Let $G$ be a finite semigroup (or monoid if that helps) and $K$ a field.

>Question: When is the semigroup algebra $KG$ local?

Here local means that there is a unique maximal right (or left) ideal.

When $G$ is a group this is true if and only if $G$ is a $p$-group when $K$ has characteristic $p$ (when the characteristic is 0, then $KG$ is only local when $G$ is trivial).

I would hope that there is a similar easy criterion for general semigroups that can be used for example in GAP to filter out all semigroups (among all semigroups with lets say at most 8 elements) with a local semigroup algebra over a given field.

An approach could be the following, if one can find a splitting field $K$ for a given $G$ (is there a good way to find a splitting field like for groups?) then the number of simple modules of $A=KG$ is $\DeclareMathOperator\rad{rad}\dim A-(\dim  (\rad(A))+\dim([A,A]))$, which can be calculated by GAP (at least when the command RadicalOfAlgebra works in the intended way even for non-unital algebras).


>Question 2: Given a local algebra by quiver and relations, is there an easy way to see whether it is isomorphic to a semigroup algebra and to find a corresponding semigroup?