MATHEMATICS

L. N. SHEVRIN

ON LOCALLY FINITE SEMIGROUPS

(Presented by Academician A. I. Mal'tsev, December 21, 1964)

The main result of the present note is the following.

Theorem. A bundle of locally finite semigroups is locally finite.

As a very special case of this theorem we obtain, for example, the well-known (see (¹, ²)) proposition on the local finiteness of the direct product of semigroups of idempotents, proved by McLean (³). From the theorem there also follows a stronger proposition, which constitutes the main result of the paper of Grillet and Riesz (⁴) (see Corollary 2 below). At the same time, the nature of the corresponding results is revealed more deeply. We shall also indicate other consequences of the theorem.

Let us recall some definitions. If a semigroup \(\Gamma\) has a decomposition into subsemigroups \(\Gamma_\alpha\) (\(\alpha\) runs through some set of indices \(A\)), which are congruence classes in \(\Gamma\), then \(\Gamma\) is called a bundle of the semigroups \(\Gamma_\alpha\). A bundle is called commutative if, for any \(\alpha, \beta, \delta \in A\), from \(\Gamma_\alpha \Gamma_\beta \subseteq \Gamma_\delta\) it follows that \(\Gamma_\beta \Gamma_\alpha \subseteq \Gamma_\delta\). A bundle is called matrix if its components can be provided with pairs of indices \(\Gamma_{\xi\eta}\) (\(\xi \in X,\ \eta \in Y\)) in such a way that
\(\Gamma_{\xi_1\eta_1}\Gamma_{\xi_2\eta_2} \subseteq \Gamma_{\xi_1\eta_2}\) for any \(\xi_1,\xi_2 \in X,\ \eta_1,\eta_2 \in Y\). We shall call a bundle left (right) if all its components are left (right) ideals.

The factor semigroup whose elements are the components of a given commutative bundle \(\Gamma\) is a commutative semigroup of idempotents and, consequently, is locally finite; therefore any finite number of elements of \(\Gamma\) is contained in a subsemigroup which is a bundle of a finite number of components of the original commutative bundle. In other words, the following is true.

Lemma 1. A commutative bundle of semigroups has a local system of subsemigroups, each of which is the union of a finite number of components of this bundle.

The concept of a local system of subsemigroups is used in the usual sense, analogous to that in group theory (see (⁵)).

Lemma 2. A commutative bundle of a finite number of semigroups has a finite series of ideals, the first term of which is one of the components of the bundle, and the remaining factors are isomorphic to semigroups obtained from other components by adjoining zero externally.

Proof. We pass to the factor semigroup corresponding to the given bundle, which, as is known, is a (lower) semilattice with respect to the following relation \(\leq\):

\[ x \leq y \leftrightarrow xy = x. \]

Let \(P\) be the indicated finite semilattice. It obviously has a zero (least element). Arrange all the elements of \(P\) in nondecreasing order of their heights (see (⁶)): \(x_1, \ldots, x_m\). It is clear that if the meet of distinct elements \(x, y \in P\) is different from each of these elements, then it has height less than both the height of \(x\) and the height of \(y\); if it is equal to one of them, then the height of this element is less than the height of the other. Hence it follows that the set of elements \(x_1,\ldots,x_i\) will be an ideal for every \(i = 1,\ldots,m\). Passing now to the corresponding components of the bundle \(X_1,\ldots,X_m\), we construct the series of ideals

\[ X_1 \subset X_1 \cup X_2 \subset \cdots \subset X_1 \cup \cdots \cup X_m. \]

The lemma is proved.

Let \(\theta\) be an abstract theoretical-semigroup property. A semigroup possessing the property \(\theta\) will, as usual, be called a \(\theta\)-semigroup. By \(L\theta\) we shall denote the property of a semigroup to possess a local system of subsemigroups that are \(\theta\)-semigroups. Consider the following conditions that the property \(\theta\) may satisfy.

A. The semigroup obtained from a \(\theta\)-semigroup by external adjunction of a zero is a \(\theta\)-semigroup.

B. An ideal extension of a \(\theta\)-semigroup by means of a \(\theta\)-semigroup is a \(\theta\)-semigroup.

C. A semigroup anti-isomorphic to a \(\theta\)-semigroup is a \(\theta\)-semigroup.

D. The left sum of \(\theta\)-semigroups is a \(\theta\)-semigroup.

E. \(\theta\) is a local property, i.e. \(L\theta=\theta\).

From Lemmas 1 and 2 the following follows.

Proposition 1. If the property \(\theta\) satisfies conditions A and B, then the commutative sum of \(\theta\)-semigroups is an \(L\theta\)-semigroup.

Proposition 2. If the property \(\theta\) satisfies conditions C and D, then the matrix sum of \(\theta\)-semigroups is a \(\theta\)-semigroup.

Proof. The matrix sum of semigroups of a given family is the left (right) sum of the right (left) sums of certain semigroups of this family. Indeed, using the notation from the definition of a matrix sum, we conclude that the semigroups \(\Gamma_{\xi\eta}\) with fixed second index form a right sum, and the unions

\[ H_\eta=\bigcup_{\xi\in X}\Gamma_{\xi\eta} \]

are the components of the decomposition of the original semigroup into a left sum.

A semigroup anti-isomorphic to a right sum of semigroups is the left sum of semigroups anti-isomorphic to the original ones; therefore, for a property \(\theta\) satisfying conditions C and D, we have: the right sum of \(\theta\)-semigroups is a \(\theta\)-semigroup. Hence, from the assertion made at the beginning of the preceding paragraph, we obtain the required result.

Since an arbitrary sum of semigroups of a given family, according to Clifford’s theorem \(\left((^{7}),\right.\) see also \(\left.(^{1,2})\right)\), is a commutative sum of the matrix sums of certain semigroups of this family, the following follows directly from Propositions 1 and 2.

Proposition 3. If the property \(\theta\) satisfies conditions A–E, then the sum of \(\theta\)-semigroups is a \(\theta\)-semigroup.

We now pass to the case of interest to us, when \(\theta\) is local finiteness. The proof of the following lemma is carried out analogously to the proof of item 3 of Theorem 2.7 of the work \((^{8})\).

Lemma 3. An ideal extension of a locally finite semigroup by means of a locally finite semigroup is itself locally finite.

Lemma 4. The left sum of locally finite semigroups is locally finite.

The proof of Lemma 4 is based on the following auxiliary lemma.

Lemma 5. Let elements \(a_1,\ldots,a_n\) of a given semigroup generate an infinite subsemigroup. Then there exists an infinite sequence \(x_1,\ldots,x_k,\ldots\), each term of which is some element among \(a_1,\ldots,a_n\), such that for any natural \(k,l,p,q\) the following condition holds:

\[ (*)\ \text{if } l\ne q,\ \text{then } x_kx_{k+1}\ldots x_{k+l}\ne x_px_{p+1}\ldots x_{p+q}. \]

Proof. Each element of the subsemigroup \(H=\{a_1,\ldots,\) \(\ldots,a_n\}\) is represented in the form of some word in the alphabet \(a_1,\ldots,a_n\), and among such representations there exist words of minimal length. Fix some such irreducible representations. By virtue of the infinitude of \(H\), at least one of the letters \(a_1,\ldots,a_n\) occurs infinitely many times

as the first letter in these words. Denote it by \(x_1\), and denote by \(H_1\) the set of all corresponding words. The set \(H_1\) is infinite; therefore at least one of the letters \(a_1, \ldots, a_n\) occurs as the second letter in the words of \(H_1\) infinitely many times. Denote it by \(x_2\). Continuing this process by induction, we obtain a sequence \(x_1, x_2, \ldots, x_k, \ldots\) and, respectively, sets \(H_1, H_2, \ldots, H_k, \ldots\). We shall show that the indicated sequence is the desired one. Suppose that for some \(k, l, p, q\) one has
\[ x_k x_{k+1}\ldots x_{k+l}=x_p x_{p+1}\ldots x_{p+q}, \]
while \(l\ne q\). Put \(s=\max(k+l, p+q)\). The set \(H_s\) is infinite; hence it contains words of arbitrarily large length. Take in \(H_s\) an arbitrary word \(h\) whose length is greater than \(s\). Each of the words \(x_kx_{k+1}\ldots x_{k+l}\), \(x_px_{p+1}\ldots x_{p+q}\) then enters into \(h\) as a segment. The lengths of these words are unequal, since \(l\ne q\). Replacing in \(h\) the longer of the indicated words by the other word, we obtain the same element, represented by a word of smaller length. This contradicts the irreducibility of the chosen representations for the elements of \(H\). The contradiction obtained proves the lemma.

Proof of Lemma 4. Let \(\Gamma\) be a left union of locally finite semigroups \(\Gamma_\alpha\) \((\alpha\in A)\), and let \(a_1,\ldots,a_n\) be arbitrary elements of \(\Gamma\). We shall show that the subsemigroup \(\{a_1,\ldots,a_n\}\) is finite. We use induction on \(n\). For \(n=1\) this assertion is trivial, since \(\Gamma\) is a periodic semigroup. Suppose the assertion has been proved for \(n-1\). Assume that the subsemigroup is infinite. Take a certain sequence \(x_1,x_2,\ldots\) of the kind considered in Lemma 5. At least one of the letters \(a_1,\ldots,a_n\) occurs in it infinitely many times; without loss of generality we assume that it is \(a_1\). In the given sequence consider all possible segments of the form \(a_1x_k\ldots x_{k+l}a_1\), where \(x_i\ne a_1\) for \(i=k,\ldots,k+l\). From condition () and from the induction hypothesis it follows that, in the segments under consideration, the numbers \(l\) are, in the aggregate, bounded by some number \(l_0\); whence, in turn, it follows that there are only finitely many distinct words of the indicated form. For each word \(a_1x_k\ldots x_{k+l}a_1\) consider the corresponding element \(b_k=x_k\ldots x_{k+l}a_1\). There are finitely many elements \(b_k\), and all of them belong to one and the same component \(\Gamma_\alpha\)—the one to which \(a_1\) belongs. But from the infinitude of the sequence under consideration and from condition () it follows that the subsemigroup generated by the indicated elements is infinite. This contradicts the local finiteness of the subsemigroup \(\Gamma_\alpha\). The contradiction obtained proves the lemma.

Since conditions A, B, and D for local finiteness are satisfied trivially, the theorem formulated at the beginning of the note follows directly from Lemmas 3 and 4 and Proposition 3.

Since a semigroup possessing a decomposition into subgroups is, as is well known (see \((^7,^1,^2)\)), a commutative union of matrix unions of groups, the following is true.

Corollary 1. A semigroup possessing a decomposition into locally finite groups is locally finite.

Since a semigroup with the identity relation \(x^r=x\) \((r>1)\) possesses, as is easy to see, a decomposition into subgroups, it follows in turn from Corollary 1 that

Corollary 2 (Green and Rees theorem). The following assertions are equivalent:

\((A_r)\) An arbitrary semigroup with the identity relation \(x^r=x\) is locally finite.

\((B_r)\) An arbitrary group with the identity relation \(x^{r-1}=1\) is locally finite.

Corollary 3 (McLean theorem). An arbitrary semigroup of idempotents is locally finite.

Let us indicate several corollaries concerning semigroups with certain types of subsemigroup structures. Taking into account the results of the paper \((^9)\), semi-

by using local finiteness of an arbitrary nilpotent semigroup and applying Lemma 3 and the theorem of the present note, we obtain

Corollary 4. A semigroup with a Dedekind structure of subsemigroups is locally finite if and only if all of its subsemigroups are locally finite.

Let us note that examples of periodic groups with a Dedekind structure of subgroups that are not locally finite are still unknown. If no such groups exist, then, by Corollary 4, any semigroup with a Dedekind structure of subsemigroups is locally finite. In any case, the following holds.

Corollary 5. An arbitrary semigroup with a distributive structure of subsemigroups is locally finite.

Taking into account the results of the paper [10], we obtain

Corollary 6. An \(RK\)-semigroup is locally finite if and only if all of its maximal subgroups are locally finite.

In conclusion we note the following question, concerning a possible generalization of the theorem of the present note and suggested by Corollary 1 from it: will an arbitrary semigroup possessing a decomposition into locally finite semigroups be locally finite?

Ural State University
named after A. M. Gorky

Received
12 XII 1964

REFERENCES

1. E. S. Lyapin, Semigroups, Moscow, 1960.
2. A. H. Clifford, G. B. Preston, The Algebraic Theory of Semigroups, Am. Math. Soc. Publ., 1961.
3. D. McLean, Am. Math. Monthly, 61, 110 (1954).
4. J. A. Green, D. Rees, Proc. Cambr. Phil. Soc., 48, 35 (1952).
5. A. G. Kurosh, Group Theory, Moscow, 1953.
6. G. Birkhoff, Lattice Theory, Moscow, IL, 1952.
7. A. H. Clifford, Proc. Am. Math. Soc., 5, No. 3, 499 (1954).
8. L. N. Shevrin, Matem. sbornik, 53 (95), No. 3, 367 (1961).
9. L. N. Shevrin, DAN, 148, No. 2, 292 (1963).
10. L. N. Shevrin, V. M. Kopytov, DAN, 145, No. 5, 1021 (1962).