Correction

The following addition must be made to my article (I. S. Ponizovskii, “On homomorphisms of commutative semigroups”), published in DAN, vol. 135, no. 5, 1960.

In order that, under the hypotheses of Theorem 1, the main Theorem 2 be true, Theorem 1 must be formulated as follows:

Let \(\mathfrak A\) be a commutative semigroup with a finite number of idempotents, \(\mathfrak A^2=\mathfrak A\), and let \(\mathfrak A\) possess a principal decreasing ideal series of length \(\mu\leqslant\omega\). Then \(\mathfrak A\) possesses a finite ideal series \(A=[\mathfrak A_\alpha]\) such that: a) all members of \(A\) are ideals of \(\mathfrak A\); b) each factor \(\mathfrak A_\alpha\) of the series \(A\) contains an ideal \(\mathfrak B_\alpha\) that is a quasi-null semigroup, and moreover \(\mathfrak G_\alpha=\mathfrak A_\alpha\setminus\mathfrak B_\alpha\) is a group; c) the identity of \(\mathfrak G_\alpha\) is an identity of \(\mathfrak A_\alpha\).

Without the requirement that the number of idempotents be finite, Theorem 2 is, generally speaking, not true, as may be seen from the following example. Let \(\mathfrak A\) consist of a zero \(0\) and a countable set of idempotents \(e_k\) \((k=1,2,\ldots)\), multiplied according to the rule: if \(k\leqslant m\), then \(e_k e_m=e_m e_k=e_m\). Then \(\mathfrak A\) is a commutative semigroup, \(\mathfrak A^2=\mathfrak A\), and \(\mathfrak A\) possesses a principal decreasing ideal series \(A'=[\mathfrak A'_n]\), where \(\mathfrak A'_n\) consists of the zero and the idempotents \(e_k\) for \(k\geqslant n\). As the series \(A\) one may simply take the series \(A'\). Let \(\mathfrak B\) be the semigroup consisting of a zero \(\theta\) and an identity \(\varepsilon\). Put

\[ \varphi e_k=\varepsilon\ (k=1,2,\ldots), \qquad \varphi 0=\theta. \]

It is easy to see that \(\varphi\) is a homomorphism of \(\mathfrak A\). However, in this case the homomorphisms \(\varphi_n\) (defined by formula (2) of the article) will be zero. Therefore their l.c.m. \(\varphi_n(1\leqslant n<\omega)\) will also be zero, whence \(\varphi\ne\) l.c.m. \(\varphi_n(1\leqslant n<\omega)\), and Theorem 2 is not true in the present case.

I. S. Ponizovskii

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