Suppose that E is a vector lattice where the ordering and the lattice operations in E are defined pointwise by a countable family \({\mathcal {F}}=\{f_i|i\in {{\mathbf {N}}}\}\) of positive linear functional of E and Z is a sublattice of E. Based on algebraic and order properties of E we give necessary and sufficient conditions in order Z to be atomic. Especially we show the existence of a basic sequence \(\{b_n\}\) of extremal points (atoms) of \(Z_+\) so that for any \(x\in Z_+\) a unique sequence \(({\widehat{x}}(n))\) of real components of x with respect to \(\{b_n\}\) exists so that \(x=\sup \{{\widehat{x}}(n)b_n\;|\;n\in {{\mathbf {N}}}\}\) and also \(x=sup_{n}\sum _{i=1}^n{\widehat{x}}(i)b_i\). These results give an answer to the problem of the existence of basic derivatives in financial markets.

中文翻译：

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金融中的原子子格和基本导数

假设Ë是其中的排序和在晶格动作的矢量格é由一个可数家族定义逐点\（{\ mathcal {F}} = \ {f_i | I \在{{\ mathbf {N}}} \ } \）正线性官能的ë和ž是一个亚晶格ë。根据E的代数和阶性质，我们给出Z成为原子的充要条件。特别是，我们显示了\（Z _ + \）的极点（原子）的基本序列\（\ {b_n \} \）的存在，因此对于任何\（x_in Z _ + \）来说，唯一序列\（（ {\ widehat {x}}（n））\）相对于\（\ {b_n \} \）的x的实数分量存在，因此{{\}中的\ {x = \ sup \ {{\ widehat {x}}（n）b_n \; | \; n \ mathbf {N}}} \} \）和\（x = sup_ {n} \ sum _ {i = 1} ^ n {\ widehat {x}}（i）b_i \）。这些结果回答了金融市场中基本衍生品存在的问题。