Arbitrary systems, will it be biological, physical, cybernetical, etc. may be described by a mathematical function, namely by a graph extension function, which we also call hierarchical function (and which mathematically shows hierarchical nature of science). This function can be also used to describe mathematical objects themselves, which in the paper is shown on the example of the action of the graph extension function on the set of integers. Anew theory of graph extensions, similar to group extension theory, is outlined. A theorem about the equivalence of different extension functions is proved. There exists an isomorphism between the modified functional graph of the cell (functional block-scheme) and themorphological graph of the cell (the graph expressing topologicalmembrane intertransformations of the cell) which expresses the most essential features of for the biology of the cell and captures one of the specific differences between living and non-living systems. It is shown that the construction of the graph of a complex organismfromthe primordial graph given by Rashevsky is nothing but an extension of the primordial graph by the graph extension function. It is described that there exist morphisms from biological graphs described by various authors to our functional graph.
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