MATHEMATICS

A. P. BIRYUKOV

SEMIGROUPS DEFINED BY IDENTITIES

(Presented by Academician A. I. Mal’cev, 11 X 1962)

Let \(\Gamma(\Phi)\) be the class of all semigroups in which the set of identities \(\Phi\) is satisfied (see \((^7)\)). The following problem naturally arises. Let \(\mathfrak H\) be a given class of semigroups. Find all sets of identities \(\Phi\) for which \(\Gamma(\Phi) \subset \mathfrak H\). In this note a solution of this problem is given for certain important classes of semigroups \(\mathfrak H\). Some results in this direction for classes of idempotent semigroups have been obtained in \((^{3-6},\,^9)\).

Let \(\Xi=\{\xi_1,\xi_2,\ldots\}\) be a countable alphabet; \(A(\xi_i), B(\xi_i),\ldots\) are words (possibly empty) over the alphabet \(\Xi\). By \(l_{\xi_j}(A)\) we denote the number of occurrences of the letter \(\xi_j\) in the word \(A(\xi_i)\); by \(\chi(A)\), the set of those \(\xi_j\in\Xi\) for which \(l_{\xi_j}(A)\ge 1\); by \(l(A)\), the length of the word \(A(\xi_i)\). Every identity is written in the form of a formal equality of two nonempty words over the alphabet \(\Xi\). At the same time, the identity \(A(\xi_i)=B(\xi_i)\) is regarded as equal to each of the identities \(\varphi A(\xi_i)=\varphi B(\xi_i)\), \(\varphi B(\xi_i)=\varphi A(\xi_i)\), if \(\varphi\) is a one-to-one mapping of \(\chi(AB)\) into \(\Xi\). In addition, every identity of the form \(A(\xi_i)=A(\xi_i)\) is regarded as equal to the empty set of identities.

A semigroup free in \(\Gamma(\Phi)\) and having \(\mathfrak X\) as its set of free generators is denoted by \(F(\mathfrak X,\Phi)\). The semigroup \(F(\mathfrak X,\Phi)\) is also called the semigroup defined over the alphabet \(\mathfrak X\) by the set of identities \(\Phi\).

In most cases the question of the membership of the class of semigroups \(\Gamma(\Phi)\) in a given class of semigroups \(\mathfrak H\) reduces to the question of the membership of the semigroups \(F(\mathfrak X,\Phi)\) (for all possible \(\mathfrak X\)) in the class \(\mathfrak H\) (for example, this assertion, obviously, holds if the class of semigroups \(\mathfrak H\) is closed under homomorphisms). Therefore, in solving the problem formulated above, semigroups defined by identities turn out to be the main object of investigation. Let \(\Phi=\{A_\gamma(\xi_i)=B_\gamma(\xi_i),\ \gamma\in\Gamma\}\) be some set of identities. Every identity of the form \(c(\xi_i)\cdot \varphi A_\gamma(\xi_i)\cdot c'(\xi_i)=c(\xi_i)\cdot \varphi B_\gamma(\xi_i)\cdot c'(\xi_i)\), where \(A_\gamma(\xi_i)=B_\gamma(\xi_i)\) is some identity from \(\Phi\); \(\varphi\) is some mapping of \(\chi(A_\gamma B_\gamma)\) into the set of nonempty words over the alphabet \(\Xi\); and \(c(\xi_i), c'(\xi_i)\) are some words (possibly empty) over the alphabet \(\Xi\), will be called an immediate consequence of \(\Phi\). An identity \(A(\xi_i)=B(\xi_i)\) will be called a consequence of \(\Phi\) if there exists a finite sequence of nonempty words
\[ A(\xi_i)\equiv A_1(\xi_i),\quad A_2(\xi_i),\ldots,\quad A_m(\xi_i)\equiv B(\xi_i) \]
(\(\equiv\) is the sign of graphical equality of words) such that the identities \(A_j(\xi_i)=A_{j+1}(\xi_i)\) \((j=1,2,\ldots,m-1)\) are immediate consequences of \(\Phi\). If every identity from \(\Psi\) is a consequence of \(\Phi\), then, briefly, this is written as \(\Phi\Rightarrow\Psi\).

Lemma 1. If in the semigroup \(F(\mathfrak X,\Phi)\) \((\mathfrak X=\{x_\alpha,\alpha\in T\})\), \(A(x_\alpha)=B(x_\alpha)\), then for any mapping \(\varphi:\chi(AB)\to\Xi\) the identity \(\varphi A(x_\alpha)=\varphi B(x_\alpha)\) is a consequence of \(\Phi\).

Proposition 1. In order that \(\Gamma(\Phi)\subset \Gamma(\Psi)\), it is necessary and sufficient that \(\Phi\Rightarrow\Psi\).

Proposition 2. Let \(\mathfrak X\) be an infinite alphabet. In order that \(\Phi\Rightarrow\Psi\), it is necessary and sufficient that \(F(\mathfrak X,\Phi)\in\Gamma(\Psi)\).

Proposition 3. In order that \(\Phi \Rightarrow \Psi\), it is necessary and sufficient that, for every finite alphabet \(\mathfrak N\), \(F(\mathfrak N,\Phi)\in \Gamma(\Psi)\).

For a set of identities

\[ \Phi=\{A_\gamma(\xi_i)=B_\gamma(\xi_i),\ \gamma\in\Gamma\} \]

we introduce the numerical characteristic \(d(\Phi)\). The greatest common divisor of the set of all positive numbers of the form
\[ d(\gamma,\xi_j)=\left|\,l_{\xi_j}(A_\gamma)-l_{\xi_j}(B_\gamma)\,\right| \quad (\gamma\in\Gamma,\ \xi_j\in\Xi,\ d(\gamma,\xi_j)>0) \]
will be denoted by \(d(\Phi)\). If \(\Phi\) is the empty set of identities, or if \(l_{\xi_j}(A_\gamma)=l_{\xi_j}(B_\gamma)\) for all \(\gamma\in\Gamma,\ \xi_j\in\Xi\), then we put \(d(\Phi)=0\).

Theorem 1. In order that the semigroup \(F(\mathfrak N,\Phi)\) be periodic, it is necessary and sufficient that \(d(\Phi)\ne 0\).

Lemma 2. If \(\mathfrak A\in\Gamma(\Phi)\) and \(\mathfrak G\) is some subgroup of the semigroup \(\mathfrak A\), then for every \(X\in\mathfrak G\)
\[ X^{d(\Phi)}=E_{\mathfrak G}, \]
where \(E_{\mathfrak G}\) is the identity of the group \(\mathfrak G\).

We introduce several classes of sets of identities.

\(\mathfrak H(\alpha,m)\) \((m\ge 1)\): \(\Phi\in\mathfrak H(\alpha,m)\) if \(\Phi\) contains either the identity
\[ \xi_1\xi_2\cdots \xi_t=\xi_1\xi_2\cdots \xi_t A(\xi_i), \]
where \(1\le t\le m-1,\ l(A)\ge 1\), or the identity
\[ \xi_1\xi_2\cdots \xi_r\xi_{r+1}A(\xi_i) = \xi_1\xi_2\xi_3\cdots \xi_r\xi_{r'}B(\xi_i), \]
where \(0\le r\le m-1,\ r'\ne r+1,\ l(A),l(B)\ge 0\).

\(\mathfrak H^*(\alpha,m)\) is the dual class (symmetric with respect to “left” and “right”) to the class of sets of identities \(\mathfrak H(\alpha,m)\).

\(\mathfrak H_\beta\): \(\Phi\in\mathfrak H_\beta\) if \(\Phi\) contains some identity \(A(\xi_i)=B(\xi_i)\) for which \(\chi(A)\ne\chi(B)\).

\(\mathfrak H(\gamma,m)\) \((m\ge 1)\): \(\Phi\in\mathfrak H(\gamma,m)\) if \(\Phi\) contains an identity of the form
\[ \xi_1\xi_2\cdots \xi_t=A(\xi_i), \]
where \(1\le t\le m\), and either \(l(A)>t\), or \(\chi(A)\ne\{\xi_1,\xi_2,\ldots,\xi_t\}\).

\(\mathfrak H_\delta\): \(\Phi\in\mathfrak H_\delta\) if \(\Phi\) contains an identity of the form
\[ \xi_1A(\xi_i)=B(\xi_i), \]
where \(\chi(A)\ne\Phi,\ \xi_1\in\chi(A)\), and either \(l_{\xi_1}(B)\ge 2\), or \(B(\xi_i)\equiv \xi_2B'(\xi_i)\) \((l(B')\ge 0)\).

\(\mathfrak H_\delta^*\) is the dual class of sets of identities.

\(\mathfrak H_\varepsilon\): \(\Phi\in\mathfrak H_\varepsilon\) if \(\Phi\) contains an identity of the form
\[ A_1(\xi_i)\xi_1A_2(\xi_i) = B_1(\xi_i)\xi_2B_2(\xi_i), \]
where \(\{\xi_1,\xi_2\}\cap \chi(A_1B_1)=\varnothing;\ l(A_1),l(A_2),l(B_1),l(B_2)\ge 0\).

\(\mathfrak H_\varepsilon^*\) is the dual class of sets of identities.

For simplicity of formulation, some of the following theorems are not stated in their full generality.

Theorem 2. In order that the semigroup \(F(\mathfrak N,\Phi)\) be idempotent, it is necessary and sufficient that \(\Phi\in\mathfrak H(\gamma,1)\), \(d(\Phi)=1\).

Theorem 3. In order that, for \(m(\mathfrak N)\ge 2\) (\(m(\mathfrak N)\) is the cardinality of \(\mathfrak N\)), the set of all idempotents of the semigroup \(F(\mathfrak N,\Phi)\) be a subsemigroup satisfying the identity \(\xi_1=\xi_2\xi_1\), it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H_\beta . \]

Corollary 1. In order that, for \(m(\mathfrak N)\ge 2\), the semigroup \(F(\mathfrak N,\Phi)\) be a semigroup with a unique idempotent, it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^*(\alpha,1)\cap \mathfrak H_\beta . \]

Corollary 2. In order that, for \(m(\mathfrak N)\ge 2\), the semigroup \(F(\mathfrak N,\Phi)\) be unitary, it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^*(\alpha,1)\cap \mathfrak H_\beta\cap \mathfrak H(\gamma,1),\quad d(\Phi)=1. \]

Theorem 4. In order that, for \(1\le m\le m(\mathfrak N)\), \(m(\mathfrak N)\ge 2\), the ideal \(F(\mathfrak N,\Phi)^m\) of the semigroup \(F(\mathfrak N,\Phi)\) be a semigroup with right cancellation, it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^*(\alpha,1)\cap \mathfrak H_\beta\cap \mathfrak H(\gamma,m). \]

Corollary 3. In order that, for \(m(\mathfrak N)\ge 2\), the semigroup \(F(\mathfrak N,\Phi)\) be a group, it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^*(\alpha,1)\cap \mathfrak H_\beta\cap \mathfrak H(\gamma,1). \]

A semigroup \(\mathfrak A\) is called \(m\)-nilpotent (or nilpotent of class \(m\)) if \(\mathfrak A^m=0\), where \(0\) is the zero of the semigroup \(\mathfrak A\) (see (8)).

Corollary 4. In order that, for \(1\le m\le m(\mathfrak N)\), \(m(\mathfrak N)\ge 2\), the semigroup \(F(\mathfrak N,\Phi)\) be \(m\)-nilpotent, it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^*(\alpha,1)\cap \mathfrak H_\beta\cap \mathfrak H(\gamma,m),\quad d(\Phi)=1. \]

Using one lemma from (²), one can prove that the following is true.

Theorem 5. In order that, for $\mathfrak m(\mathfrak N)\geqslant 2$, the semigroup $F(\mathfrak N,\Phi)$ be inverse, it is necessary and sufficient that
\[ \Phi \in \mathfrak H(\alpha,1)\cap \mathfrak H^{*}(\alpha,1)\cap \mathfrak H(\gamma,1). \]

Theorem 6. In order that, for $\mathfrak m(\mathfrak N)\geqslant 2$, the semigroup $F(\mathfrak N,\Phi)$ be completely simple without zero, it is necessary and sufficient that
\[ \Phi\in \mathfrak H_\beta\cap \mathfrak H(\gamma,1). \]

Associate with the set of identities $\Phi$ the set of identities
\[ \overline{\Phi}=\{\Phi,\ \xi_1=\xi_2^{d(\Phi)}\xi_1,\ \xi_1=\xi_1\xi_2^{d(\Phi)}\}. \]
According to Corollary 3, all semigroups in $\Gamma(\Phi)$ are groups if and only if $d(\Phi)\neq 0$. Suppose that $\Phi$ does not contain the commutativity identity (i.e. the identity $\xi_1\xi_2=\xi_2\xi_1$). Then:

Theorem 7. 1) Let $\Phi\in \mathfrak H_\beta$. In order that, for $\mathfrak m(\mathfrak N)\geqslant 2$, the semigroup $F(\mathfrak N,\Phi)$ be commutative, it is necessary and sufficient that
\[ d(\Phi)\neq 0,\quad \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^{*}(\alpha,1)\cap \mathfrak H(\gamma,2) \]
and that the group $F(\mathfrak N,\overline{\Phi})$ be commutative.

2) Let $\Phi\notin \mathfrak H_\beta$. In order that, for $\mathfrak m(\mathfrak N)\geqslant 2$, the semigroup $F(\mathfrak N,\Phi)$ be commutative, it is necessary and sufficient that
\[ d(\Phi)\neq 0,\quad \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^{*}(\alpha,1)\cap [\mathfrak H(\gamma,1)\cup(\mathfrak H(\gamma,2)\cap \mathfrak H_\delta\cap \mathfrak H_\delta^{*})] \]
and that the group $F(\mathfrak N,\overline{\Phi})$ be commutative.

Corollary 5. 1) Let $\Phi\in \mathfrak H_\beta$, $d(\Phi)\leqslant 2$. In order that the semigroup $F(\mathfrak N,\Phi)$ be commutative, it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^{*}(\alpha,1)\cap \mathfrak H(\gamma,2). \]

2) Let $\Phi\notin \mathfrak H_\beta$, $d(\Phi)\leqslant 2$. In order that the semigroup $F(\mathfrak N,\Phi)$ be commutative, it is necessary and sufficient that
\[ \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H^{*}(\alpha,1)\cap [\mathfrak H(\gamma,1)\cup(\mathfrak H(\gamma,2)\cap \mathfrak H_\delta\cap \mathfrak H_\delta^{*})]. \]

Consider identities of the form
\[ \xi_1\xi_2\ldots \xi_m=\xi_1\xi_2\ldots \xi_r\xi_{m+1}\xi_{m+2}\ldots \xi_{m+t}\xi_{m-s+1}\ldots \xi_m, \tag{1} \]
where $0\leqslant r,s\leqslant m$, $t=\max\{1,m-(s+t)\}$. It is clear that identities of the form (1) include such frequently used identities as
\[ \xi_1=\xi_2,\quad \xi_1=\xi_2\xi_1,\quad \xi_1=\xi_1\xi_2\xi_1,\quad \xi_1\xi_2=\xi_3\xi_4,\quad \xi_1\xi_2=\xi_1\xi_3\xi_2. \]

Theorem 8. In order that the identity (1) follow from the set of identities $\Phi$, it is necessary and sufficient that
\[ d(\Phi)=1,\quad \Phi\in \mathfrak H(\alpha,r+1)\cap \mathfrak H^{*}(\alpha,s+1)\cap \mathfrak H_\beta\cap \mathfrak H(\gamma,m). \]

Corollary 6. If $\mathfrak m(\mathfrak N)<\infty$, $d(\Phi)=1$, $\Phi\in \mathfrak H(\gamma,m)\cap \mathfrak H_\beta$, then the semigroup $F(\mathfrak N,\Phi)$ is finite.

Let
\[ \Phi_1=\{\xi_1=\xi_1^2,\ \xi_1\xi_2\xi_3\xi_4=\xi_1\xi_3\xi_2\xi_4\},\quad \Phi_2=\{\xi_1=\xi_1^2,\ \xi_1\xi_2\xi_3=\xi_2\xi_1\xi_3\}. \]
Semigroups from $\Gamma(\Phi_1)$, $\Gamma(\Phi_2)$ were used by V. V. Vagner (¹) for the study of generalized groups.

Theorem 9. 1) Let $\Phi\in \mathfrak H_\beta$. In order that $\Phi\Rightarrow \Phi_1$, it is necessary and sufficient that
\[ d(\Phi)=1,\quad \Phi\in \mathfrak H(\gamma,1). \]

2) Let $\Phi\notin \mathfrak H_\beta$. In order that $\Phi\Rightarrow \Phi_1$, it is necessary and sufficient that
\[ d(\Phi)=1,\quad \Phi\in \mathfrak H(\gamma,1)\cap \mathfrak H_\varepsilon\cap \mathfrak H_\varepsilon^{*}. \]

Theorem 10. 1) Let $\Phi\in \mathfrak H_\beta$. In order that $\Phi\Rightarrow \Phi_2$, it is necessary and sufficient that
\[ d(\Phi)=1,\quad \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H(\gamma,1). \]

2) Let $\Phi\notin \mathfrak H_\beta$. In order that $\Phi\Rightarrow \Phi_2$, it is necessary and sufficient that
\[ d(\Phi)=1,\quad \Phi\in \mathfrak H(\alpha,1)\cap \mathfrak H(\gamma,1)\cap \mathfrak H_\varepsilon. \]

In conclusion I express my sincere gratitude to E. S. Lyapin for his guidance in carrying out this work.

Received
5 X 1962

REFERENCES

¹ V. V. Vagner, Matem. sborn., 52, 596 (1960).
² S. Green, D. Rees, Proc. Cambr. Phil. Soc., 48, 35 (1952).
³ N. Kimura, Proc. Japan Acad., 33, 642 (1957).
⁴ N. Kimura, Proc. Japan Acad., 34, 113 (1958).
⁵ N. Kimura, Pacif. J. Math., 8, 257 (1958).
⁶ N. Kimura, Proc. Japan Acad., 34, 121 (1958).
⁷ E. S. Lyapin, Semigroups, 1960.
⁸ L. N. Shevrin, Matem. sborn., 53, 343 (1961).
⁹ M. Yamada, N. Kimura, Proc. Japan Acad., 34 (1958).