On the varieties generated by ai-semirings of order two. (Q896240)

There are six non-isomorphic semirings of order two with idempotent addition. The variety \(\mathbf S_{\mathbf 2}\) generated by these semirings is finitely based by the identities \(xyzt\sim xzyt\), \((xy)^2\sim xy\) and \(x+yz\sim x+yz+xz+yx\). The authors show that the lattice \(L(\mathbf S_{\mathbf 2})\) of all subvarieties of \(\mathbf S_{\mathbf 2}\) is a Boolean algebra of order \(2^6\) and present a finite identity basis for each subvariety. This means that \(\mathbf S_{\mathbf 2}\) is hereditarily finitely based.