Does an operation have to be completely defined and closed on a set for it to be an algebraic structure?

Question

If we have an arbitrary operation $+$, and for example a set $\{a,b,c,d,e\}$, is that set still an algebraic structure if $+$ is only defined on $a+b=e$, and $a+c=5$? (where $5$ is not in the set). that means e.g. that $b+d$ is undefined.