UDC 519.48

MATHEMATICS

L. M. GLUSKIN

ON POSITIONAL OPERATIVES

(Presented by Academician V. M. Glushkov, 23 II 1968)

In the present note simple positional operatives are studied \((^{1,2})\). In particular, the theory of completely simple semigroups of Rees—Sushkevich—Clifford \((^{3,4})\) is generalized to this class of algebras.

1. Let \(\mathfrak P=\{\rho_k\}_{k=1}^n\) be a family of permutations \(\rho_k\) of the set \(J_{2n-1}=\{1,2,\ldots,2n-1\}\), with \(\rho_1=\rho_n=\varepsilon\) (\(\varepsilon\) is the identity permutation). An operative \(S\) is called a \(\mathfrak P\)-operative if, for any \(k\in J_n=\{1,2,\ldots,n\}\), \(x_i\in S\),

\[ x_1x_2\cdots x_{k-1}(x_kx_{k+1}\cdots x_{k+n-1})x_{k+n}\cdots x_{2n-1} = x_{\rho_k1}x_{\rho_k2}\cdots x_{\rho_k(2n-2)}x_{\rho_k(2n-1)}. \tag{1} \]

\(S\) is called a \(\pi\)-operative if, for every \(k\in J_n\),
\(\rho_k\{k,k+1,\ldots,k+n-1\}=\{k,k+1,\ldots,k+n-1\}\). In other words, with a \(\pi\)-operative \(S\) one can associate a family \(\pi=\{\sigma_k,\pi_k\}_{k=1}^n\) of permutations \(\sigma_k,\pi_k\) of the set \(J_n\) such that \(\pi_1=\pi_n=\sigma_1=\sigma_n=\varepsilon\), \(\sigma_k k=k\), and for any \(x_i,y_i\in S\)

\[ x_1x_2\cdots x_{k-1}(y_1y_2\cdots y_n)x_{k+1}\cdots x_n = x_{\sigma_k1}x_{\sigma_k2}\cdots x_{\sigma_k(k-1)}y_{\pi_k1}\cdots y_{\pi_kn}x_{\sigma_k(k+1)}\cdots x_{\sigma_kn}. \]

A \(\pi\)-operative is called associative if \(\sigma_k=\pi_k=\varepsilon\) for all \(k\in J_n\); alternative if \(n\) is odd and all \(\sigma_k=\pi_{2j+1}=\varepsilon\), \(\pi_{2j}=\tau\), where \(\tau i=n-i+1\).

A subset \(A\) of an operative \(S\) which is a \((k)\)-ideal \((^{1,2})\), for \(k=1,2,\ldots,n\), is called an ideal; for \(k=1\) and \(k=n\), a two-sided ideal. An operative \(S\) is called \(i\)-simple if it contains no ideals distinct from zero and \(S\) and is not isomorphic to the operative \(V_n=\{a,0\}\), where \(0\) is zero and \(a^n=0\).

1. Theorem 1. Every \(i\)-simple \(\mathfrak P\)-operative \(S\) with zero \(0\) is the union of its two-sided ideals \(A_\gamma\): \(S=\bigcup_\gamma A_\gamma\), the number of which \(|\Gamma|\) is a divisor of the number \(n-1\); \(A_\gamma\cap A_\delta=\{0\}\) for \(\gamma\ne\delta\). If \(T\) is an \(i\)-simple \(\mathfrak P\)-operative without zero, then \(T\) contains no proper two-sided ideals.

Let \(a\) be some fixed element of \(S\setminus\{0\}\) and let \(a=a_1a_2\cdots a_n\), where \(a_j\in S\). Denote by \(A_j\) that one of the two-sided ideals \(A_\gamma\) which contains the element \(a_j\); by \(\xi_{ki}\), the least positive residue of the number \(\rho_k^{-1}(i+k-1)\) modulo \(n-1\).

Theorem 2. For any \(k,j\in J_{n-1}=\{1,2,\ldots,n-1\}\),

\[ A_k=A_{\xi_{k1}}A_{\xi_{k2}}A_{\xi_{k3}}\cdots A_{\xi_{kn}},\qquad A_{\xi_{k1}}=A_{\xi_{kn}}=A_k, \]

\[ A\,A_{k_2}A_{k_3}\cdots A_{k_n}=\{0\},\qquad \text{if } k_j\ne \xi_{kj}. \]

1. Introduce the following equivalence \(\delta\) on the set \(J_{n-1}\): \(j\delta k\) if and only if \(A_j=A_k\).

Theorem 3. \((\xi_{j}\xi_{ki})\delta(\xi_{\xi k i})\) for any \(i,j,k\in J_n\); all equivalence classes of \(\delta\) consist of one and the same number of elements.

Denote by \(l\) the least positive number such that \(A_{l+1}=A_1\) and \(\pi_{l+1}=\varepsilon\). For associative operatives the equivalence \(\delta\) turns out to be congruence modulo \(l\). If for an alternative operative \(\delta\) is not congruence

mod \(l\), then \(A_i\) can be chosen so that \(j\delta k\) if and only if \(j \equiv k(\operatorname{mod} l)\) or \(j+k \equiv 1(\operatorname{mod} l)\).

1. The theorem below makes it possible to carry over to \(\pi\)-operatives the theory of ideal factors of semigroups \((^{3,4})\):

Theorem 4. Let \(M\) be a minimal ideal\(^*\) of a \(\pi\)-operative \(S\). Then either \(M^n=\{0\}\), or \(M\) is an \(i\)-simple operative.

1. An \(i\)-simple operative \(S\) is called \(j\)-simple if \(S^{k-1}aS^{n-k}\ne\{0\}\) for all \(a\in S\setminus\{0\}\), \(k\in J_n\).

Theorem 5. If \(S\) is a \(j\)-simple \(\mathfrak{p}\)-operative, then all \(A_\gamma\) (see item 2) are its minimal two-sided ideals.

The two-sided ideals \(A_\gamma\) of a \(j\)-simple operative \(S\) will be called its components. A \(\pi\)-operative \(S\) is called a \(\pi^*\)-operative if \(\pi_k\{1,n\}=\{1,n\}\) for any \(k\in J_n\). In particular, an associative and an alternative are \(\pi^*\)-operatives.

Theorem 6. Every \(i\)-simple \(\pi^*\)-operative is \(j\)-simple.

1. A \(j\)-simple operative \(S\) is called \(c\)-simple if each of its components contains a minimal left and a minimal right ideal; a \(c\)-simple \(\mathfrak{p}\)-operative is called \(c_1\)-simple (\(c_2\)-simple) if it contains exactly one component (respectively, if each of its left or right ideals is a two-sided ideal).

Theorem 7. In order that an \(i\)-simple associative be \(c\)-simple, it is necessary and sufficient that it contain a minimal left ideal and a minimal right ideal.

Theorem 8. Every \(c\)-simple \(\mathfrak{p}\)-operative without zero is isomorphic to the suboperative of all nonzero elements of some \(c_1\)-simple \(\mathfrak{p}\)-operative with zero.

Theorem 9. Every \(c_2\)-simple \(\mathfrak{p}\)-operative is similar \((^{1,2})\) to a nonzero \(c_2\)-simple \(\pi^*\)-operative in which all \(\sigma_k=\varepsilon\).

1. A subset \(B\) of an operative \(S\) is called its biideal if \(BS^{2n-3}B\subseteq B\).

Theorem 10. In order that a \(j\)-simple \(\mathfrak{p}\)-operative be \(c\)-simple, it is necessary and sufficient that each of its components contain at least one minimal biideal.

Theorem 11. In order that an \(i\)-simple \(\pi^*\)-operative (in particular, an associative or an alternative) be \(c\)-simple, it is necessary and sufficient that it contain a minimal biideal.

The last criterion is apparently new even for \(n=2\) (i.e., for semigroups). The theorem below complements one of the results of the articles \((^{1,2})\).

Theorem 12. Every two-sided invertible element of a \(\mathfrak{p}\)-operative is \((k)\)-invertible for any \(k\in J_n\). In order that all elements of a \(\mathfrak{p}\)-operative \(S\) be two-sided invertible, it is necessary and sufficient that \(S\) have no biideals distinct from the operative \(S\) itself.

Theorem 13. In order that an associative \(S\) be an \(n\)-group, it is necessary and sufficient that it contain no biideals distinct from \(S\).

For \(n=2\) the last theorem is given in \((^{4})\).

1. Let \(\Gamma, I, \Lambda\) be nonempty sets of indices; \(I=\bigcup_{\gamma\in\Gamma} I_\gamma\), \(\Lambda=\bigcup_{\gamma\in\Gamma}\Lambda_\gamma\), where the \(I_\gamma\) (respectively, \(\Lambda_\gamma\)) are pairwise disjoint subsets of the set \(I\) (\(\Lambda\)). The set \(B=\bigcup_{\gamma\in\Gamma} I_\gamma\times\Lambda_\gamma\) will be called the quasidiagonal of the Cartesian product \(I\times\Lambda\).

Let, further, \(G_0=G\cup\{0\}\) be a group with an externally adjoined zero \((^3)\). On the Cartesian product \(G_0\times B\) define the equivalence
\[ \sim:\quad (g;i,\lambda)\approx(g';i',\lambda') \]
if and only if \((g;i,\lambda)=(g';i',\lambda')\) or \(g=g'=0\). Denote by \(G\circ B\) the quotient set of the Cartesian pro-

\(^*\) The concepts of a minimal ideal, of a chain, etc. are defined as in the theory of semigroups.

of the product \(G_0 \times B\) by the equivalence \(\sim\). One may assume that \(G \circ B\) consists of all triples \((g;i,\lambda)\), where \(g \in G_0,\ (i,\lambda)\in B\), with all triples \(0=(0;i,\lambda)\) identified with one another. If \((i,\lambda)\) is a fixed element of \(B'\) and \(F \subseteq G_0\), then by \((F;i,\lambda)\) we denote the set of all elements \((g;i,\lambda)\), where \(g\in F\).

It turns out that for an arbitrary \(c\)-simple \(p\)-operative \(S\) with zero there exist a group \(G\) and a quasidiagonal \(B\) such that \(S=G\circ B\). In this case \(0\) is the zero of the operative \(S\), and the set \((G_0;i,\lambda)\), for any \((i,\lambda)\in B\), is its minimal biideal. If \(S\) is \(c_1\)-simple, then \(B=I\times \Lambda\).

1. A matrix (an \(\Lambda I\)-matrix) over a group with zero \(G_0\) is an arbitrary function \(P\) assigning to each pair of elements \(\lambda\in\Lambda,\ i\in I\) an element \(p_{\lambda i}\in G_0\).

Let an operative \(S\) be defined on the set \(G\circ B\) (Sec. 8); for any \(k=1,2,\ldots,n\) denote by \(v_k,\ v'_k\) a pair of variables, one of which is equal to \(i_k(\in I)\), the other to \(\lambda_k(\in\Lambda)\), with \((i_k,\lambda_k)\) ranging over the whole set \(B\), \(v_1=x_1,\ v'_n=i_n\). Suppose that for each \(k\in J_{n-1}\) a regular \((3,4)\)-matrix \(p_k(v_k,v'_{k+1})\) over the semigroup \(G_0\) is defined, and for any indices \(\gamma\in\Gamma,\ k\in J_{n-1}\) an automorphism (or inverse automorphism) \(\psi_k^{(\gamma)}\) of the semigroup \(G_0\). Moreover, if the group \(G\) is not abelian and \(v'_k=i_k,\ v_k=\lambda_k\) (respectively \(v'_k=\lambda_k,\ v_k=i_k\)), then \(\psi_k^{(\gamma)}\), for any \(\gamma\in\Gamma\), is an automorphism (respectively an inverse automorphism) of the semigroup \(G_0\). We shall call \(S\) a Rees operative over the group \(G\) if, for any \((x_j;i_j,\lambda_j)\in S\),

\[ (x_1;i_1,\lambda_1)(x_2;i_2,\lambda_2)\ldots(x_n;i_n,\lambda_n)=(z;i_1,\lambda_n), \tag{2} \]

where, for \(i_1\in I_\gamma\),

\[ z=x_1\cdot p_1(v_1,v'_2)\cdot \psi_2^{(\gamma)}x_2\cdot p_2(v_2,v'_3)\ldots \psi_{n-1}^{(\gamma)}x_{n-1}\cdot p_{n-1}(v_{n-1},i_n)\cdot x_n. \]

Theorem 14. Every \(c\)-simple \(\pi\)-operative \(S\) is isomorphic to a Rees operative over some group \(G\). If the group \(G\) is not abelian, then \(S\) is either associative or alternative.

For \(c\)-simple \(p\)-operatives Theorem 14 is, generally speaking, not true (cf. Sec. 8 and Theorem 9).

1. Let
   \[ B=\bigcup_{\gamma=1}^{t} I_\gamma\times\Lambda_\gamma; \]
   \(G\) be a group; \(P=(p_{\lambda i})\) a regular \(\Lambda I\)-matrix over \(G_0\) for which from \(p_{\lambda i}\ne0\) and \(\lambda\in\Lambda_\gamma\) it follows that \(i\in I_{\gamma+1}\) (from \(\lambda\in\Lambda_t\) it follows that \(i\in I_1\)); \(a\) an element of \(G\); \(\psi\) an automorphism of the semigroup \(G_0\) such that \(\psi^t=\lambda_a^{-1}\), where \(n-1=lt,\ \lambda_a x=a^{-1}xa\). Denote
   \[ j*\gamma=\left[\frac{1}{t}(j+\gamma-2)\right] \]
   (here \([m]\) is the integer part of the number \(m\)). Next let \(S(G^{(l)},B,P)\) be the operative defined on the set \(G\circ B\) (Secs. 8, 9) with operation (2), where for \(i_1\in I_\gamma\)

\[ z=\psi^{1*\gamma}(x_1\cdot p_{x_1 i_2})\cdot \psi^{2*\gamma}(x_2\cdot p_{x_2 i_3})\cdot\ldots\cdot \psi^{(n-1)*\gamma}(x_{n-1}\cdot p_{x_{n-1}i_n})\cdot G\cdot X_n. \]

Theorem 15. \(S(G^{(l)},B,P)\) is a \(c\)-simple associative operative with zero. Every \(c\)-simple associative operative with zero is isomorphic to some operative \(S(G^{(l)},B,P)\).

From this there easily follows a description of \(c\)-simple associatives without zero (they all turn out to be bundles of mutually isomorphic \(n\)-groups) and, in particular, of \(c\)-simple associatives without proper right ideals \((^5)\). The author has found all isomorphisms of \(c\)-simple associatives.

1. We restrict ourselves here to the description only of \(c_1\)-simple alternatives. Let \(n=2l+1\); \(G\) be a group; \(B=I\times\Lambda\); \(a,s,c\) be elements of \(G\); \(\psi\) an automorphism; \(\xi\) an inverse automorphism of the semigroup \(G_0\) such that

\[ \psi^l=\lambda_a^{-1},\qquad \xi^2=\lambda_c,\qquad \xi\psi=\lambda_{asc}\psi^{\,l-1}\xi,\qquad \psi a=a,\qquad \xi a=\left\{\prod_{j=1}^{l-1}\psi^j(sc)\right\}\cdot asc; \]

\(P=(p_{\varkappa\lambda})\) and \(Q=(q_{ij})\) be, respectively, a regular \(\Lambda\Lambda\)-matrix and an \(II\)-matrix over \(G_0\), satisfying, for any \(i,j\in I,\ \varkappa,\lambda\in\Lambda\), the condi-

to the conditions \(\xi p_{\lambda\mu}=c^{-1}p_{\chi\lambda}\), \(\xi q_{ji}=\psi(s\cdot q_{ij})\). Let, furthermore, \(S(G,B,P,Q,\xi)\) be an operative defined on the set \(G\circ B\) with action (2), where

\[ z=\left\{\prod_{j=1}^{l}\psi^{j-1}(x_{2j-1}\cdot p_{x_{2j-1}x_{2j}}\cdot \xi x_{2j}\cdot q_{i_{2j}i_{2j+1}})\right\}\cdot a\cdot x_n . \]

Theorem 16. \(S(G,B,P,Q,\xi)\) is a \(c_1\)-simple alternative. Every \(c_1\)-simple alternative is isomorphic to some operative \(S(G,B,P,Q,\xi)\).

For \(n=3\) this result was found in papers \((^6,^7)\).

1. In describing a \(\pi\)-operative of Rees \(S\) over an abelian group \(G\) (see Theorem 14), as in Sec. 11, we shall restrict ourselves only to \(c_1\)-simple operatives (after Secs. 10–11 one may assume that \(S\) is neither associative nor alternative). For such an operative \(S\), all \(\sigma_k=\varepsilon\). The action in \(S\) is still determined by formula (2), where

\[ \begin{aligned} z&=z'\cdot p_1(x_1,v_2')\cdot p_2(v_2,v_3')\cdot\ldots\cdot p_{n-1}(v_{n-1},i_n),\\ z'&=x_1\cdot\psi_2x_2\cdot\psi_3x_3\cdot\ldots\cdot\psi_{n-1}x_{n-1}\cdot x_n . \end{aligned} \tag{3} \]

Formula (3) defines on the group \(G\) the \(\pi\)-commutator \((^1,^2)\), and thereby singles out all dependencies among the automorphisms \(\psi_k\). All matrices \(p_k(v_k,v'_{k+1})\) satisfy one of the following three conditions:

1) The matrices \(p_k\) do not depend on the indices \(\lambda\in\Lambda\) (i.e. have the form \(p_k(i_k,i_{k+1})\), in particular \(p_k(i_k)\), \(p_k(i_{k+1})\), or \(p_k=c\).

2) The matrices \(p_k\) do not depend on the indices \(i\in I\).

3) \(p_1=p_1(v_2')\), \(p_{n-1}=p_{n-1}(v_{n-1})\); if the matrix \(p_k\) depends on \(i_k\) (on \(\lambda_k\)), then \(p_{k-1}\) and \(p_{k+1}\) do not depend on \(\lambda_k\) (respectively on \(i_k\)).

Thus the functions \(p_1,p_2,\ldots,p_{n-1}\) depend not on \(2n-2\) variables \(i_k,\lambda_k\) (see Secs. 9–11), but only on \(n-1\) variables. One can also reduce the number of the functions \(p_k\) themselves (as in Secs. 10–11); we shall not do this here.

1. Let \(S\) be a \(c_1\)-simple Rees operative (see Secs. 6, 9—in this case \(|\Gamma|=1\)); let \(N\) be a normal divisor of the group \(G\) such that \(\psi_jN=N\) for every \(j\in J_n\) \((\psi_j=\psi_j^{(\gamma)})\); let \(\sigma_I\) be an equivalence on the set \(I\) such that, for any matrices \(p_k,p_l\) having the form \(p_k=p_k(i,v')\) or \(p_l=p_l(v,i)\), it follows from \(i\sigma_I i'\) that \(p_k(i,v')\in p_k(i',v')N\) and \(p_l(v,i)\in p_l(v,i')N\); analogously an equivalence \(\sigma_\Lambda\) is defined on the set \(\Lambda\). Let, further, \(\rho=\rho(N,\sigma_I,\sigma_\Lambda)\) be the following equivalence on the operative \(S\): \((x;i,\lambda)\rho(y;j,\chi)\) if and only if \(x=y=o\) or \(y\in xN\), \(i\sigma_I j\), \(\lambda\sigma_\Lambda\chi\).

It follows from Sec. 9 that \(\rho\) is a congruence of the operative \(S\).

The converse of this assertion is as follows. If \(\rho\) is a congruence on a \(c_1\)-simple \(\pi\)-operative \(S\), then an operative of Rees isomorphic to \(S\) can be chosen so that \(\rho\) has the form \(\rho\{N,\sigma_I,\sigma_\Lambda\}\). We shall not enumerate here the isomorphisms of Rees operatives.

For \(n=2\) (i.e. for semigroups) this result is contained in paper \((^8)\) (see also \((^4)\)). From this there easily follows a description of the homomorphisms of \(\pi\)-operatives all of whose elements are two-sided invertible \((^1,^2)\).

Kharkov Institute
of Radio Electronics

Received
19 II 1968

REFERENCES

\(^1\) L. M. Gluskin, DAN, 157, No. 4, 767 (1964).
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\(^3\) E. S. Lyapin, Semigroups, Moscow, 1960.
\(^4\) A. H. Clifford, G. B. Preston, The Algebraic Theory of Semigroups, 1961.
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