MATHEMATICS

E. G. SHUTOV

DEFINING RELATIONS OF FINITE SEMIGROUPS OF PARTIAL TRANSFORMATIONS

(Presented by Academician A. I. Mal'tsev on 15 II 1960)

1°. In the present note a system of defining relations is found for the semigroup of all partial transformations of a finite set containing \(n \ge 4\) elements, and also for its subsemigroup of all one-to-one partial transformations.

Terms: a generating set of a semigroup, an irreducible generating set, a relation of a semigroup with respect to a generating set, a consequence of relations, and a system of defining relations of a semigroup are used in the usual sense (see, for example, \((^2)\)). If a relation \(u=v\) of a semigroup is a consequence of the relations \(\Sigma\) of this semigroup, then we shall say that \(u\) is reduced to \(v\) by means of \(\Sigma\).

2°. Let \(\Omega\) be the set of the numbers \(1,2,\ldots,n\), \(n \ge 4\); let \(\Delta_1\) and \(\Delta_2\) be subsets of \(\Omega\), where \(\Delta_1\) and \(\Delta_2\) may be empty. A mapping of \(\Delta_1\) into \(\Delta_2\) is called a partial transformation of the set \(\Omega\). If, under the partial transformation \(a\), the number \(i\) is mapped to \(k\), we shall write \(ai=k\). The totality \(W_n\) of all partial transformations of the set \(\Omega\) and the totality \(V_n\) of all one-to-one partial transformations of the set \(\Omega\), with respect to the usual multiplication of partial transformations, are semigroups. Denote by \(H_n\) the semigroup of all transformations of the set \(\Omega\), and by \(S_n\) the group of all one-to-one transformations in \(H_n\). Introduce the following notation:

\[ a_1= \begin{pmatrix} 2&3&\cdots&n\\ 2&3&\cdots&n \end{pmatrix}, \qquad a= \begin{pmatrix} 1&2&3&\cdots&n\\ 1&1&3&\cdots&n \end{pmatrix}, \]

\[ c_i= \begin{pmatrix} 1&2&\cdots&i-1&i&i+1&\cdots&n\\ i&2&\cdots&i-1&1&i+1&\cdots&n \end{pmatrix} \qquad (2\le i\le n). \]

It is known that the set \(M_1\) of all \(c_2,c_3,\ldots,c_n\) is an irreducible generating set of the group \(S_n\), and the set \(M_2\) of all \(c_2,c_3,\ldots,c_n,a\) is an irreducible generating set of the semigroup \(H_n\) \((^1)\). Let \(M_3\) be the set of all \(c_2,c_3,\ldots,c_n,a_1\), and \(M_4\) the set of all \(c_2,c_3,\ldots,c_n,a,a_1\). It is not hard to show that \(M_3\) and \(M_4\) are irreducible generating sets, respectively, of the semigroups \(V_n\) and \(W_n\).

3°. Let \(c_2^2=e,\ c_i a_1 c_i=a_i\). Consider the following system of relations of the semigroup \(V_n\) with respect to the set \(M_3\):

1. Defining relations of the group \(S_n\) with respect to \(M_1\) (see \((^2)\)).
2. \(a_1e=ea_1=a_1,\quad a_1a_2=a_2a_1,\quad a_1^2=a_1.\)
3. \(a_2c_i=c_i a_2,\quad a_i c_2=c_2a_i\quad (3\le i\le n).\)
4. \(c_2a_1a_2=a_1a_2.\)

\[ (\Sigma_1) \]

$4^\circ$. The following three lemmas can be proved.

Lemma 1. If $ui=k$ $(u\in S_n,\ i\in\Omega)$, then the relation

\[ ua_i=a_ku \]

of the semigroup $V_n$ is a consequence of the relations $(\Sigma_1)$.

Lemma 2. The relations

\[ a_i a_k=a_k a_i,\qquad a_i^2=a_i\qquad (1\leq i,k\leq n) \]

of the semigroup $V_n$ are consequences of the relations $(\Sigma_1)$.

Lemma 3. Let $i_1,i_2,\ldots,i_n$ be a permutation of the numbers $1,2,\ldots,n$, and let $2\leq m\leq n$. If, for $u,v$ from $S_n$, we have

\[ ui_k=vi_k\qquad (m+1\leq k\leq n), \]

then the relation

\[ ua_{i_1}a_{i_2}\cdots a_{i_m}=va_{i_1}a_{i_2}\cdots a_{i_m} \]

of the semigroup $V_n$ is a consequence of the relations $(\Sigma_1)$.

$5^\circ$. Theorem 1. The system of relations $(\Sigma_1)$ is a system of defining relations for the semigroup $V_n(2^0)$ with respect to the generating set $M_3(2^0)$.

Proof. By Lemmas 1 and 2, every word of the semigroup $V_n$ with respect to $M_3$ is reduced, using the relations $(\Sigma_1)$, to a word of the form

\[ ua_{i_1}a_{i_2}\cdots a_{i_m}\qquad (u\in S_n,\ i_1<i_2<\cdots<i_m,\ 0\leq m\leq n). \]

In view of what was said above, to prove the theorem it suffices to prove that every relation of the semigroup $V_n$ of the form

\[ ua_{i_1}a_{i_2}\cdots a_{i_m}=va_{j_1}a_{j_2}\cdots a_{j_{m_1}}, \tag{1} \]

where $u,v\in S_n$; $i_1<i_2<\cdots<i_m$; $j_1<j_2<\cdots<j_{m_1}$, $0\leq m,m_1\leq n$, is a consequence of the relations $(\Sigma_1)$. Let $i_1,i_2,\ldots,i_n$ be a permutation of the numbers $1,2,\ldots,n$. It is easy to see that if the relation (1) holds in the semigroup $V$, then

\[ m=m_1,\qquad i_k=j_k,\qquad ui_r=vi_r\qquad (1\leq k\leq m,\ m+1\leq r\leq n). \]

It follows that the relation (1) has the form

\[ ua_{i_1}a_{i_2}\cdots a_{i_m}=va_{i_1}a_{i_2}\cdots a_{i_m}, \tag{2} \]

where $ui_k=vi_k$ $(m+1\leq k\leq n,\ 0\leq m\leq n)$. If $m=0,1$, then $u$ and $v$ are identical, and therefore in this case relation (2) is a consequence of the relations $(\Sigma_1)$. Let $2\leq m\leq n$. Then, by Lemma 3, relation (2) is a consequence of the relations $(\Sigma_1)$.

$6^\circ$. Consider the following system of relations for the semigroup $W_n(2^0)$ with respect to the generating set $M_4(2^0)$:

1. Defining relations of the semigroup $H_n$ with respect to $M_2(?)$.
2. Relations 2 and 3 from the system of relations $(\Sigma_1)$.
3. $a_2a=a,\quad a_1a=aa_1a_2,\quad a_3a=aa_3.$
4. $aa_2=a_2.$

\[ (\Sigma_2) \]

$7^\circ$. The following two lemmas can be proved.

Lemma 4. Let $u\in H_n$; $i\in\Omega$; and let $i_1,i_2,\ldots,i_m$ be the set of all those elements of $\Omega$ which under $u$ are mapped to $i$. Then the relation

\[ a_i u=ua_{i_1}a_{i_2}\cdots a_{i_m} \]

of the semigroup $W_n$ is a consequence of the relations $(\Sigma_2)$.

Lemma 5. Let \(i_1, i_2, \ldots, i_n\) be a permutation of the numbers \(1, 2, \ldots, n\), \(1 \leqslant m \leqslant n\). If, for \(u, v\) in \(H_n\), one has

\[ u i_k = v i_k \qquad (m+1 \leqslant k \leqslant n), \]

then the relation

\[ u a_{i_1} a_{i_2} \ldots a_{i_m} = v a_{i_1} a_{i_2} \ldots a_{i_m} \]

of the semigroup \(W_n\) is a consequence of the relations \((\Sigma_2)\).

\(8^\circ\). By Lemmas 2, 4, 5, analogously to the proof of Theorem 1, the following theorem can be proved:

Theorem 2. The system of relations \((\Sigma_2)\) is a system of defining relations of the semigroup \(W_n\) \((2^\circ)\) with respect to the generating set \(M_4\) \((2^\circ)\).

Udmurt State
Pedagogical Institute
named after the Decade of the UASSR

Received
4 II 1960

CITED LITERATURE

\(^{1}\) N. N. Vorob’ev, Scientific Notes of the Leningrad State Pedagogical Institute named after A. I. Herzen, 89, 61 (1958).
\(^{2}\) A. Ya. Aizenshtat, Mathematical Collection, 45 (87), No. 3 (1958).