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In recent years, the algebraic and combinatorial properties
of inclusion matrices of $t$-subsets versus $k$-subsets of
a $v$-set $X$ have been the focus of attention by some authors.
We first make a comparative study of the known bases for the $Z$-modules
generated by trades (the elements of the null space of these matrices),
and then, by utilizing these bases, we note that a large family
of new bases can by produced.
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