Geodesic
Duality for some free modes
Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 1, pp. 45-61

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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.
Keywords: duality, modes, affine spaces and their subreducts, barycentric algebras, convex sets, simplices, hypercubes 
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Pszczoła, Krzysztof; Romanowska, Anna; Smith, Jonathan. Duality for some free modes. Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 1, pp. 45-61. http://geodesic.mathdoc.fr/item/DMGAA_2003_23_1_a4/