Flexible constraint satisfiability and a problem in semigroup theory

We examine some flexible notions of constraint satisfaction, observing some relationships between model theoretic notions of universal Horn class membership and robust satisfiability. We show the $\mathtt{\text{NP}}$-completeness of $2$-robust monotone 1-in-3 3SAT in order to give very small examples of finite algebras with $\mathtt{\text{NP}}$-hard variety membership problem. In particular we give a $3$-element algebra with this property, and solve a widely stated problem by showing that the $6$-element Brandt monoid has $\mathtt{\text{NP}}$-hard variety membership problem. These are the smallest possible sizes for a general algebra and a semigroup to exhibit $\mathtt{\text{NP}}$-hardness for the membership problem of finite algebras in finitely generated varieties.