Semigroup analogue to the classification of finite simple groups?

Question

Is there a semigroup analogue to the classification of finite simple groups?

If so what are some of the major results?

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Classification of Finite Simple Groups

Special Classes of Semigroups

One of my semigroup equation sequences in OEIS, g(f(x)) = f(f(f(x)))

Answer 1

In group theory, the classification of finite simple groups reduces the classification of all finite groups to the so-called extension problem. Roughly speaking, the extension problem consists in describing a group in terms of a particular normal subgroup and quotient group.

There is no semigroup analogue to this theory, but a weaker classification scheme exists. A semigroup $S$ divides a semigroup $T$ if $S$ is a homomorphic image of a subsemigroup of $T$. The Krohn–Rhodes theorem states that every finite semigroup $S$ divides a wreath product of finite simple groups, each dividing $S$, and copies of the 3-element monoid $\{1, a, b\}$ in which $aa = ba = a$ and $ab = bb = b$.

The best reference on this theory is the (advanced) book

[1] J. Rhodes, B. Steinberg. The $q$-theory of finite semigroups. Springer Verlag (2008). ISBN 978-0-387-09780-0.