The function thus defined is a semigroup morphism.
We turn the set of ...... into a category by defining the morphisms to be ......
By a partial automorphism of $A$ we understand an isomorphism $f:B\to C$ between two subalgebras $B$ and $C$ of $A$.
The induced homomorphism is multiplication by 2.
If we know a covering space $E$ of $X$ then not only do we know that ...... but we can also recover $X$ (up to homeomorphism) as $E/G$.
Associated with each Steiner system is its automorphism group, that is, the set of all ......
The last statement of Lemma 2 yields the algebra isomorphism $A=B$.
an orientation preserving homeomorphism
a complete set of representatives of the isomorphism classes of $A$-modules