UDC 519.40

MATHEMATICS

V. N. Salii

Transformative Semigroups \((\mathfrak G,\circ,\xi,\chi_1,\chi_2)\)

(Presented by Academician A. I. Mal'cev, 10 V 1967)

It is known that not all properties of (partial) transformations of a set can be expressed in terms of the operation of their multiplication. Consequently, in considering an abstract semigroup isomorphic to a given semigroup of transformations, we cannot recover many important connections between the corresponding transformations. Among such connections are the most important binary relations defined in semigroups of transformations: the projection relations \(\chi_1\) (first) and \(\chi_2\) (second) \(\bigl((\varphi_1,\varphi_2)\in\chi_i\) if and only if \(pr_i\varphi_1\subset pr_i\varphi_2\bigr)\), and the relation of semicompatibility of transformations \(\xi\) \(\bigl((\varphi_1,\varphi_2)\in\xi\) if and only if \(\varphi_1\) and \(\varphi_2\) act identically on \(pr_1\varphi_1\cap pr_1\varphi_2\); for transformations their compatibility is usually denoted by an infix sign).

The desire to give as complete an abstract description as possible of semigroups of transformations leads to the natural concept of a relativized semigroup \({}^{(3)}\): a system \((\mathfrak G,\circ,\rho_1,\ldots,\rho_n)\), where \((\mathfrak G,\circ)\) is a semigroup, and \(\rho_1,\ldots,\rho_n\) are certain binary relations given on \(\mathfrak G\). If \((\mathfrak G,\circ,\rho_1,\ldots,\rho_n)\) and \((\mathfrak G^*,\circ^*,\rho_1^*,\ldots,\rho_n^*)\) are two relativized semigroups, then a mapping \(P:\mathfrak G\to\mathfrak G^*\) is called a homomorphism of the first into the second if \(P\) is a homomorphism of \((\mathfrak G,\circ)\) into \((\mathfrak G^*,\circ^*)\), and, moreover, \((g_1,g_2)\in\rho_i\) if and only if \((P(g_1),P(g_2))\in\rho_i^*\). The study of relativized semigroups of transformations was initiated by V. V. Wagner \({}^{(1)}\). He also posed the problem of an abstract characterization of relativized semigroups of transformations \((\mathfrak G,\circ,\xi,\chi_1,\chi_2)\).

Relativized semigroups isomorphic to relativized semigroups of transformations of the form \((\mathfrak G,\circ,\xi,\chi_1,\chi_2)\) will be called transformative semigroups. Attempts to describe the class of transformative semigroups led to the isolation of the classes of transformative semigroups of the first kind \((\mathfrak G,\circ,\xi,\chi_1)\) \({}^{(2)}\), projection semigroups of the first and second kinds \((\mathfrak G,\circ,\chi_1)\) and \((\mathfrak G,\circ,\chi_2)\), respectively \({}^{(5)}\), general projection semigroups \((\mathfrak G,\circ,\chi_1,\chi_2)\) \({}^{(3)}\), and relativized semigroups \((\mathfrak G,\circ,\xi)\) \({}^{(6)}\). In the present note an elementary (but infinite) system of axioms is given for transformative semigroups \((\mathfrak G,\circ,\xi,\chi_1,\chi_2)\). The problem is solved by the same method with whose aid the author obtained an abstract characterization of projection semigroups \({}^{(3)}\) and an elementary axiomatics of transformative semigroups of the first kind \({}^{(4)}\).

Let \((\mathfrak G,\circ,\xi,\chi_1,\chi_2)\) be some relativized semigroup. Define a ternary predicate \(\Pi\) as follows. Put

\[ \Pi_0(g_1,g_2,g_3)\leftrightarrow (\exists \bar g_1,\bar g_2,\ldots,\bar g_{m-1},\bar g_m,g) \bigl((\bar g_1,\bar g_2)\in\xi\land\cdots\land(\bar g_{m-1},\bar g_m)\in\xi \land\{\bar g_1g,\bar g_2,\ldots,\bar g_{m-1},\bar g_m\}\subset \chi_1\langle g_1\rangle\cup\chi_1\langle g_2\rangle \land(\bar g_m g,g_3)\in\chi_1\bigr); \]

\[ \Pi_n(g_1,g_2,g_3)\leftrightarrow (\exists \bar g_1,\bar g_2,\ldots,\bar g_{m-1},\bar g_m,g) \bigl((\bar g_1,\bar g_2)\in \]

\[ \in \xi \wedge \cdots \wedge (\bar g_{m-1}, \bar{\bar g}_{m}) \in \xi \wedge \Pi_{n-\bar g_1}(g_1,g_2,\bar g_1 g) \wedge \Pi_{n-1}(g_1,g_2,\bar g_2) \wedge \cdots \]
\[ \cdots \wedge \Pi_{n-1}(g_1,g_2,\bar g_m) \wedge (\bar g_m g,g_3) \in \chi_1; \]
\[ \Pi(g_1,g_2,g_3) \leftrightarrow (\exists n)\,\Pi_n(g_1,g_2,g_3) \]

(the symbol \(g\) may be empty).

We construct inductively, over the alphabet with letters \(a_g, b_g, c_{(g_1,g_2)}\) \((g,g_1,g_2\in \mathfrak G)\), the following set of words \(\Gamma\).

First \(\Gamma_1\) is defined:

1. \(c_{(g_1,g_2)}\in \Gamma_1\).

2. \(\Pi(g_1,g_2,g_3)\to c_{(g_1,g_2)}b_{g_3}\in \Gamma_1\).

3. \(\Pi(g_1,g_2,g_3)\wedge (\exists \bar g_3,\ldots,\bar{\bar g}_3,\tilde g_3)\bigl((g_3,\bar g_3)\in \xi \wedge \cdots \wedge (\bar{\bar g}_3,\tilde g_3)\in \xi \wedge\)
   \[ \wedge \Pi(g_1,g_2,\bar g_3)\wedge \cdots \wedge \Pi(g_1,g_2,\bar{\bar g}_3)\wedge (\tilde g_3,g_4)\in \chi_2 \to c_{(g_1,g_2)}b_{g_3}a_{g_4}\in \Gamma_1. \]

4. \(\gamma b_{g_1}a_{g_2}\in \Gamma_1 \wedge (\exists \bar g_2,\bar{\bar g}_2,\bar g_3,\bar{\bar g}_3,g)\bigl((\bar g_3g,g_3)\in \chi_1 \wedge (g_2\bar g_2,\bar g_3)\in\)
   \[ \in \chi_1 \wedge \gamma b_{g_1\bar g_2}\in \Gamma_1 \wedge (\bar g_2\bar{\bar g}_2,\bar{\bar g}_3g)\in \chi_1 \wedge \gamma b_{g_1\bar{\bar g}_2}\in \Gamma_1 \wedge (\bar{\bar g}_3,\bar g_3)\in \xi\bigr)\to \]
   \[ \to \gamma b_{g_1}a_{g_2}b_{g_3}\in \Gamma_1 \]
   (\(\bar g_2,\bar{\bar g}_2,g\) may be empty symbols).

5. \(\gamma b_{g_1}a_{g_2}b_{g_3}\in \Gamma_1 \wedge (\exists \tilde g_3,\ldots,\bar{\bar g}_3,\bar g_3)\bigl((g_3,\tilde g_3)\in \xi \wedge \cdots \wedge (\bar{\bar g}_3,\bar g_3)\in \xi \wedge\)
   \[ \wedge \gamma b_{g_1}a_{g_2}b_{\tilde g_3}\in \Gamma_2 \wedge \cdots \wedge \gamma b_{g_1}a_{g_2}b_{\bar g_3}\in \Gamma_1 \wedge \bigl((\bar g_3,g_4)\in \chi_2 \vee (\exists \bar g_2)(\bar g_3=g_2\bar g_2 \wedge \]
   \[ \wedge \gamma b_{g_1\bar g_2}a_{g_4}\in \Gamma_1)\bigr)\bigr)\to \gamma b_{g_1}a_{g_2}b_{g_3}a_{g_4}\in \Gamma_1 \]
   (\(\bar g_2\) may be an empty symbol).

6. There are no other words in \(\Gamma_1\).

Suppose now that \(\Gamma_n\) has been constructed. Then:

1. \(\gamma b_{g_1}a_{g_2}b_{g_3}\in \Gamma_n \wedge g_3=g_2\bar g_2\to \gamma b_{g_1\bar g_2}\in \Gamma_{n+1}\) (\(\bar g_2\) may be empty).

2. \(\gamma b_{g_1}a_{g_2}b_{g_3}a_{g_4}\in \Gamma_n \wedge g_3=g_2\bar g_2\to \gamma b_{g_1\bar g_2}a_{g_4}\in \Gamma_{n+1}\) (\(\bar g_2\) may be empty).

3. \(c_{(g_1,g_2)}b_{g_3}\in \Gamma_{n+1}\wedge (\exists g_3,\ldots,\bar{\bar g}_3,\tilde g_3)\bigl((g_3,\bar g_3)\in \xi \wedge \cdots \wedge (\bar{\bar g}_3,\tilde g_3)\in\)
   \[ \in \xi \wedge c_{(g_1,g_2)}b_{\bar g_3}\in \Gamma_{n+1}\wedge \cdots \wedge c_{(g_1,g_2)}b_{\bar{\bar g}_3}\in \Gamma_{n+1}\wedge (\tilde g_3,g_4)\in \chi_2\bigr)\to \]
   \[ \to c_{(g_1,g_2)}b_{g_3}a_{g_4}\in \Gamma_{n+1}. \]

4—5. As in the definition of \(\Gamma_1\), with \(\Gamma_1\) replaced everywhere by \(\Gamma_{n+1}\).

1. There are no other words in \(\Gamma_{n+1}\).

Finally, put

\[ \Gamma=\bigcup_{n=1}^{\infty}\Gamma_n. \]

On the set \(\Gamma\) define a binary relation \(\varepsilon_1\), setting \((\gamma_1,\gamma_2)\in \varepsilon_1\) if and only if

\[ \gamma_1=\gamma b_{g_1}a_{g_2}b_{g_3}\ldots a_{g_{2n}}b_{g_{2n+1}}(a_{g_{2n+2}}), \quad \gamma_2=\gamma b_{\tilde g_1}a_{g_2}b_{\tilde g_3}\ldots a_{g_{2n}}b_{\tilde g_{2n+1}}(a_{g_{2n+2}}) \]

and \((g_{2k+1},\tilde g_{2k+1})\in \xi\) \((k=0,1,\ldots,n)\).

Lemma 1. If \(\mathbf I\Delta_{\mathfrak G}\subset \xi\) and \(\Pi_{\xi}^{-1}\circ \subset \xi\) (i.e. \(\xi\) is reflexive and symmetric), then
\[ \bar\varepsilon_1=\bigcup_{n=1}^{\infty}\varepsilon_1^n \]
is an equivalence relation.

Using \(\bar\varepsilon_1\), we introduce in \(\Gamma\) one more binary relation. Namely, let
\[ \gamma_1=\bar\gamma b_{g_1}a_{g_2}b_{g_3}\ldots b_{g_{2n-1}}a_{g_{2n}}b_{g_{2n+1}}\bar{\bar\gamma}, \]

\[ \gamma_2=\tilde{\gamma} b_{g_1}a_{g_2}b_{g_3}\ldots b_{g_{2m-1}}a_{g_{2m}}b_{g_{2m+1}}\bar{\gamma} \quad\text{and}\quad g_{2n+1}=g_{2n}\tilde{\delta}_{2n},\quad g_{2n-1}\tilde{\delta}_{2n} \]
\[ =g_{2n-2}\tilde{\delta}_{2n-2},\ldots,\quad g_3\tilde{\delta}_4=g_2\tilde{\delta}_2,\quad \bar{g}_{2m+1}=\bar{g}_{2m}\bar{\delta}_{2m},\quad \bar{g}_{2m-1}\bar{\delta}_{2m} \]
\[ =\bar{g}_{2m-2}\bar{\delta}_{2m-2},\ldots,\quad \bar{g}_3\bar{\delta}_4=\bar{g}_2\bar{\delta}_2 \]
\[ (\tilde{\delta}_{2n},\tilde{\delta}_{2n-2},\ldots,\tilde{\delta}_4,\tilde{\delta}_2,\bar{\delta}_{2m},\bar{\delta}_{2m-2},\ldots,\bar{\delta}_4,g_2 \]
may be empty symbols),
\[ (\bar{\gamma} b_{g_1\bar{g}_2}\gamma,\gamma b_{g_1\bar{g}_2}\bar{\gamma})\in\bar{\varepsilon}_1. \]
In this and only in this case we regard \((\gamma_1,\gamma_2)\in\bar{\varepsilon}_2\). Putting
\[ \bar{\varepsilon}_2=\bigcup_{n=0}^{\infty}\varepsilon_2^n, \]
we construct the fundamental relation on the set \(\Gamma\):
\[ \varepsilon=\bigcup_{n=1}^{\infty}\bar{\varepsilon}_2\circ\bar{\varepsilon}_1\ \cup\ \bar{\varepsilon}_1\circ\bar{\varepsilon}_2. \]
By Lemma 1 and the definition of \(\bar{\varepsilon}_2\), \(\varepsilon\) will be an equivalence relation. \(\varepsilon\)-equivalent words will be regarded as different writings of one and the same word.

Introduce the notation
\[ \gamma\circ b_g= \begin{cases} \gamma_0 b_{g_1}a_{g_2}b_g, & \text{if } \gamma=\gamma_0 b_{g_1}a_{g_2},\\ \gamma_0 b_{g_1g}, & \text{if } \gamma=\gamma_0 b_{g_1},\\ c_{(g_1,g_2)}b_g, & \text{if } \gamma=c_{(g_1,g_2)}; \end{cases} \]
and \(\gamma\circ a_g=\gamma_0 b_{g_1}a_g\), if \(\gamma=\gamma_0 b_{g_1}\).

Lemma 2. If
\[ \mathrm{III}\quad (g_1,g_2)\in\xi\ \wedge\ (g_3,g_4)\in\xi \to (g_1g_3,g_2g_4)\in\xi \]
(i.e. \(\xi\) is stable), then from \((\gamma_1,\gamma_2)\in\varepsilon\) it follows, under the condition \((g_1,g_2)\in\xi\), that
\[ (\gamma_1\circ b_{g_1},\gamma_2\circ b_{g_2})\in\varepsilon, \]
if both these words belong to \(\Gamma\).

Now put \((\gamma_1,\gamma_2)\in\omega\) if and only if:
1. \(\gamma_2=c_{(g_2,g_2)}\) and \(\gamma_1\circ b_{g_2}\in\Gamma\).
2. \(\gamma_2=c_{(g_2,g_2)}b_{g_2}\) and \(\gamma_1\circ a_{g_2}\in\Gamma\).

Lemma 3. If:
\[ \mathrm{IV}\quad c_{(g_1,g_1)}b_{g_2}\in\Gamma\to (g_1,g_2)\in\chi_1, \]
\[ \mathrm{V}\quad \chi_1^2\subset\chi_1 \quad\text{(i.e. \(\chi_1\) is transitive),} \]
\[ \mathrm{VI}\quad (g_1,g_2)\in\chi_1\to(g_3g_1,g_3g_2)\in\chi_1 \quad\text{(i.e. \(\chi_1\) is left regular),} \]
\[ \mathrm{VII}\quad c_{(g_1,g_1)}b_{g_1}a_{g_2}\in\Gamma\to(g_1,g_2)\in\chi_2, \]
\[ \mathrm{VIII}\quad \chi_2^2\subset\chi_2 \quad\text{(i.e. \(\chi_2\) is transitive),} \]
then the binary relation \(\omega\) is transitive.

Proof. Let \((\gamma_1,\gamma)\in\omega\) and \((\gamma,\gamma_2)\in\omega\).

1. \(\gamma_2=c_{(g_2,g_2)}\), \(\gamma=c_{(g,g)}\), and \(c_{(g,g)}b_{g_2}\in\Gamma\). According to IV, \((g,g_2)\in\chi_1\);

a) \(\gamma_1=\gamma_0 a_{g_1}\), \(\gamma_0a_{g_1}b_g\in\Gamma\). By V and the definition of \(\Gamma\), \(\gamma_0a_{g_1}b_{g_2}\in\Gamma\);

b) \(\gamma_1=\gamma_0b_{g_1}\), \(\gamma_0b_{g_1g}\in\Gamma\). In view of VI, \((g_1g,g_1g_2)\in\chi_1\), whence \((\gamma_1,\gamma_2)\in\omega\).

1. \(\gamma_2=c_{(g_2,g_2)}\), \((\gamma,c_{(g,g)}b_g)\in\varepsilon\), \(\gamma\circ b_{g_2}\in\Gamma\). By Lemma 2,
   \[ (\gamma\circ b_{g_2},\ c_{(g,g)}b_g\circ b_{g_2})\in\varepsilon. \]
   Here
   \[ c_{(g,g)}b_{gg_2}=c_{(g,g)}b_g\circ b_{g_2} \]
   belongs to \(\Gamma\) together with \(\gamma\circ b_{g_2}\), by the definition of \(\Gamma\). Then IV gives \((g,gg_2)\in\chi_1\). If \(\gamma_1=\gamma_0b_{g_1}\), then, by the condition, \(\gamma_0b_{g_1}a_g\in\Gamma\). Obviously,
   \[ \gamma_0b_{g_1}a_gb_{gg_2}\in\Gamma, \]
   and then also
   \[ \gamma_0b_{g_1}\circ b_{g_2}=\gamma_0b_{g_1g_2}\in\Gamma. \]

2. \(\gamma_2=c_{(g_2,g_2)}b_{g_2}\), \((\gamma,c_{(g,g)}b_g)\in\varepsilon\), \(\gamma\circ a_{g_2}\in\Gamma\). Then
   \[ c_{(g,g)}b_ga_{g_2}\in\Gamma, \]
   i.e., by VII, \((g_1,g_2)\in\chi_2\). Since \(\gamma_1=\gamma_0b_{g_1}\), from \(\gamma_0b_{g_1}a_g\in\Gamma\), according to VIII, we obtain
   \[ \gamma_0b_{g_1}a_{g_2}\in\Gamma. \]
   The lemma is proved.

Lemma 4. If
\[ \mathrm{IX}\quad (g_1g_2,g_1)\in\chi \]
(i.e. \(\chi\) is right-negative), then
\[ \gamma\in\omega^{-1}\langle c_{(g_1g_2,g_1g_2)}\rangle \leftrightarrow \gamma\in\omega^{-1}\langle c_{(g_1,g_1)}\rangle\ \wedge\ \gamma\circ b_{g_1}\in\omega^{-1}\langle c_{(g_2,g_2)}\rangle. \]

Lemma 5. If
\[ \mathrm{X}\quad \Delta_{\mathfrak{G}}\subset\chi_2 \quad\text{(i.e. \(\chi_2\) is reflexive),} \]
\[ \mathrm{XI}\quad (g_1g_2,g_2)\in\chi_2 \quad\text{(i.e. \(\chi_2\) is left-negative),} \]
\[ \mathrm{XII}\quad \Delta_{\mathfrak{G}}\subset\chi_1 \quad\text{(i.e. \(\chi_1\) is reflexive),} \]
then
\[ \omega^{-1}\langle c_{(g,g)}\rangle\circ b_g = \omega^{-1}\langle c_{g,g}b_g\rangle. \]

Proof. Let \(\gamma_0 \in \omega^{-1}\langle c_{(g,g)}\circ b_g\rangle\), and moreover \(\gamma_0=\gamma_0\circ b_g\). If \(\gamma_0=\gamma_0\circ a_{g_0}\), then \(\gamma \in \omega\langle c_{(g,g)}b_g\rangle\) by X; and if \(\gamma_0=\gamma_0 b_{g_0}\), then we apply XI. Conversely, assuming \(\gamma\in \omega\langle c_{(g,g)}b_g\rangle\), i.e. \(\gamma\circ a_g\in \Gamma\), we have, in view of XII, \(\gamma a_g b_g\in\Gamma\), but \(\gamma a_g b_g=\gamma a_g\circ b_g\), and \(\gamma a_g\in \omega^{-1}\langle c_{(g,g)}\rangle\), and \((\gamma,\gamma a_gb_g)\in\varepsilon\).

Theorem 1. If in a relativized semigroup \((\mathfrak G,\circ,\xi,\chi_1,\chi_2)\) the binary relations \(\xi,\chi_1,\chi_2\) satisfy conditions I–XII and XIII
\[ (c_{(g_1,g_2)}b_{g_1},\,c_{(g_1,g_2)}b_{g_2})\in\varepsilon \to (g_1,g_2)\in\xi, \]
then it admits a homomorphism into some transformative semigroup of transformations.

Proof. Construct a mapping \(P\) of the set \(\mathfrak G\) into the set \(\mathfrak F(\Gamma\times\Gamma)\) of all transformations of \(\Gamma\), putting
\[ pr_1P(g)=\omega^{-1}\langle c_{(g,g)}\rangle \]
and
\[ P(g)(\gamma)=\gamma\circ b_g. \]
Lemma 4 and the stability of \(\xi\) (condition III) ensure the equality
\[ P(g_1g_2)=P(g_2)\circ P(g_1) \]
(we recall that equality of words from \(\Gamma\) is understood in the sense of their \(\varepsilon\)-equivalence). Further,
\[ pr_1P(g_1)\subset pr_1P(g_2) \]
is equivalent, by definition, to the inclusion
\[ \omega^{-1}\langle c_{(g_1,g_1)}\rangle \subset \omega^{-1}\langle c_{(g_2,g_2)}\rangle, \]
which, in turn, in view of Lemma 3 and the reflexivity of \(\chi_2\) (condition XII), is replaced by the relation
\[ c_{(g_1,g_1)}\in\omega^{-1}\langle c_{(g_2,g_2)}\rangle, \]
i.e.
\[ c_{(g_1,g_1)}b_{g_2}\in\Gamma. \]
But the latter, according to IV, is equivalent to
\[ (g_1,g_2)\in\chi_1. \]
Analogously, Lemmas 4 and 5, the reflexivity of \(\chi_2\) (condition X), and condition VII ensure the equivalence of the relations
\[ pr_2P(g_1)\subset pr_2P(g_2) \]
and
\[ (g_1,g_2)\in\chi_2. \]
Let now \((g_1,g_2)\in\xi\). Then from Lemma 2 it follows that, if
\[ \gamma\in pr_1P(g_1)\cap pr_1P(g_2), \]
then
\[ (P(g_1)(\gamma),\,P(g_2)(\gamma))\in\varepsilon, \]
i.e. \(P(g_1)P(g_2)\). Conversely, if \(P(g_1)\) and \(P(g_2)\) are half-compatible, then, in particular,
\[ P(g_1)(c_{(g_1,g_2)})=P(g_2)(c_{(g_1,g_2)}), \]
i.e.
\[ (c_{(g_1,g_2)}b_{g_1},\,c_{(g_1,g_2)}b_{g_2})\in\varepsilon, \]
whence, by XIII, \((g_1,g_2)\in\xi\). The theorem is proved.

Theorem 2. In order that the relativized semigroup \((\mathfrak G,\circ,\xi,\chi_1,\chi_2)\) be transformative, it is necessary and sufficient that the binary relations \(\xi,\chi_1,\chi_2\) satisfy conditions I–XIII and XIV
\[ \xi\cap\chi_1\cap\chi_1^{-1}\subset \Delta_{\mathfrak G}. \]

Proof. Sufficiency. Under condition XIV the mapping \(P\) defined in the preceding theorem will be one-to-one. Indeed, if
\[ P(g_1)=P(g_2), \]
then
\[ (g_1,g_2)\in\chi_1\cap\chi_1^{-1} \]
(since
\[ pr_1P(g_1)=pr_1P(g_2) \])
and
\[ (g_1,g_2)\in\xi \]
(since \(P(g_1)P(g_2)\)). Hence
\[ g_1=g_2 \]
according to XIV.

The necessity of all the conditions except IV, VIII, XIII is known \((^3,^4)\). The proof of the necessity of IV, VIII, XIII involves certain technical difficulties.

Saratov State University

Received
30 IV 1967

CITED LITERATURE

1. V. V. Wagner, Mat. sbornik, 38 (80), 203 (1956).
2. V. V. Wagner, Izv. vyssh. uchebn. zaved., Matematika, 4, 36 (1960).
3. V. N. Salii, Izv. vyssh. uchebn. zaved., Matematika, 3, 144 (1966).
4. V. N. Salii, Volzhsk. matem. sbornik, issue 5, 284 (1966).
5. B. M. Schein, Mat. sbornik, 65 (107), 188 (1964).
6. B. M. Schein, Mat. sbornik, 75 (113), 65 (1966).