MATHEMATICS

L. M. GLUSKIN

ON POSITIONAL OPERATIVES

(Presented by Academician A. I. Mal'tsev on February 29, 1964)

In the present note we study algebras with one \(n\)-ary operation subject to certain conditions of associativity type.

1. A set \(S\) with one everywhere-defined \(n\)-ary operation is called an \(n\)-operative. We shall write this operation in the form

\[ (x_1x_2\ldots x_n)=x \qquad (x_i,\ x\in S). \]

Let

\[ ((x_1x_2\ldots x_n)x_{n+1}\ldots x_{2n-1})=(x_1x_2\ldots x_{2n-1}). \]

If \(A_i\) are nonempty subsets of the \(n\)-operative \(S\), then by \((A_1A_2\ldots A_n)\) we denote the set of all products \((a_1a_2\ldots a_n)\), where \(a_i\in A_i\); if, in addition, one of the sets \(A_i\) (for example, \(A_1\)) consists of a single element \(a\), then we shall write \((A_1A_2\ldots A_n)=(aA_2\ldots A_n)\). If \(A_{t+1}=A_{t+2}=\ldots=A_{t+k}=A\), then we denote

\[ (A_1A_2\ldots A_n)=(A_1A_2\ldots A_tA^kA_{t+k+1}\ldots A_n); \]

\(A^0\) is the empty symbol.

Let \(S\) be an \(n\)-operative, and let \(\varphi\) be a permutation of the indices \(1,2,\ldots,n\). The set \(S\), with respect to the \(n\)-ary operation

\[ [x_1x_2\ldots x_n]=(x_{\varphi1}x_{\varphi2}\ldots x_{\varphi n}), \]

is also an \(n\)-operative, which we denote by \(S_\varphi\). If \(\varphi1=1\), \(\varphi n=n\), then the operative \(S_\varphi\) will be called similar to the operative \(S\).

1. Let \(\rho=\{\rho_k\}_{k=1}^n\) be a system of (one-to-one) permutations of the indices \(1,2,\ldots,2n-1\) such that \(\rho_1=\rho_n=\varepsilon\) (\(\varepsilon\) is the identity permutation). An \(n\)-operative \(S\) is called an \(n\rho\)-operative if for any \(k\) \((1\le k\le n)\) and all \(x_i\in S\),

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

Let \(\pi=\{\sigma_k,\pi_k\}_{k=1}^n\) be a system of pairs of permutations of the \(n\) symbols \(1,2,\ldots,n\) such that \(\sigma_1=\sigma_n=\pi_1=\pi_n=\varepsilon\), \(\sigma_k k=k\) for all \(k\). An \(n\)-operative \(S\) is called an \(n\pi\)-operative if for all \(x_i,y_j\in S\) and any \(k\) \((1\le k\le n)\),

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

\[ \ldots y_{\pi_k n}x_{\sigma_k(k+1)}\ldots x_{\sigma_k n}). \]

An \(n\pi\)-operative will be called: an \(n\)-associative, if \(\sigma_k=\pi_k=\varepsilon\) for \(k=1,2,\ldots,n\); an \(n\)-alternative, if \(n\) is odd and all \(\sigma_k=\pi_{2j+1}=\varepsilon\), \(\pi_{2j}=\tau\), where \(\tau i=n-i+1\) for \(i=1,2,\ldots,n\); an \(n\pi\)-commutative, if \(\sigma_k=\varepsilon\) for every \(k\) \((1\le k\le n)\) and

\[ (x_1x_2\ldots x_{n-1}x_n)=(x_nx_2\ldots x_{n-1}x_1) \]

for any \(x_i\in S\). For \(n=2\) there exists a unique \(n\rho\)-operative, which is a 2-associative, i.e. an ordinary semigroup \((^1)\). A 3-alternative is V. V. Wagner’s semigroup \((^{2,3})\). Some results on \(n\)-associatives (for arbitrary \(n\)) can be found in notes \((^{4-6})\).*

1. An element \(a\) of an \(n\)-operative \(S\) is called \((j)\)-invertible if

\[ (S^{j-1}aS^{n-j})=S. \]

An element \(a\) that is simultaneously \((1)\)-invertible and \((n)\)-invertible is called two-sided invertible.

* The terms “operative” and “associative” were taken by me from works \((^{7,8})\). In \((^8)\) only 2-operatives are considered.

One can give an example of an \(n\)-associative \(S\) containing an element \(a\) that is \((k)\)-invertible for \(k=1,2,\ldots,n-1\), but is not \((n)\)-invertible.

Theorem 1. Every two-sided invertible element of a pl-operative \(S\) is \((k)\)-invertible for every \(k\).

Theorem 2. The set of all two-sided invertible elements of a pl-operative \(S\) is its suboperative.

Theorem 3. If a pr-operative \(S\) contains a suboperative \(T\), all elements of which are two-sided invertible in \(S\), then \(S\) is similar to some pl-operative \(S_\varphi\).

By the definition of similarity (see item 1), in this case all elements of \(T_\varphi\) are two-sided invertible in \(S_\varphi\).

1. Let \(A\) be a finite group of order \(k\), \(n=kt+1\), where \(t\) is an integer \(>0\); let \(\nu\) be a one-to-one mapping of the set \(\{1,2,\ldots,n\}\) onto \(A\). Instead of \(\nu(i)=a\in A\) we shall write \(a=a_i\). We shall call \(\nu\) a numbering of the group \(A\) if \(a_1=e\) (\(e\) is the identity of the group \(A\)) and from \(i\equiv j\pmod{k}\) it follows that \(a_i=a_j\). The group \(A\) with numbering \(\nu\) will be denoted by \(A_\nu\). The cyclic group generated by an element \(a\) of order \(k\), with numbering \(a_i=a^{i-1}\) \((i=1,2,\ldots,n)\), will be denoted by \([a]\).

To each element \(a_j\in A_\nu\) there corresponds a permutation \(\tau_j\) of the set \(\{1,2,\ldots,n-1\}\) such that: a) \(a_{\tau_j i}=a_j a_i\); b) if \(ks<i\le k(s+1)\), then also \(ks<\tau_j i\le k(s+1)\). Let \(\pi_j\) \((j=1,2,\ldots,n)\) be the following permutation of the set \(\{1,2,\ldots,n\}\): \(\pi_j i=\tau_j^{-1}(i+j-1)\), if \(i\le n-j\), \(\pi_j i=\tau_j^{-1}(i+j-n)\), if \(n-j<i<n\), \(\pi_j n=n\). The system of pairs of permutations \(\{\sigma_j,\pi_j\}_{j=1}^n\), where all \(\sigma_j=\varepsilon\), will be denoted by \(\pi_{A_\nu}\).

Theorem 4. If a pl-operative \(S\) contains at least one two-sided invertible element, then \(S\) is either an \(n\)-associative, or an \(n\)-alternative, or a pl-commutative operative.

1. Let \(D\) be a semigroup, whose operation we shall denote by \(*\); let \(a\) be an arbitrary two-sided invertible element of \(D\); let \(\Psi\) be a finite subset of the extended automorphism group of the semigroup \(D\) (i.e., the elements \(\psi_i\) of the set \(\Psi\) are automorphisms or inverse automorphisms of the semigroup \(D\)); \(\psi_i a=a\) for all \(\psi_i\in\Psi\); it is possible that \(\psi_i=\psi_j\) for \(i\ne j\). Introduce on the set \(D\) an \(n\)-ary operation: for arbitrary \(x_i\in D\)

\[ (x_1x_2\ldots x_n)=x_1*\psi_2x_2*\psi_3x_3*\ldots *\psi_{n-1}x_{n-1}*a*x_n. \]

The \(n\)-operative obtained in this way will be denoted by \(S=S(D,a,\Psi,n)\). If \(\Psi\) is a group and the indices of \(\psi_i\) correspond to some numbering \(\nu\) of it (see item 4), then we denote \(S=S(D,a,\Psi_\nu,n)\).

Let, further, \(\lambda_a\) be the inner automorphism of the semigroup \(D\) generated by the element \(a\): \(\lambda_a x=a^{-1}*x*a\). If \(\psi\) is an automorphism, \(\psi a=a\), \(\psi^{\,n-1}=\lambda_a^{-1}\), \(\psi_i=\psi^{i-1}\) \((i=1,2,\ldots,n-1)\), then we denote \(S=S(D,a,\psi,n)\). If \(n=2m+1\), \(\psi\) is an automorphism, \(\xi\) is an involution, \(\psi^m=\lambda_a^{-1}\), \(\xi\psi=\lambda_a\psi^{m-1}\xi\), \(\psi a=\xi a=a\), \(\psi_{2j+1}=\psi^j\), \(\psi_{2j}=\psi^{j-1}\xi\), then we denote \(S=S(D,a,\psi,\xi,n)\). \(S(D,a,\psi,n)\) is \(n\)-associative, \(S(D,a,\psi,\xi,n)\) is \(n\)-alternative; if \(D\) is commutative, then \(S(D,a,\Psi_\nu,n)\) is pl\(_{\Psi_\nu}\)-commutative (see items 2, 4).

1. Theorem 5. If an \(n\)-associative \(S\) contains at least one two-sided invertible element, then it is isomorphic to some operative \(S(D,a,\psi,n)\), where \(\psi\) is an automorphism of the semigroup \(D\) and \(\psi a=a\).

Theorem 6. If an \(n\)-alternative \(S\) contains at least one two-sided invertible element, then it is isomorphic to some operative \(S(D,a,\psi,\xi,n)\), where \(\psi\) is an automorphism of the semigroup \(D\), \(\xi\) is its involution, \(\psi a=\xi a=a\).

Theorem 7. If a pl-commutative \(S\) contains at least one \((1)\)-invertible element, then it is similar to the operative \(S(D,a,\Psi_\nu,n)\) (see items 4, 5).

where \(D\) is a commutative semigroup, \(\Psi\) is a subgroup of the group of its automorphisms, \(\psi_i a = a\) for any \(\psi_i \in \Psi\).

This theorem is very close to Evans’ results \((^9)\).

1. A subset \(A\) of an \(n\)-operative \(S\) is called an \((r)\)-ideal \((^{10})\) if \((S^{r-1} A S^{n-r}) \subseteq A\). A \((1)\)-ideal is called a right ideal, and an \((n)\)-ideal a left ideal. If an \(n p\)-operative \(S\) contains no proper left or right ideals, then every element of it is two-sided invertible.

From Theorems 3–7 it follows:

Theorem 8. Every \(n p\)-operative (\(n\ell\)-operative) without proper left and right ideals is similar (respectively isomorphic) to some operative
\[ S = S(D, a, \Psi, n), \]
where \(D\) is a group; moreover \(S\) satisfies the conditions of one of Theorems 5–7. In particular, every \(n\)-associative without proper left and right ideals is isomorphic to some operative
\[ S(D, a, \psi, n), \]
where \(D\) is a group and \(\psi\) is its automorphism.

An \(n\)-associative \(S\) is called an \(n\)-Dörnte group \((^{11,12})\) if
\[ (x_1 x_2 \ldots x_{k-1} S x_{k+1} \ldots x_n) = S \]
for any \(k\) \((1 \leq k \leq n)\) and any \(x_i \in S\).

Theorem 9. Every \(n\)-Dörnte group is isomorphic to some operative
\[ S(D, a, \psi, n), \]
where \(D\) is a group, \(\psi\) is its automorphism, \(\psi a = a\).

1. An element \(e\) of an \(n\)-operative \(S\) is called unitary (in the terminology of B. V. Wagner \((^2)\), biunitary) if
   \[ (x e^{\,n-1}) = (e^{\,n-1} x) = x \]
   for any \(x \in S\).

From Theorems 1–7 it follows:

Theorem 10. If an \(n p\)-operative (\(n\ell\)-operative) contains a unitary element, then it is similar (respectively isomorphic) to an operative
\[ S(D, e, \Psi, n), \]
where \(D\) is a semigroup with identity \(e\), and the set \(\Psi\) satisfies the conditions of one of Theorems 5–7.

Theorem 11. An \(n p\)-operative \(S\) can be embedded in an \(n p\)-operative \(\overline S\) with unitary element \(S\) if and only if \(S\) is similar to an \(n\)-associative, an \(n\)-alternative, or to some \(n\ell\)-commutative.

Theorem 12. An \(n\ell\)-operative \(S\) can be embedded in an \(n\ell\)-operative \(\overline S\) with unitary element \(e\) if and only if \(S\) is \(n\)-associative or \(n\)-alternative, or is similar to some \(n\ell_\Psi\)-commutative.

Theorem 13. Every \(n\)-alternative \(S\) can be embedded in a semigroup (see item 2).

An element \(e\) of an operative \(S\) is called an identity if
\[ (e^{k-1} x e^{n-k}) = x \]
for any \(x \in S\) and every \(k\).

From Theorem 8 follows the following theorem:

Theorem 14. If an \(n p\)-operative (\(n\ell\)-operative) \(S\) contains an identity, then it is similar (respectively isomorphic) to the operative
\[ S(D, e, \varepsilon, n) \]
(see item 5), where \(D\) is a semigroup with identity \(e\), and \(\varepsilon\) is its identity automorphism.

The structure of an \(n\)-associative with identity was studied in the note \((^6)\).

Theorem 15. An \(n p\)-operative (\(n\ell\)-operative) \(S\) can be embedded in an \(n p\)-operative \(\overline S\) with identity \(e\) if and only if \(S\) is similar to an \(n\)-associative (respectively is an \(n\)-associative).

An \(n p\)-operative \(S\) for \(n > 2\) may contain several unitary elements or several identities. The set of all identities of an \(n\)-associative \(S\) (as also the set of all biunitary elements of a semigroup—see \((^2)\)) is a suboperative of \(S\).

Theorems 11, 12, and 15 can be somewhat strengthened, namely: the operative \(\overline S\) can be chosen so that the element \(e\) is its unique unitary element.

1. Somewhat aside from the preceding results are the following two theorems.

Theorem 16. Let \(A\) be a \((k)\)-ideal of an \(n\ell\)-operative \(S\), \(k \ne 1\), \(k \ne n\). If
\[ (A^n) = A, \]
then \(A\) is a \((j)\)-ideal of the operative \(S\) for any \(j\).

Theorem 17. Let \(A\) be simultaneously a left and a right ideal of an \(n\ell\)-ope-

operative \(S\). If \((A^n)=A\), then \(A\) is a \((k)\)-ideal of the operative \(S\) for every \(k\).

1. In connection with the results of Sections 4, 6–8, the question arises of isomorphism of operatives \(S(D,a,\Psi,n)\) in the similarity of \(n\pi\)-operatives.

Let \(D\) and \(D'\) be semigroups with identities \(e\) and \(e'\), and let \(a\) and \(a'\) be their invertible elements; \(\Psi\) and \(\Psi'\) are subsets of their extended automorphism groups; \(c\) is an invertible element of \(D'\); \(f\) is an isomorphism of the semigroup \(D\) onto \(D'\). Denote \(c_1=e'\), \(c_k=\psi'_2 c * \psi'_3 c * \cdots * \psi'_k c\) \((k=2,3,\ldots,n-1,\ \psi'_i\in\Psi')\), \(\xi_k x=c_k^{-1} * x * c_k\) \((x\in D')\). Suppose that \(f\) and \(c\) satisfy the conditions: \(fa=c_{n-1} * a' * c\) and \(\psi'_k=\xi_{k-1} f\psi_k f^{-1}\), if \(\psi_k\) is an inverse automorphism \((\psi_k\in\Psi)\); \(\psi'_k=\xi_k f\psi_k f^{-1}\), if \(\psi_k\) is an automorphism. Then the mapping \(\varphi(x)=c * f(x)\) is an isomorphism of the operative \(S(D,a,\Psi,n)\) onto \(S(D',a',\Psi',n)\). We shall say that \(\varphi\) is generated by the isomorphism \(f\) and the element \(c\in D'\).

Theorem 18. Every isomorphism of the operative \(S(D,a,\Psi,n)\) onto \(S(D',a',\Psi',n)\) is generated by some isomorphism \(f\) of the semigroup \(D\) onto \(D'\) and an element \(c\in D'\).

1. Let \(S\) be an arbitrary \(n\)-operative. The operative \(S'\), defined on the set \(S\) with the operation

\[ [x_1x_2\cdots x_n]=(x_nx_{n-1}\cdots x_2x_1)\qquad (x_i\in S) \]

will be called the inverse \(^{(2)}\) with respect to the operative \(S\).

Theorem 19. Let \(S\) be an arbitrary \(n\pi\)-operative containing a two-sided invertible element and not being \(n\pi\)-commutative. If the operative \(S_\varphi\) (see Section 1) is an \(n\pi\)-operative, then either \(S_\varphi=S\), or \(S_\varphi=S'\), where \(S'\) is the operative inverse with respect to \(S\). In particular, if the \(n\pi\)-operative \(S_\varphi\) is similar to the operative \(S\), then \(S_\varphi=S\).

Kommunarsk Mining and Metallurgical Institute

Received
12 II 1964

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