UDC 519.48

MATHEMATICS

I. S. IVANOV

FREE \(T\)-SUMS OF MULTIOPERATOR FIELDS

(Presented by Academician A. I. Mal’cev on 9 VI 1965)

The theory, constructed by L. A. Skornyakov \((^{2,3})\), of free \(T\)-extensions of linear algebras is transferred in the present note to a broader class of multioperator algebras introduced by A. G. Kurosh \((^1)\). Theorem 2 gives a positive solution to the question of the existence of isomorphic continuations for any two free \(T\)-decompositions of a multioperator, in particular nonassociative, field. For a nonassociative field this question was not considered in \((^3)\), although it is natural after Theorem 2 and Theorem § 5 of \((^3)\). Theorem 1 generalizes the analogous theorem of \((^3)\). The incorrectness of the original proof of this theorem (in \((^3)\)) is noted.

Below, a nonempty system of operations \(\Omega=\{\omega,\ldots\}\), \(n(\omega)\ge 2\), and a field \(P\) are assumed fixed.

§ 1. An \(\Omega\)-algebra \(A\) over the field \(P\) \((^1)\) is called an \(\Omega\)-algebra with divisions (a multioperator field, or \(\Omega\)-field) if, for each \(\omega\in\Omega\), every equation of the form

\[ a_1\ldots a_{i-1}xa_{i+1}\ldots a_n\omega=c,\qquad 0\ne a_j,\ c\in A, \tag{*} \]

is solvable (uniquely solvable) in \(A\) for \(i=1,2,\ldots,n=n(\omega)\).

A single-valued mapping \(\theta\) of an \(\Omega\)-algebra \(A\) into an \(\Omega\)-algebra \(K\) augmented formally by the symbol \(\infty\) is called a \(T\)-homomorphism \((^2)\) if \(\theta^{-1}(0)\) is nonempty and the following conditions are satisfied:

Г1. If \(a_i\in A\), \(\xi_j\in P\), \(\theta(a_i)\ne\infty\), \(i=1,2,\ldots,n\), then
\[ \theta(\xi_1a_1+\xi_2a_2)=\xi_1\theta(a_1)+\xi_2\theta(a_2),\quad \theta(a_1\ldots a_n\omega)=\theta(a_1)\ldots\theta(a_n)\omega,\quad \omega\in\Omega,\ n=n(\omega). \]

Г2. If \(\theta(a_i)\ne 0\), \(i=1,2,\ldots,n\), and for at least one \(1\le i_0\le n\)
\[ \theta(a_{i_0})=\infty, \]
then
\[ \theta(a_1\ldots a_{i_0}\ldots a_n\omega)=\infty. \]

It is immediately verified that the \(T\)-homomorphic image of an \(\Omega\)-algebra with divisions is an \(\Omega\)-algebra with divisions. The successive performance of two \(T\)-homomorphisms is again a \(T\)-homomorphism, if, by definition, one sets \(\theta(\infty)=\infty\).

§ 2. Let a \(T\)-homomorphism \(\theta\) and an equation \((*)\), not solvable in the \(\Omega\)-algebra \(A\), be fixed. A basis \(\Sigma=\Phi\cup\Psi\) of the additive vector space \(A^+\) of the \(\Omega\)-algebra \(A\) will be called \((\theta,*)\)-regular if: 1) \(\Phi\) is a basis of the subspace \(E(\Phi)=\{a\in A^+\mid \theta(a)\ne\infty\}\); 2) for all \(a\in E(\Psi)\setminus 0\), \(\theta(a)=\infty\); 3) the subspace
\[ L=E(\{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n\}), \]
where the \(a_j\) are the terms of equation \((*)\), has the basis set
\[ \Sigma\cap L=\Sigma'. \]

In the basis \(S\), constructed over the set \(\Sigma\cup\lambda\) \((^2)\), of the free sum
\[ A_\lambda=A*F(\lambda) \]
of the \(\Omega\)-algebra \(A\) and the free \(\Omega\)-algebra \(F(\lambda)\) with one free generator \(\lambda\), we fix
\[ b_1=a'_1\ldots a'_{i-1}\lambda a'_{i+1}\ldots a'_n\omega,\qquad a'_j\in\Sigma', \]
\[ E(\{a'_1,\ldots,a'_{i-1},a'_{i+1},\ldots,a'_n\})=L. \]
The word \(b_1\) occurs as a summand in the \(S\)-canonical representation of the element
\[ p=a_1\ldots a_{i-1}\lambda a_{i+1}\ldots a_n\omega-c \]
corresponding to equation \((*)\). Then, under the natural epimorphism
\[ A_\lambda\to B=A_\lambda/I, \]
where \(I\) is the ideal of the \(\Omega\)-algebra \(A_\lambda\) generated by the element \(p\), the set
\[ S^{b_1}=\{x\in S\mid l_{b_1}(x)=0\} \]
of elements of \(b_1\)-length zero \((^2)\) is mapped one-to-one onto a basis of the vector space \(B^+\) of the \(\Omega\)-algebra \(B\). In the \(\Omega\)-algebra \(B\supset A\), equation \((*)\) is solvable (see \((^2)\)). Using

basis \(S^{b_1}\), one can carry over to the case of multioperator algebras the proof of Lemma 6 of \((^2)\), stating that every \(T\)-homomorphism of an \(\Omega\)-algebra \(A\) into an \(\Omega\)-algebra \(K\) with divisions can be extended to the obtained extension \(B\). Hence, as in \((^2)\), we obtain:

Proposition 1. Every \(\Omega\)-algebra \(A\) over a field \(P\) possesses a free \(T\)-extension \(\mathfrak A \supset A\) in the sense of \((^2)\), uniquely determined up to an isomorphism identical on \(A\). If, moreover, for any \(0 \ne x_j \in A\), \(\omega \in \Omega\), one has \(x_1 \ldots x_n\omega \ne 0\), then \(\mathfrak A\) is an \(\Omega\)-field.

Corollary 1. Every \(\Omega\)-algebra can be embedded in an \(\Omega\)-algebra with divisions.

This corollary generalizes B. H. Neumann’s theorem \((^4)\), proved for linear algebras.

Proposition 1 makes it possible to define the concepts (see \((^{2,3})\)): free \(\Omega\)-field, free \(T\)-sum of \(\Omega\)-fields. The following propositions are true and are immediate generalizations of theorems from \((^{2,3})\).

Proposition 2. Every \(\Omega\)-field is a \(T\)-homomorphic image of some free \(\Omega\)-field.

Proposition 3. An \(\Omega\)-subfield \(\mathfrak B\) of the free \(T\)-sum \(\sum_T^{T}\mathfrak A_\sigma * \mathfrak F\) is decomposable into the free \(T\)-sum

\[ \mathfrak B = \sum_T^{T}(\mathfrak B \cap \mathfrak A_\sigma) * \mathfrak F_1, \]

where \(\mathfrak F\) and \(\mathfrak F_1\) are free \(\Omega\)-fields.

For free \(T\)-sums of multioperator fields, besides properties analogous to I—V in \((^3)\), the following also hold:

A1 (validity of Mal'cev’s postulate). If in each of the \(\Omega\)-fields \(\mathfrak A_\alpha\) an \(\Omega\)-subfield \(\mathfrak B_\alpha\) is taken and \(\mathfrak A = \sum_T^{T}\mathfrak A_\alpha\), then the \(\Omega\)-subfield \(\mathfrak B\) generated by all \(\mathfrak B_\alpha\) is their free \(T\)-sum and \(\mathfrak B \cap \mathfrak A_\alpha = \mathfrak B_\alpha\).

A2. If \(M\) is a finite subset of the \(\Omega\)-field \(\mathfrak A = \sum_T^{T}\mathfrak A_\alpha\), then there exists a finite set of indices \(\alpha_1,\ldots,\alpha_n\) such that

\[ M \subset \sum_{i=1}^{n\,T}\mathfrak C_{\alpha_i}, \]

where \(\mathfrak C_{\alpha_i}\) is an \(\Omega\)-subfield of the \(\Omega\)-field \(\mathfrak A_{\alpha_i}\). Each \(\mathfrak C_{\alpha_i}\) has, as an \(\Omega\)-field, a finite number of generators.

§ 3. Let, for the \(\Omega\)-field \(\mathfrak A\), there be a free \(T\)-decomposition:

\[ \mathfrak A = \sum_T^{T}\mathfrak A_\sigma. \]

Then, by the definition of the free \(T\)-sum,

\[ \Sigma^*\mathfrak A_\sigma = A_1 \subset \ldots \subset A_\beta \subset \ldots \subset A_\theta = \mathfrak A. \]

In this transfinite sequence of extensions, for every \(0 \ne x \in \mathfrak A\) there is uniquely determined a height \(h(x)\)—such an indefinite ordinal number \(\alpha \geqslant 1\) that \(x \in A_\alpha \setminus A_{\alpha-1}\) (here \(A_0\) is to be understood as the empty set), and a degree \(s_\alpha(x)\) \((^2)\).

The \(\Omega\)-subfield generated by a set \(Z \subset \mathfrak A\) will be denoted by \(T(Z)\). Suppose that, as an \(\Omega\)-field, \(\mathfrak A\) is generated by some finite subset of it. An ordered set \(X=\{x_i \in \mathfrak A \mid x_i \ne 0,\ i=1,\ldots,m\}\) will be called a finite system of generators (f.s.g.) of the \(\Omega\)-field \(\mathfrak A\), if \(T(X)=\mathfrak A\) and \(h(x_i)\leq h(x_{i+1})\), \(i=1,2,\ldots,m-1\). If the subset \(X_\alpha=X\cap(A_\alpha\setminus A_{\alpha-1})\) of the f.s.g. \(X\) is nonempty, then by \(s_\alpha(X)\) we shall denote the sum \(\sum_i s_\alpha(x_i)\), \(x_i\in X_\alpha\).

On the set \(\mathfrak P\) of all f.s.g. of the \(\Omega\)-field \(\mathfrak A\) we define a quasi-ordering. Let \(X=\{x_1,\ldots,x_m\}\), \(Y=\{y_1,\ldots,y_n\}\in\mathfrak P\). By definition, we set \(X\leq Y\) if one of the following conditions is satisfied: 1) \(m<n\), 2) \(m=n,\ h(x_i)\leq h(y_i)\), \(i=1,2,\ldots,n\), and at least one of the inequalities is strict; 3) \(m=n,\ h(x_i)=h(y_i),\ s_{\alpha_i}(X)\leq s_{\alpha_i}(Y)\) for all \(\alpha_i=h(x_i)\), \(i=1,2,\ldots,n\). It is easily verified that the relation thus defined is reflexive and transitive and therefore is a partial ordering on the set \(\mathfrak P\) of classes of equivalent f.s.g.: \(X\sim Y\) if \(m=n\), \(h(x_i)=h(y_i)\), \(s_{\alpha_i}(X)=s_{\alpha_i}(Y)\) for all \(\alpha_i=h(x_i)\), \(i=1,2,\ldots,n\). Since the minimality condition holds for \(\mathfrak P\), we thus arrive at the definition of minimal f.s.g. (m.f.s.g.) of \(\Omega\)-fields \(\mathfrak A\). It is clear that

the number of elements in any two m.k.s.o. is the same and does not depend on the \(T\)-decomposition under consideration.

Denote by \(n(X)\) and \(r(X)\), respectively, the number of elements of a finite set \(X\) of the \(\Omega\)-group \(\mathfrak A=\Sigma^T \mathfrak A_\sigma\) and of its subset \(X\setminus \left(\bigcup_\sigma \mathfrak A_\sigma\right)\).

Let, further,
\[ \bar A_0=\{x\in A_1\mid s_1(x)=0\}, \]
where \(A_1=\Sigma^* \mathfrak A_\sigma\); \(X_0=X\cap \bar A_0\). If \(X\) is a k.s.o. of the \(\Omega\)-group \(\mathfrak A\) and there exists a finite set \(\bar X_0\subset \bar A_0\) such that
\[ T(\bar X_0)=T(X_0),\qquad r(\bar X_0)<r(X_0),\qquad n(\bar X_0)\leq n(X_0)+1, \]
then the passage to the k.s.o. \(X'=(X\setminus X_0)\cup \bar X_0\) will be called an \(r\)-reduction. Any k.s.o. \(X\), in no more than \(r(X_0)\) steps, is brought to an \(r\)-irreducible form that admits no \(r\)-reductions.

Let \(Y\) be an m.k.s.o. of the \(\Omega\)-group \(\mathfrak A\), and let \(X\) be the \(r\)-irreducible k.s.o. obtained from \(Y\). Denote it by \(X=\pi(Y)\) and call it a regular k.s.o. of the \(\Omega\)-group \(\mathfrak A\).

As an analogue of Grushko’s theorem, the following has been obtained.

Proposition 4. The elements of a regular k.s.o. \(X=\pi(Y)\) belong to separate \(T\)-summands.

The proof of this proposition is based on a specialization of the methods, developed in papers \((^1,^3,^5)\), for constructing bases in the \(\Omega\)-subalgebra of a free sum of \(\Omega\)-algebras.

Corollary 2. An \(\Omega\)-group with \(n\) generators decomposes into a free \(T\)-sum of no more than \(2n\) \(T\)-indecomposable finitely generated \(\Omega\)-subgroups.

Corollary 3. A free \(\Omega\)-group with \(n\) free generators is not decomposable into a free \(T\)-sum containing more than \(n\) summands.

Hence, using property A2, we arrive at theorem *:

Theorem 1. The cardinality of the set of free generators of a free \(\Omega\)-group is an invariant of this \(\Omega\)-group.

Theorem 2. Any two free \(T\)-decompositions of an \(\Omega\)-group have isomorphic refinements.

To prove Theorem 2, in view of Proposition 3 and properties III and IV for free \(T\)-sums (see \((^3)\)), it suffices to establish an isomorphism of the free \(\Omega\)-groups \(\mathfrak F_1\) and \(\mathfrak F_2\) for any two free \(T\)-decompositions:
\[ \mathfrak C *_T \mathfrak F_1=\mathfrak C *_T \mathfrak F_2, \]
where \(\mathfrak C\) is an arbitrary \(\Omega\)-group.

Let \(M_1\) and \(M_2\) be some fixed systems of free generators of the free \(\Omega\)-groups \(\mathfrak F_1\) and \(\mathfrak F_2\), and suppose
\[ \operatorname{card} M_1<\operatorname{card} M_2, \tag{**} \]
and \(M_2\) is an infinite set. Then, using A2, by the usual argument we arrive at a contradiction. If \(\mathfrak C\) is generated, as an \(\Omega\)-group, by a finite set and if \(M_2\) is finite, then \((**)\) leads to a contradiction with Corollary 2. But when \(M_1,M_2\) are finite, by virtue of property A2 one may restrict oneself only to the case considered, that of a finitely generated \(\Omega\)-group \(\mathfrak C\). Theorem 2 is proved.

The work was carried out under the supervision of A. G. Kurosh, to whom I take this opportunity to express my deep gratitude.

Moscow State University
named after M. V. Lomonosov

Received
29 V 1965

REFERENCES

\(^1\) A. G. Kurosh, Sibirsk. matem. zhurn., 1, No. 1, 62 (1960).
\(^2\) L. A. Skornyakov, Matem. sborn., 42, No. 4, 425 (1957).
\(^3\) L. A. Skornyakov, Matem. sborn., 44, No. 3, 297 (1958).
\(^4\) B. H. Neumann, Proc. London Math. Soc., 1, No. 1, 241 (1951).
\(^5\) A. G. Kurosh, Matem. sborn., 20, No. 3, 239 (1947).

* In paper \((^3)\) this is Theorem § 5, formulated for a free nonassociative ring. The proof given there is unconvincing, since it contains an incorrect reference (p. 309) to paper \((^5)\). As for Theorem 5 of \((^3)\), which is also affected by the noted incorrectness, the question of the validity of this theorem has not been clarified. Corollary 2 of the present note contains a weaker assertion.