The Wikipedia page Metabolic pathway defines a pathway as "a metabolic pathway is a linked series of chemical reactions occurring within a cell". This means that any sequence of reactions can be labeled a metabolic pathway. However, as metabolism was being uncovered, groups of reactions were assigned specific labels, such as glycolysis, Krebs Cycle, or Serine biosynthesis. Often the categorization was based on common chemistry or identification of an input and output. For example, serine biosynthesis starts at 3-phosphoglycerate and ends at serine. This is a somewhat ad hoc means for defining pathways, particularly when pathways are dynamic structures, changing as environmental result in changes in gene expression. For example, the Kreb Cycle is often not cyclic as depicted in textbooks. In E. coli and other bacteria, it is only cyclic during aerobic growth on acetate or fatty acids. Instead, under anaerobiosis, its enzymes function as two distinct biosynthetic pathways producing succinyl-CoA and α-ketoglutarate.
It has therefore been proposed to define a pathway as either a single elementary mode or some combination of elementary modes. The added advantage is that the set of elementary modes is unique and non-decomposable to simpler pathways. A single elementary mode can therefore be thought of as an elementary pathway. Note that the set of elementary modes will change as the set of expressed enzymes change during transitions from one cell state to another.
Elementary modes, therefore, provide an unambiguous definition of a pathway.