Description of Homomorphisms Why do homomorphisms get called "structure-preserving maps/functions" or get said to "preserve the multiplication" when they only induce structural preservation for a subalgebra of the algebra that they map to?  
If the question doesn't come as clear here, consider the algebras described by the following operation tables
G  a  b
a  a  b
b  b  b

F  1  2  3
1  1  2  3
2  2  2  2
3  3  2  3.

Of course here, redundantly, we can say that $G: \{a, b \} \times \{a, b \} \to \{a, b\}$, $F: \{1, 2, 3\} \times \{1, 2, 3\} \to \{1, 2, 3\}$ just from those tables.  Define a unary function $H:a \to 1, b \to 3$.  One can check that for all $x, y$ in $\{a, b\}$, $HGab=FHaHb$ [or equivalently $abGH=aHbHG$ or equivalently $H(G(a,b))=F(H(a),H(b))$].  Thus, we have a homomorphism here (more specifically a monomorphism, since $H$ only qualifies as an injection).  But, it comes as clear that $(\{a, b\}, H)$ satisfies for all $x$, $Hxx=x$, while $(\{1, 2, 3\}, F)$ does not satisfy for all $x$ $Fxx=x$, though the subalgebra $(\{1, 3\}, F)$ of $(\{1, 2, 3\}, F)$ does satisfy for all $x$ $Fxx=x$.  
Why then do homomorphisms get called structural-preserving maps, or get said to preserve the multiplication?  Does this consist of an error on people's part, or do they have something else in mind?  Why not say that homomorphisms between algebras consist of sub-transferred maps, or sub-induced maps, or preserve some sub-multiplication? 

"Structure" refers to the domain, not to the range. Homomorphisms preserve structure in the domain when they map things into the range. That is, one should think of a homomorphism $f : A \to B$ as directed from $A$ to $B$, and in such a way that the structure in $A$ gets preserved "along the way" to $B$. 
