Abstract
In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L N ∋ (l, λ) →l · λ ∈L such that for alll ∈L and λ, μ ∈L N the relation (l · λ)μ =l(λ · μ) holds, where λ · μ ∈L N is given by (λ · μ)(i) = λ(i)μ,i ∈N. We assume thatL has a unit, that is an element δ ∈L N withl · δ =l, for alll ∈L, and δ · λ = λ, for all λ ∈L N. Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX ⊗Y are introduced and their properties are investigated.
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Bos, W. and Wolff, G.: Affine Räume I,Mitt. Math. Sem. Gießen 129, 1978, 1–115.
Bos, W. and Wolff, G.: Affine Räume II,Mitt. Math. Sem. Gießen 130, 1978, 1–83.
Gudder, S.P. and Schroeck, F.: Generalized convexity.SIAM J. Math. Anal. 11, 1980, 984–1001.
Kneser, H.: Konvexe Raüme,Arch. d. Matth. 3, 1952, 198–206.
Ostermann, F. and Schmidt, J.: Der baryzentrische Kalkül als axiomatische Grundlage der affinen Geometrie,J. Reine u. Angew. Math. 224, 1966, 44–57.
Pareigis, B.:Categories and Functors. Academic Press, New York-London, 1970.
Pareigis, B.: Non-additive ring and module theory I. General theory of monoids,Publ. Math. (Debrecen) 24, 1977, 189–204.
Pumplün, D.: Regularly ordered Banach spaces and positively convex spaces,Results in Math. 7, 1984, 85–112.
Pumplün, D. and Röhrl, H.: Banach spaces and totally convex spaces I,Comm. in Alg. 12, 1984, 953–1019.
Pumplün, D. and Röhrl, H.: Banach spaces and totally convex spaces II,Comm. in Alg. 13, 1985, 1047–1113.
Rodé, G.: Superkonvexe Analysis,Arch. Math. 34, 1980, 452–462.
Röhrl, H.: Convexity theories I. γ-convex spaces, In:Constantin Carathéodory. An International Tribute. World Scientific Publ., pp. 1175–1209, 1991.
Semadeni, Z.: Monads and their Eilenberg-Moore algebras in functional analysis,Queen's Papers Pure Appl. Math. 33, 1973.
Stone, M.H.: Postulates for the barycentric calculus,Ann. Mat. Pura Appl. 29, 1949, 25–30.
Wickenhäuser, A.: Positively convex spaces, Diplomarbeit FU Hagen, 1987.
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Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet
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Pareigis, B., Röhrl, H. Left linear theories — A generalization of module theory. Appl Categor Struct 2, 145–171 (1994). https://doi.org/10.1007/BF00873297
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DOI: https://doi.org/10.1007/BF00873297