Chain

In ordered sets

The same as a totally ordered set: in a general partially ordered set, a subset which is totally ordered with respect to the induced order. The rank of a partially ordered set is the maximal cardinality of a chain.

In algebraic topology

A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$.

References

[1]	N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966) Zbl 0047.41402
[2]	P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) Zbl 0091.36306