Does there exist a semigroup such that every element factorizes in this way, which nonetheless lacks a left identity? If a semigroup $S$ has a left identity-element, then for any $y \in S$ we can write $y = xy$ for some $x \in S$. Just take $x$ to be any of the left identities, of which there is at least one, by hypothesis. Does there exist a semigroup $S$ such that every $y \in S$ factorizes in this way, which nonetheless lacks a left identity-element?

Every idempotent semigroup has this property, because $y=yy$, but not all have left identities.