 1999.