  The classes of sequentially Cohen-Macaulay and sequentially homotopy
Cohen-Macaulay complexes and posets are studied. First, some different versions
of the definitions are discussed and the homotopy type is determined. Second,
it is shown how various constructions, such as join, product and rank-selection
preserve these properties. Third, a characterization of sequential
Cohen-Macaulayness for posets is given. Finally, in an appendix we outline
connections with ring-theory and survey some uses of sequential
Cohen-Macaulayness in commutative algebra.
