Last updates: 7 December 2009
Let
$S$
be a set with an associative operation with identity. Show that the identity is unique.
(This tells us that any commutative monoid has only one heart.) | |
Let
$S$
be a set with an associative operation with identity. Let
$s\in S$
and assume that
$s$
has an inverse in
$S$.
Show that the inverse of
$s$
is unique. (This tell us that any element of an abelian group has only one mate.) | |
Let
$S$
be a set with identity. Let
$s\in S$
and assume that
$s$
has an inverse in
$S$.
Show that the inverse of the inverse of
$s$
is equal to
$s$. (This tells us that
$-(-s)=s$.) | |
Let
$S$
be an abelian group. Show that if
$a+c=b+c$ then
$a=b$. | |
Let
$S$
be a ring. Show that if
$s\in S$ then
$s\cdot 0=0$. |
[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)