The semigroups \(B_2\) and \(B_0\) are inherently nonfinitely based, as restriction semigroups. (Q2854964)

A semigroup \(S\) endowed with two unary operations \(x\mapsto x^*\) and \(x\mapsto x^+\) is called a restriction semigroup if it satisfies the identities \(x^+x=x\), \((x^+y)^+=x^+y^+\), \(x^+y^+=y^+x^+\), \(xy^+=(xy)^+x\), \((x^+)^*=x^+\), \(xx^*=x\), \((yx^*)^*=y^*x^*\), \(x^*y^*=y^*x^*\), \(y^*x=x(yx)^*\), \((x^*)^+=x^*\). Each inverse semigroup becomes a restriction semigroup under the operations \(x^+=xx^{-1}\), \(x^*=x^{-1}x\).NEWLINENEWLINE The author shows that, quite surprisingly, every finite restriction semigroup on which the two unary operations do not coincide is inherently nonfinitely based. This applies in particular to the 5-element Brandt semigroup \(B_2=\langle a,b\mid a^2=b^2=0,\;aba=a,\;bab=b\rangle\) (with restriction operations induced by inversion) and to its 4-element restriction subsemigroup \(B_0=B_2\setminus\{b\}\). The author also gives neat characterizations of the restriction semigroup varieties generated by \(B_2\) and \(B_0\).