# Identities of finite inverse semigroups

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?

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.

• Thank you! The book is 6 years old. Is it true that the strongest result in this area is by Kadourek about finite inverse semigroups with solvable subgroups? – user158834 Jun 9 at 15:37
• I'm not aware of anything more recent but I don't really follow this very closely. You might email Sapir. He probably follows more closely. This problem has been open for a long long time so I doubt much has changed. – Benjamin Steinberg Jun 9 at 17:43