MATHEMATICS

I. S. Ponizovskii

ON HOMOMORPHISMS OF COMMUTATIVE SEMIGROUPS

(Presented by Academician A. I. Mal’cev on 1 VII 1960)

The problem of determining all homomorphisms of an arbitrary semigroup is still far from being solved. At present the solution of this problem is known only for certain, though interesting, classes of semigroups (L. M. Gluskin \((^1)\) described the homomorphisms of completely simple semigroups, Preston \((^2)\) the homomorphisms of inverse semigroups, and A. I. Mal’cev \((^{3,4})\) the homomorphisms of symmetric and matrix semigroups). In the present note it is shown how the description of all homomorphisms of a commutative semigroup of fairly general type can be reduced to the description of homomorphisms of commutative semigroups which are called quasinilpotent in this note. Together with Tamura’s results \((^5)\), this leads, for example, to a description of all homomorphisms of an arbitrary finite commutative semigroup.

1. Let \(\mathfrak A\) be a commutative semigroup with zero \(0\).

We shall call a system \(A=[\mathfrak A_\alpha]\) of subsemigroups, totally ordered by decrease (where \(\alpha\) ranges over the set of ordinal numbers from \(1\) to \(\mu\)), a decreasing ideal series of \(\mathfrak A\) if the following conditions are satisfied: a) the intersection of any collection of semigroups from \(A\) belongs to \(A\); b) \(\mathfrak A_1=\mathfrak A,\ \mathfrak A_\mu=0\); c) \(\mathfrak A_{\alpha+1}\) is an ideal of \(\mathfrak A_\alpha\) \((1\leq \alpha<\mu)\).

The factor semigroups \(\overline{\mathfrak A}_\alpha=\mathfrak A_\alpha/\mathfrak A_{\alpha+1}\) will be called the factors of \(A\). The number \(\mu\) will be called the length of the given series. A principal decreasing ideal series of \(\mathfrak A\) will mean a decreasing ideal series \(A=[\mathfrak A_\alpha]\) \((1\leq \alpha\leq \mu)\) of the semigroup \(\mathfrak A\), in which each member \(\mathfrak A_{\alpha+1}\) is a maximal ideal of \(\mathfrak A\) contained in \(\mathfrak A_\alpha\) and distinct from \(\mathfrak A_\alpha\) (i.e., if \(I\) is an ideal of \(\mathfrak A\) and \(\mathfrak A_\alpha \supset I \supset \mathfrak A_{\alpha+1}\), then either \(I=\mathfrak A_\alpha\), or \(I=\mathfrak A_{\alpha+1}\)).

Let \(\nu\) be an ordinal number. Define \(\mathfrak A^\nu\) as follows. Put \(\mathfrak A^1=\mathfrak A\). Suppose that \(\mathfrak A^\lambda\) have already been defined for all \(\lambda<\nu\). Then put

\[ \mathfrak A^\nu=\bigcap_{\lambda<\nu}\mathfrak A^\lambda,\quad \text{if } \nu \text{ is a limit ordinal,} \]

\[ \mathfrak A^\nu=\mathfrak A^{\nu-1}\cdot \mathfrak A,\quad \text{if } \nu \text{ is not a limit ordinal.} \]

We shall call a semigroup \(\mathfrak A\) quasinilpotent if, for some ordinal number \(\tau\), \(\mathfrak A^\tau=0\). If \(\mathfrak A^n=0\), where \(n\) is finite, then we shall call \(\mathfrak A\) nilpotent. The semigroup \(\mathfrak A\) is quasinilpotent if and only if \(\mathfrak A\) has a decreasing ideal series all of whose factors are nilpotent.

1. The following proposition is the basis of the proposed description of homomorphisms of commutative semigroups.

Theorem 1. Let \(\mathfrak A\) be a commutative semigroup, \(\mathfrak A^2=\mathfrak A\), and suppose that \(\mathfrak A\) has a principal decreasing ideal series of length \(\mu\leq \omega^*\). Then \(\mathfrak A\) has a decreasing ideal series \(A=[\mathfrak A_\alpha]\) such that:

a) all members of \(A\) are ideals of \(\mathfrak A\); b) each factor \(\overline{\mathfrak A}_\alpha\) of the series \(A\) contains

* By \(\omega\) is denoted the first infinite ordinal number.

ideal \(\overline{\mathfrak B}_\alpha\), which is a quasi-nilpotent semigroup, and moreover \(\overline{\mathfrak G}_\alpha=\overline{\mathfrak A}_\alpha\setminus \overline{\mathfrak B}_\alpha\) is a group; c) the identity of \(\overline{\mathfrak G}_\alpha\) is the identity of \(\overline{\mathfrak A}_\alpha\).

1. Let the semigroup \(\mathfrak A\) satisfy the conditions of Theorem 1. By Theorem 1, \(\mathfrak A\) has a series

\[ \mathfrak A=\mathfrak A_1\supset \mathfrak A_2\supset \cdots \supset \mathfrak A_\sigma=0 \tag{1} \]

such that assertions a), b), c) of Theorem 1 hold for it. In particular, each factor \(\overline{\mathfrak A}_\alpha=\mathfrak A_\alpha/\mathfrak A_{\alpha+1}\) \((1\leq \alpha<\sigma)\) of the series (1) has an identity. This means that in \(\mathfrak A_\alpha\setminus \mathfrak A_{\alpha+1}\) there exists an element \(e_\alpha\) that is an identity for all elements of \(\mathfrak A_\alpha\setminus \mathfrak A_{\alpha+1}\). Below, \(\omega_\alpha\) always denotes the ideal homomorphism of \(\mathfrak A\) generated by the \(\mathfrak A_\alpha\)-term of the series (1).

Let \(\overline{\varphi}_\alpha\) be a homomorphism of the factor \(\overline{\mathfrak A}_\alpha=\mathfrak A_\alpha/\mathfrak A_{\alpha+1}\) of the series (1). If \(x\in \mathfrak A\), then \(xe_\alpha\in \mathfrak A_\alpha\) (\(e_\alpha\) is the same as above) and \(\omega_{\alpha+1}(xe_\alpha)\in \overline{\mathfrak A}_\alpha\). Therefore \(\overline{\varphi}_\alpha\omega_{\alpha+1}(xe_\alpha)\) is meaningful. Put

\[ \varphi_\alpha x=\overline{\varphi}_\alpha\omega_{\alpha+1}(xe_\alpha) \tag{2} \]

for any \(x\in \mathfrak A\). It is easy to verify that \(\varphi_\alpha\) is a homomorphism of \(\mathfrak A\).

The following proposition holds (the main one in the note).

Theorem 2. Every homomorphism \(\varphi\) of a semigroup \(\mathfrak A\) satisfying the conditions of Theorem 1 has the form \(*\)

\[ \varphi=\text{g.c.d. }\varphi_\alpha \qquad (1\leq \alpha<\nu), \]

where each homomorphism \(\varphi_\alpha\) is obtained from some homomorphism \(\overline{\varphi}_\alpha\) of the factor \(\overline{\mathfrak A}_\alpha\) of the series (1) by formula (2).

Theorem 2 reduces the question of describing the homomorphisms of the semigroup \(\mathfrak A\) to the question of describing the homomorphisms of the factors of the series (1). The structure of these factors is described by Theorem 1.

Theorem 3 shows how, knowing the homomorphisms of quasi-nilpotent semigroups, one can describe the homomorphisms of the factors of the series (1), and hence completely describe the homomorphisms of \(\mathfrak A\).

Theorem 3. Let \(\mathfrak C\) be a commutative semigroup, \(\mathfrak B\) an ideal of \(\mathfrak C\) that is a quasi-nilpotent semigroup, and \(\mathfrak G=\mathfrak C\setminus \mathfrak B\) a group whose identity is an identity of \(\mathfrak C\).

Every homomorphism \(\varphi\) of the semigroup \(\mathfrak C\) induces homomorphisms \(\pi,\rho\), respectively, of the ideal \(\mathfrak B\) and the group \(\mathfrak G\):

\[ \begin{aligned} \varphi x&=\pi x \quad &&\text{for } x\in \mathfrak B,\\ \varphi x&=\rho x \quad &&\text{for } x\in \mathfrak G. \end{aligned} \tag{3} \]

Here only two cases are possible:

I. \(\varphi\mathfrak B\cap \varphi\mathfrak G\) is empty; \(\pi,\rho\) are connected by the conditions: a) from \(\pi x=\pi y\) \((x,y\in \mathfrak B)\) it follows that \(\pi(xz)=\pi(yz)\) for any \(z\in \mathfrak G\); b) from \(\rho z=\rho u\) \((z,u\in \mathfrak G)\) it follows that \(\pi(xz)=\pi(xu)\) for any \(x\in \mathfrak B\).

II. In \(\mathfrak B\) there is a normal subsemigroup \(\mathfrak N\) such that \(\pi\mathfrak N=\rho\mathfrak G\), and from \(\rho z=\pi N\) \((z\in \mathfrak G,\ N\in \mathfrak N)\) it follows that \(\pi(xz)=\pi(xN)\) for any \(x\in \mathfrak B\); in this case \(\varphi\mathfrak C=\varphi\mathfrak B\).

Conversely, if \(\pi,\rho\) are respectively homomorphisms of \(\mathfrak B,\mathfrak G\) and either I a), b), or II is satisfied, then formulas (3) determine a homomorphism \(\varphi\) of the semigroup \(\mathfrak C\), if one sets

\[ \varphi x\cdot \varphi z=\varphi(xz) \qquad (x\in \mathfrak B,\ z\in \mathfrak G). \]

1. If \(\mathfrak A^2\ne \mathfrak A\), then, adjoining to \(\mathfrak A\) an identity externally, we obtain a semigroup \(\mathfrak A'\), and \({\mathfrak A'}^2\ne \mathfrak A'\). The description of the homomorphisms of \(\mathfrak A\) is equivalent

\(*\) By \(\text{g.c.d. }\varphi_\alpha\) \((1\leq \alpha<\nu)\) is denoted the common greatest divisor of the homomorphisms \(\varphi_\alpha\).

in this case, to the description of the homomorphisms of \(\mathfrak A'\). Therefore it is enough to be able to describe the homomorphisms of all such semigroups \(\mathfrak A\) for which \(\mathfrak A^2=\mathfrak A\); this is done in Theorem 2.

1. A finite quasi-nilpotent semigroup is nilpotent. The structure of finite nilpotent (not necessarily commutative) semigroups is described in (5); from this description one very easily obtains a description of all homomorphisms of finite nilpotent semigroups. Together with Theorems 2 and 3 this gives a description of all homomorphisms of an arbitrary finite commutative semigroup.

Received
22 VI 1960

REFERENCES

1. L. M. Gluskin, Uch. zap. Kharkovsk. gos. ped. inst., 18, 41 (1956).
2. G. B. Preston, J. London Math. Soc., 29, 386 (1954).
3. A. I. Mal’tsev, Matem. sborn., 31(73), No. 1, 136 (1952).
4. A. I. Mal’tsev, DAN, 90, No. 3, 333 (1953).
5. T. Tamura, Osaka Math. J., 10, No. 2, 191 (1958).