An inverse semigroup is an algebra with two operations: binary $\cdot$ and unary $^{-1}$ such that $\cdot$ is associative and $xx^{-1}x=x, xx^{-1}yy^{-1}=yy^{-1}xx^{-1}$. The Brandt semigroup with 1, $B\_2^1$, is the inverse semigroup of $2\times 2$-matrices consisting of 0, I, and the four matrix units $e\_{i,j}$, $i,j=1,2$ where $e\_{i,j}$ is the matrix with $(i,j)$-entry 1 and other entries 0, $e\_{i,j}^{-1}=e\_{j,i}$. It is known (Kleiman) that the identities of $B\_2^1$ are not finitely based. 


**Question.** Is it known that the identities of any finite inverse semigroup containing $B\_2^1$ as an inverse subsemigroup are not finitely based?



Identities of finite inverse semigroups
I believe this is a well known open question. It has this property as a semigroup but it is not clear as an inverse semigroup. Mark Sapir showed it is contained in a finitely based locally finite variety of inverse semigroups. Your question is problem 3.10.13 in his book Combinatorial algebra: syntax and semantics.


