  The Bruhat order on a Coxeter group is often described by examining
subexpressions of a reduced expression. We prove that an analogous description
applies to the Bruhat order on double cosets. This establishes the
compatibility of the Bruhat order on double cosets with concatenation, leading
to compatibility between the monoidal structure and the ideal of lower terms in
the singular Hecke 2-category. We also prove other fundamental properties of
this ideal of lower terms.
