It is widely believed in finite semigroup theory that asymptotically almost all finite semigroups $S$, up to isomorphism, are 3-nilpotent, i.e., they satisfy $\#\{abc\,:\,a,b,c\in S\} = 1$. My understanding is that this has not been proved. This is the subject of the MO question "Are most semigroups nilpotent of degree 3?"
The unique element $z\in SSS$ for a 3-nilpotent semigroup $S$ is a zero element because for any $a$, $az=azzz \in SSS = \{z\}$ and likewise $za=z$. So 3-nilpotent semigroups have a zero.
Here is a therefore weaker statement:
(1) Asymptotically almost all isomorphism classes of finite semigroups of order $n$ contain a zero element.
Or even weaker:
(2) Asymptotically almost all isomorphism classes of finite semigroups of order $n$ contain a left-zero or right-zero element.
Are there any known proofs of (1) or (2)?
In any of these statements, "isomorphism" could be replaced by "equivalence:=(anti-)isomorphism", and these are equivalent statements, as the counterexample semigroups could only get at most twice as dense.
Are there any nice classes of finite semigroups for which this is true in a nontrivial way? For example, do most finite band semigroups have a zero?
I figure that it's possible these weaker statements are easier to prove because they only depend on looking at the minimal ideal ("kernel") of $S$, the structure of which is well understood by the Rees-Suschkewitsch theorem. In particular, the minimal ideal is a $\mathcal{D}$-class, and there are left zeros iff the minimal $\mathcal{D}$-class has singleton $\mathcal{H}$-classes and only one $\mathcal{L}$-class. Likewise, there are right zeros iff the minimal $\mathcal{D}$-class has singleton $\mathcal{H}$-classes and only one $\mathcal{R}$-class. Combining these, $S$ has a zero iff the minimal $\mathcal{D}$-class is a singleton.
