Let $\cal W$ be the collection of subspaces of a vector space $V$.
Question: Is $\cal W$ an abelian group under the sum of spaces?
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The law is well-defined, and commutative.
There is a neutral element, namely, the null-subspace, and $$(U_1+U_2)+U_3=U_1+(U_2+U_3)$$ for all $U_1,U_2,U_3$ subspaces of $V$. But a non-zero subspace $U$ doesn't have an inverse, because $U+U'\supset U$, as $0\in U'$. So we have a monoid, but not a group.