MATHEMATICS

I. S. Ponizovskii

A NOTE ON COMMUTATIVE SEMIGROUPS

(Presented by Academician A. I. Mal’tsev on 22 IX 1961)

1. If \(c\) is a two-sided ideal of the semigroup \(\mathfrak{B}\), \(\mathfrak{D}=\mathfrak{B}\setminus c\), then, following \(\left({}^{1}\right)\), we shall assume that \(\mathfrak{B}/c=\mathfrak{D}=\mathfrak{D}\cup \bar{0}\), where \(\bar{0}\) is the zero of \(\mathfrak{B}/c\). We shall call \(\mathfrak{B}\) an extension of \(c\) by means of \(\mathfrak{D}\).

2. By \(\mathfrak{K}\) we denote the class of commutative semigroups with zero, containing a finite number of nonregular elements \(\left(\left({}^{1}\right),\ \text{p. }104\right)\). It can be shown that the class \(\mathfrak{K}\) consists of those and only those commutative semigroups with zero which have a finite ideal composition series (in the sense of Rees \(\left({}^{2}\right)\)). In particular, the class \(\mathfrak{K}\) contains all finite commutative semigroups with zero. We note that the requirement that a zero be present in the semigroups of the class \(\mathfrak{K}\) is inessential and is adopted only to simplify the formulations. A commutative semigroup \(\mathfrak{A}\) will be called elementary if
   \[ \mathfrak{A}=\mathfrak{G}\cup\mathfrak{N},\quad \mathfrak{G}\cap\mathfrak{N}=\varnothing^*, \]
   where \(\mathfrak{G}\) is a group with identity \(e\), \(\mathfrak{N}\) is a nilpotent finite ideal of \(\mathfrak{A}\), and \(e\) is the identity of \(\mathfrak{A}\). Elementary semigroups belong to \(\mathfrak{K}\); they are described without difficulty: their description is readily reduced to the description of finite nilpotent semigroups, and the latter are described in \(\left({}^{3}\right)\). An ideal series of a commutative semigroup \(\mathfrak{A}\) will be called reduced if all its factors are elementary semigroups.

3. Theorem 1. Let \(\mathfrak{A}\in\mathfrak{K}\), \(\mathfrak{A}^{2}=\mathfrak{A}\). Then:

\(\alpha)\) \(\mathfrak{A}\) has a reduced series;

\(\beta)\) all terms of any reduced series of \(\mathfrak{A}\) are ideals of \(\mathfrak{A}\);

\(\gamma)\) if
\[ \begin{gathered} \mathfrak{A}=\mathfrak{A}_{1}\supset \mathfrak{A}_{2}\supset\cdots\supset \mathfrak{A}_{n}\supset \mathfrak{A}_{n+1}=0,\\ \mathfrak{A}=\mathfrak{B}_{1}\supset \mathfrak{B}_{2}\supset\cdots\supset \mathfrak{B}_{m}\supset \mathfrak{B}_{m+1}=0 \end{gathered} \tag{1} \]
are reduced series of \(\mathfrak{A}\), then \(m=n\), and between the factors of these series one can establish a one-to-one correspondence \(\rho\) such that, if \(\mathfrak{A}_{i}/\mathfrak{A}_{i+1}\) and \(\mathfrak{B}_{j}/\mathfrak{B}_{j+1}\) correspond under \(\rho\), then
\[ \mathfrak{A}_{i}\setminus\mathfrak{A}_{i+1}=\mathfrak{B}_{j}\setminus\mathfrak{B}_{j+1}. \]

From \(\gamma)\) it follows, in particular, that any two reduced series of \(\mathfrak{A}\in\mathfrak{K}\), \(\mathfrak{A}^{2}=\mathfrak{A}\), are isomorphic.

1. Let \(\mathfrak{A}\in\mathfrak{K}\), \(\mathfrak{A}^{2}=\mathfrak{A}\), and let (1) be a reduced series of \(\mathfrak{A}\), \(p_i=\mathfrak{A}_{i}\setminus\mathfrak{A}_{i+1}\), \(i\leq n\). Since \(\mathfrak{A}_{i}/\mathfrak{A}_{i+1}\) is an elementary semigroup (by the definition of a reduced series), \(\mathfrak{A}_{i}/\mathfrak{A}_{i+1}\) contains an identity. Hence there exists \(e_i\in p_i\) such that \(e_i\) is an identity for the elements of \(p_i\).

Denote by \(a_k\) the ideal homomorphism of \(\mathfrak{A}\) generated by \(\mathfrak{A}_k\), a term of (1) (\(a_k\) has meaning in view of Theorem 1 \(\beta\)). Put, for \(x\in p_i\), \(i\leq k\leq n\):
\[ \varphi_{ki}x=a_{k+1}(xe_k). \]

\(\varphi_{ki}\) maps \(p_i\) into \(\bar p_k=\mathfrak{A}_k/\mathfrak{A}_{k+1}\). We collect the mappings \(\varphi_{ki}\) into the matrix
\[ \Phi= \begin{pmatrix} \varphi_{11} & 0 & \cdots & 0\\ \varphi_{21} & \varphi_{22} & \cdots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \varphi_{n1} & \varphi_{n2} & \cdots & \varphi_{nn} \end{pmatrix}. \]

* By \(\varnothing\) the empty set is denoted.

Consider the collection \(V(\mathfrak A)\) of all \(n\)-dimensional vectors \(X_i\) of the form
\(X_i=(0,\ldots,0,x_i,\varphi_{i+1,i}x_i,\ldots,\varphi_{ni}x_i)\) (the zeros stand in the first \(i-1\) places of \(X_i\)), \(x_i\in \mathfrak p_i\). The zero standing in the \(k\)-th place of \(X_i\) is regarded as the zero of \(\overline{\mathfrak p}_k\). If \(X\in V(\mathfrak A)\), then by \(\{X\}_k\) we denote the \(k\)-th component of \(X\). From the definition of \(X_i\) and \(\varphi_{ki}\) it follows that \(\{X\}_k\in \overline{\mathfrak p}_k\) for all \(X\in V(\mathfrak A)\) and \(k=1,2,\ldots,n\). Therefore one may define multiplication of vectors \(X,Y\in V(\mathfrak A)\):

\[ \{X\cdot Y\}_k=\{X\}_k\cdot \{Y\}_k . \tag{2} \]

Theorem 2. With respect to multiplication (2), \(V(\mathfrak A)\) is a semigroup isomorphic to \(\mathfrak A\).

Thus every semigroup \(\mathfrak A\in \mathfrak K\), \(\mathfrak A^2=\mathfrak A\), can be represented isomorphically by a semigroup of vectors whose components belong to elementary semigroups. At the same time, Theorem 2 implies:

Theorem 3. The specification of the factors \(\overline{\mathfrak p}_i=\mathfrak A_i/\mathfrak A_{i+1}\) and of the matrix \(\Phi\) completely determines the semigroup \(\mathfrak A\).

Theorem 2 also leads to the proposition:

Theorem 4. If an arbitrary semigroup \(\mathfrak A\) has a finite ideal series all of whose factors are elementary semigroups, then \(\mathfrak A\) is commutative and \(\mathfrak A\in\mathfrak K\).

In view of Theorem 3 the notation has meaning:
\(\mathfrak A=(\overline{\mathfrak p}_1,\ldots,\overline{\mathfrak p}_n,\Phi)\).

1. Let \(\mathfrak A,\mathfrak B\in\mathfrak K\), \(\mathfrak A^2=\mathfrak A\), \(\mathfrak B^2=\mathfrak B\),
   \(\mathfrak A=(\overline{\mathfrak p}_1,\ldots,\overline{\mathfrak p}_n,\Phi)\),
   \(\mathfrak B=(\overline{\mathfrak q}_1,\ldots,\overline{\mathfrak q}_m,\Psi)\).

Suppose that there is a non-identical substitution \(\Lambda\) of the set \(1,2,\ldots,n\) such that for every \(i=1,2,\ldots,n\) there exists an isomorphism \(\lambda_i\) of \(\overline{\mathfrak p}_i\) onto \(\overline{\mathfrak q}_{\Lambda i}\). Denote here and in Theorem 5 by \(L\) the square matrix of degree \(n\) such that in the \(i\)-th column of \(L\), in the row with number \(\Lambda i\), there stands \(\lambda_i\), while the remaining elements of the \(i\)-th column of \(L\) are zeros. Naturally, \(L^{-1}\) is defined.

Theorem 5. Let \(\mathfrak A,\mathfrak B\in\mathfrak K\), \(\mathfrak A^2=\mathfrak A\), \(\mathfrak B^2=\mathfrak B\),
\(\mathfrak A=(\overline{\mathfrak p}_1,\ldots,\overline{\mathfrak p}_n,\Phi)\),
\(\mathfrak B=(\overline{\mathfrak q}_1,\ldots,\overline{\mathfrak q}_m,\Psi)\). For \(\mathfrak A\) and \(\mathfrak B\) to be isomorphic it is necessary and sufficient that \(m=n\) and that there be found a matrix \(L\) such that \(\Psi=L\Phi L^{-1}\) (multiplication of the nonzero elements of \(L,\Phi,L^{-1}\) is understood in the sense of composition of mappings).

1. We shall describe briefly how semigroups of the class \(\mathfrak K\) can be constructed. It is enough to restrict ourselves to semigroups \(\mathfrak A=\mathfrak K\), \(\mathfrak A^2=\mathfrak A\), since one can always adjoin an identity \(e\) to a semigroup \(\mathfrak A\), and if \(\mathfrak A'=\mathfrak A\cup e\) and \(\mathfrak A\in\mathfrak K\), then \(\mathfrak A'\in\mathfrak K\).

6.1. Let \(\mathfrak p\) be a set with a partial binary operation (multiplication), i.e. an operation defined not on all of \(\mathfrak p\times\mathfrak p\). If \(x,y\in\mathfrak p\) and \(x\cdot y\) is not defined in \(\mathfrak p\), we shall write \(x\cdot y\notin\mathfrak p\). Construct \(\overline{\mathfrak p}=\mathfrak p\cup \overline{0}\), putting:
\(x\cdot y=\overline{0}\) if \(x\cdot y\notin\mathfrak p\), and
\(x\cdot\overline{0}=\overline{0}\cdot x=\overline{0}\cdot\overline{0}=\overline{0}\), for \(x,y\in\mathfrak p\). We shall call \(\overline{0}\) the zero of \(\overline{\mathfrak p}\). We shall say that \(\mathfrak p\) is a semigroupoid if \(\overline{\mathfrak p}\) is a semigroup.

6.2. Let \(\mathfrak p\) be a semigroupoid, and let \(\mathfrak q\) be a set with a partial single-valued binary operation—multiplication. A mapping \(\varphi:\mathfrak p\to\mathfrak q\) will be called a homomorphism of \(\mathfrak p\) if the implication holds:
\(x,y,x\cdot y\in\mathfrak p \to \varphi x\cdot \varphi y\in\mathfrak q,\ \varphi x\cdot \varphi y=\varphi(x\cdot y)\).
If the homomorphism \(\varphi:\mathfrak p\to\mathfrak q\) can be extended to a homomorphism \(\overline{\varphi}:\overline{\mathfrak p}\to\overline{\mathfrak q}\), with \(\varphi\overline{0}\) the zero of \(\overline{\mathfrak q}\), then \(\varphi\) will be called an exact homomorphism of \(\mathfrak p\). From the definition itself it follows that exact homomorphisms of \(\mathfrak p\) are completely determined by homomorphisms of the semigroup \(\overline{\mathfrak p}\).

If \(\mathfrak R\) is a subset of \(\mathfrak q\), then, putting for \(x,y\in\mathfrak R\)
\(x\circ y=x\cdot y\) if \(x\cdot y\in\mathfrak R\), we turn \(\mathfrak R\) into a set with a partial operation. We shall say that the operation in \(\mathfrak R\) is induced by the operation in \(\mathfrak q\). If \(\varphi\) is an exact homomorphism of \(\mathfrak p\) into \(\mathfrak q\), then \(\varphi\mathfrak p\) will be a semigroupoid with respect to the operation induced in \(\varphi\mathfrak p\) by the operation in \(\mathfrak q\). A homomorphism of \(\mathfrak p\) that is a one-to-one mapping will be called a monomorphism of \(\mathfrak p\).

Theorem 6. Every homomorphism \(\varphi\) of a semigroupoid \(\mathfrak p\) is representable in the form \(\varphi=\mu\psi\), where \(\psi\) is an exact homomorphism of \(\mathfrak p\), and \(\mu\) is a monomorphism of the semigroupoid \(\psi\mathfrak p\).

This theorem reduces the question of describing homomorphisms of a semigroupoid \(\mathfrak p\) into the given structure to the description of abstract homomorphisms (i.e. two-sided stable equivalences) of the semigroup \(\bar{\mathfrak p}\) and to the description of monomorphisms of a certain semigroupoid \(\psi\mathfrak p\) into the given structure.

6.3. Clifford \((^{4})\), in somewhat different terms, proved:

Theorem. Let \(\mathfrak c\) be a semigroup with identity, \(\mathfrak D\) a semigroupoid, and \(\mathfrak p\) a homomorphism of \(\mathfrak D\) into \(\mathfrak c\). Define in \(\mathfrak B=\mathfrak c\cup\mathfrak D\) the multiplication “\(\circ\)”:

\[ X,Y\in\mathfrak c\to X\circ Y=XY;\qquad X,Y,XY\in\mathfrak D\to X\circ Y=XY; \]

\[ X\in\mathfrak D,\quad Y\in\mathfrak c\to X\circ Y=\varphi X\cdot Y,\qquad Y\circ X=Y\cdot\varphi X; \]

\[ X,Y\in\mathfrak D,\quad XY\notin\mathfrak D\to X\circ Y=\varphi X\cdot\varphi Y. \]

Then with respect to the multiplication “\(\circ\)” \(\mathfrak B\) is a semigroup that is an extension of \(\mathfrak c\) by means of the semigroup \(\overline{\mathfrak D}\).

Every extension of \(\mathfrak c\) by means of \(\overline{\mathfrak D}\) can be obtained in the manner described in the theorem.

6.4. The construction of semigroups \(\mathfrak A\in\mathfrak K\), \(\mathfrak A^{2}=\mathfrak A\), which we propose consists in the successive application of the operation of extension of an elementary semigroup \(\mathfrak c\) by means of a semigroup \(\mathfrak p\in\mathfrak K\). From Theorems 1 and 4 it follows that in this way all semigroups \(\mathfrak A\in\mathfrak K\), \(\mathfrak A^{2}=\mathfrak A\), and only they, will be obtained.

Since \(\mathfrak c\) is an elementary semigroup, \(\mathfrak c\) contains an identity. Therefore, by 6.3, the description of the indicated extension is reduced to the description of homomorphisms of \(\mathfrak p\) into \(\mathfrak c\). This, in turn, in view of Theorem 7, is reduced to the description of abstract homomorphisms of the semigroup \(\bar{\mathfrak p}\) and monomorphisms of semigroupoids of the form \(\psi\mathfrak p\) (\(\psi\) is an exact homomorphism of \(\mathfrak p\)) into \(\mathfrak c\). The semigroup \(\bar{\mathfrak p}\) should be regarded as known; then its homomorphisms can be found as indicated in \((^{4})\). Let \(\mathfrak q=\psi\mathfrak p\); it can be shown that \(\bar{\mathfrak q}\in\mathfrak K\). As a result, everything is reduced to the description of monomorphisms of semigroupoids \(\mathfrak q\), \(\bar{\mathfrak q}\in\mathfrak K\), into elementary semigroups. This problem, in turn, can be reduced to the description of monomorphisms into elementary semigroups of semigroupoids \(\mathfrak R\) such that \(\overline{\mathfrak R}\) is a nilpotent semigroup. Such monomorphisms are described in Theorem 6.

6.5. We shall say that \(\mathfrak R\) is a nilpotent semigroupoid of degree \(n\), if \(\overline{\mathfrak R}^{\,n}=0\). Let \(\mathfrak A\) be an arbitrary semigroup, \(\Omega\subset\mathfrak c\) (in \(\Omega\) the operation in \(\mathfrak A\) is not taken into account). Denote by \(\Omega_{n}\) the collection of all words in the alphabet \(\Omega\) whose length does not exceed \(n\). Then \(\Omega_{n}\) is a nilpotent semigroupoid of degree \(n\) with respect to the usual multiplication of words.

Let \(\bar I=I\cup\bar 0\) be a two-sided ideal of \(\overline{\Omega}_{n}\), \(I\cap\Omega=\varnothing\) (the case \(I=\varnothing\) is not excluded). Put \((\Omega,n,I)=\Omega_{n}\setminus I\); \((\Omega,n,I)\) is also a nilpotent semigroupoid of degree \(n\) with respect to multiplication of words. The embedding of \(\Omega\) in \(\mathfrak A\) induces a homomorphism \(f:(\Omega,n,I)\to\mathfrak A\).

Theorem 7. A one-to-one mapping \(\nu\) of a nilpotent semigroupoid \(\mathfrak R\) of degree \(n\) into a semigroup \(\mathfrak A\) is a monomorphism of \(\mathfrak R\) if and only if there exist a semigroupoid \((\Omega,n,I)\), \(\Omega\subset\mathfrak A\), and an exact homomorphism \(\varphi\) of \((\Omega,n,I)\) onto \(\mathfrak R\) such that \(f=\nu\varphi\) (\(f\) is the homomorphism of \((\Omega,n,I)\) into \(\mathfrak A\) induced by the embedding of \(\Omega\) in \(\mathfrak A\)).

1. The description of the construction of semigroups of the class \(\mathfrak K\), together with the theorem on their isomorphism (Theorem 5), gives a description of semigroups of the class \(\mathfrak K\).

Received
18 IX 1961

REFERENCES

\({}^{1}\) E. S. Lyapin, Semigroups, 1960.
\({}^{2}\) D. Rees, Proc. Cambridge Phil. Soc., 36, 387 (1940).
\({}^{3}\) T. Tamura, Osaka Math. J., 10, No. 2, 191 (1958).
\({}^{4}\) A. H. Clifford, Trans. Am. Math. Soc., 68, No. 2, 165 (1950).
\({}^{5}\) I. S. Ponizovskii, DAN, 135, No. 5, 1058 (1960).