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Convexity theories 0 fin. Foundations.

Heinrich Kleisli, Helmut Röhrl (1996)

Publicacions Matemàtiques

In this paper we study big convexity theories, that is convexity theories that are not necessarily bounded. As in the bounded case (see [4]) such a convexity theory Γ gives rise to the category ΓC of (left) Γ-convex modules. This is an equationally presentable category, and we prove that it is indeed an algebraic category over Set. We also introduce the category ΓAlg of Γ-convex algebras and show that the category Frm of frames is isomorphic to the category of associative, commutative, idempotent...

Distributive laws and Koszulness

Martin Markl (1996)

Annales de l'institut Fourier

Distributive law is a way to compose two algebraic structures, say 𝒰 and 𝒱 , into a more complex algebraic structure 𝒲 . The aim of this paper is to understand distributive laws in terms of operads. The central result says that if the operads corresponding respectively to 𝒰 and 𝒱 are Koszul, then the operad corresponding to 𝒲 is Koszul as well. An application to the cohomology of configuration spaces is given.

Duality for some free modes

Krzysztof J. Pszczoła, Anna B. Romanowska, Jonathan D.H. Smith (2003)

Discussiones Mathematicae - General Algebra and Applications

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.

Free algebras in varieties

Jan Pavlík (2010)

Archivum Mathematicum

We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to one of equational classes defined by equation arrows. Free algebras in the varieties are investigated and their existence is proved under the assumptions of accessibility.

Hu's Primal Algebra Theorem revisited

Hans-Eberhard Porst (2000)

Commentationes Mathematicae Universitatis Carolinae

It is shown how Lawvere's one-to-one translation between Birkhoff's description of varieties and the categorical one (see [6]) turns Hu's theorem on varieties generated by a primal algebra (see [4], [5]) into a simple reformulation of the classical representation theorem of finite Boolean algebras as powerset algebras.

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