A sufficient condition for the non-finite basis property of semigroups.
The author proves that a semigroup \(S\) has no finite identity basis whenever there is an integer \(n\geq 2\) such that (i) \(S\) does not satisfy the identity \(x^ny^nx^n=x^n(y^nx^n)^{n+1}\); (ii) \(S\) satisfies the identities \(xy_1^ny_2^n\cdots y_k^nx=xy_k^ny_{k-1}^n\cdots y_1^nx\) for \(k=2,3,\ldots\) and the identities \(x^{n+2}=x^2\), \(x^{n+1}yx=xyx=xyx^{n+1}\), \((x^ny^n)^n=(y^nx^n)^n\). The last identity in (ii) can be omitted provided that for any distinct \(x,y\in S\), there exist idempotents \(e,f\in S\) such that \(ex\neq ey\) and \(xf\neq yf\). As an application, the author gives an alternative proof of the result by \textit{W. T. Zhang} and \textit{Y. F. Luo} [Bull. Aust. Math. Soc. 84, No. 3, 484-491 (2011; Zbl 1243.20072)] that the identities of the 6-element \(J\)-trivial semigroup \(L=\langle d,e\mid d^2=d,\;e^2=e,\;ded=0\rangle\) have no finite basis. Moreover, if \(T\) is any group of finite exponent or any commutative semigroup of index at most 2, then the product \(L\times T\) is not finitely based. The author also shows that the variety generated by \(L\) has a largest proper subvariety and this subvariety is not finitely based as well.
scientific article; zbMATH DE number 6111284
scientific article