Suppose $\mathcal{M} = (M, +, 0)$ is a cancelable commutative monoid. Let $G$ be the maximal subgroup of $M$, i.e.

$$G = \{ a \in M \colon (\exists b \in M)\, a + b = 0 \}.$$

For $a, b \in M$ say $a \preceq b$ if $(\exists c \in M)\, a + c = b$ and say $a \equiv b$ if $a \preceq b$ and $b \preceq a$. Note for all $a$, $\{b \in M \colon a \equiv b\} = \{a + g \colon g \in G\}$. In particular, if $a \equiv b$ and $c \equiv d$ then $a + b \equiv c + d$ and so there is a monoid $\mathcal{M}/\equiv$.

Is it always the case that $\mathcal{M}\cong (\mathcal{M}/ \equiv) \times G$ as monoids?

What if $G$ is finite?

What if for all $a \in M$, $\{b \in M\colon b\preceq a\}$ is finite?