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.1080/00927879408825152 . [2] Panaite , F. , Oystaeyen Van , F. 2007 A structure theorem for quasi-Hopf comodule algebras Proc. Amer. Math. Soc. 135 1669 – 1677 10.1090/S0002-9939-07-08712-6 . [3

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Abstract  

The paper is concerned with endomorphism algebras for weak Doi-Hopf modules. Under the condition “weak Hopf-Galois extensions”, we present the structure theorem of endomorphism algebras for weak Doi-Hopf modules, which extends Theorem 3.2 given by Schneider in [1]. As applications of the structure theorem, we obtain the Kreimer-Takeuchi theorem (see Theorem 1.7 in [2]) and the Nikshych duality theorem (see Theorem 3.3 in [3]) in the case of weak Hopf algebras, respectively.

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We deal with structure theory of Ambrose algebras, getting structure theorems, analogous to the classical ones of Wedderburn.

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The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely, `A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for any adjunct representation of a dismantlable lattice the number of chains as well as the number of times a pair of elements occurs remains the same. If a dismantlable lattice L has n elements and n+k edges then it is proved that the number of irreducible elements of L lies between n-2k-2 and n-2. These results are used to enumerate the class of lattices with exactly two reducible elements, the class of lattices with n elements and upto n+1 edges, and their subclasses of distributive lattices and modular lattices.

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Generalized random processes are classified by various types of continuity. Representation theorems of a generalized random process on

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{M p} on a set with arbitrary large probability, as well as representations of a correlation operator of a generalized random process on
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{M p} and L r(R), r > 1, are given. Especially, Gaussian generalized random processes are proven to be representable as a sum of derivatives of classical Gaussian processes with appropriate growth rate at infinity. Examples show the essence of all the proposed assumptions. In order to emphasize the differences in the concept of generalized random processes defined by various conditions of continuity, the stochastic differential equation y′(ω; t) = f(ω; t) is considered, where y is a generalized random process having a point value at t = 0 in the sense of Lojasiewicz.

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References [1] Kaoutit , L. E. and Torrecillas , J. G. , Comatrix corings: Galois corings , descent theory, and structure

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. 54 31 43 De Guzman, I. P., Structure theorems for alternative H* -algebras, Math. Proc. Camb. Phil. Soc. 94 (1983), 437-446. MR 0720794

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References [1] Lev , V. 1997 Addendum to “Structure theorem for multiple addition” J. Number Theory 65

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