S. I. Adian

ON THE IDENTITY PROBLEM IN ASSOCIATIVE SYSTEMS OF A SPECIAL KIND

(Presented by Academician P. S. Novikov, 7 VII 1960)

1. An associative system of a special kind, given by generators

\[ a_1,\ a_2,\ldots,\ a_n \tag{1} \]

and defining relations

\[ A_i = 1 \qquad (i=1,2,\ldots,k), \tag{2} \]

will be called an associative system of rank \(l\), if all words \(A_i\) in the relations (2) have length \(l\).

Theorem 1. If the natural numbers \(k\) and \(l\) are such that in every group with \(k\) nontrivial defining relations

\[ B = 1 \qquad (i=1,2,\ldots,k), \]

whose left-hand sides have lengths not exceeding \(l\), the identity problem is solvable, then in every associative system of rank \(l\) with \(k\) defining relations the identity problem is also solvable.

Let the numbers \(k\) and \(l\) satisfy the condition of Theorem 1, and let the associative system \(\mathfrak{B}\) of rank \(l\) be given by the generators (1) and the defining relations (2). In what follows, unless special reservations are made, the words considered by us will be words in the alphabet (1), and equalities of words will be equalities in the system \(\mathfrak{B}\). By the sign \(\equiv\) we shall denote equality in the free associative system. We shall call a left (right) wing of a word \(A\) such a nonempty word \(B\) that \(A \equiv BC\) (\(A \equiv CB\)) for some \(C\). In the case when \(C\) is nonempty, the word \(B\) will be called a proper left (right) wing of the word \(A\).

We shall call a word \(A\) left-sided (right-sided) with respect to the system of relations (2) if \(A \equiv B_1B_2\ldots B_r\), where each \(B_j\) is a left (right) wing of some word \(A_i\) from the relations (2). We shall call a word \(A\) two-sided with respect to the system of relations (2) if it is both left-sided and right-sided with respect to this system of relations.

2. First operation of extending the system of defining relations. If, in a word \(A_i\) from the system of relations (2), one performs a cyclic permutation of a left or right wing \(B\), which is a two-sided word with respect to (2), then the word thus obtained is also equal to 1 in the system \(\mathfrak{B}\). The first operation of extending the system of defining relations (2) will consist in adding to these relations all those relations which are obtained from them by cyclic permutation of left or right wings that are two-sided words with respect to this system of relations. To the system of relations obtained as a result of one application of the first extension operation, one may again apply this operation, first introducing the notions of left-sided, right-sided, and two-sided word

with respect to this new system of relations, etc. Since all the relations newly added in this process have left-hand sides of length \(l\), after a finite number of steps we shall obtain a system of defining relations

\[ A_i = 1 \quad (i = 1, 2, \ldots, k, \ldots, k_1), \tag{3} \]

closed with respect to the application of the first extension operation.

A word \(A\), two-sided with respect to the system of relations (3), whose length is \(\leq l\), will be called elementary if no proper left or right wing of it is a two-sided word with respect to (3). A word \(A\), two-sided with respect to the system of relations (3), will be called a whole word with respect to (3) if it is representable (in the free semigroup) as a product of elementary two-sided words with respect to (3). For every system of relations (3) that is closed with respect to the application of the first extension operation, the following is proved:

Lemma 1. Every two-sided word with respect to the system of relations (3) is a whole word with respect to (3) and has a unique representation (in the free semigroup) as a product of elementary two-sided words.

Let

\[ B_1, B_2, \ldots, B_s \tag{4} \]

be the set of all elementary two-sided words with respect to the system of relations (3). From Lemma 1 it follows that each \(A_i\) from the relations (3) has a unique representation \(A_i \equiv \varphi_i(B_1, B_2, \ldots, B_s)\), and the relations (3), including the relations (2), can be represented in the form

\[ \varphi_i(B_1, B_2, \ldots, B_s) = 1 \quad (i = 1, 2, \ldots, k, \ldots, k_1). \tag{5} \]

3. The generating operation. Let us be given a set \(\mathfrak M\) of relations of the system \(\mathfrak B\) and sets

\[ \mathfrak N_1, \mathfrak N_2, \ldots, \mathfrak N_s \tag{6} \]

of words of the system \(\mathfrak B\), satisfying the following conditions:

1) The union of all the sets \(\mathfrak N_i\) contains all elementary two-sided words with respect to the system of relations \(\mathfrak M\).

2) Any two elements of one and the same \(\mathfrak N_i\) are equal in the system \(\mathfrak B\).

3) Two sets \(\mathfrak N_i\) and \(\mathfrak N_j\) for \(i \ne j\) have no common elements.

4) The system of relations \(\mathfrak M\) is closed with respect to the first extension operation.

5) The set \(\mathfrak M\) consists of those and only those relations which are obtained from certain relation schemes

\[ \varphi_i(x_1, x_2, \ldots, x_s) = 1 \quad (i = 1, 2, \ldots, k, \ldots, k_1) \tag{7} \]

as the result of independently replacing all occurrences of each \(x_j\) by arbitrary elements of the set \(\mathfrak N_j\); moreover, two different occurrences of the same \(x_j\) may be replaced either by one and the same element or by different elements of the corresponding set \(\mathfrak N_j\).

6) It is possible to choose substitutions for the variables \(x_1, x_2, \ldots, x_s\) so that the schemes (7), for \(i = 1, 2, \ldots, k\), give the relations (2) of the system \(\mathfrak B\).

Consider the group \(F\), given by generators \(b_1, b_2, \ldots, b_s\) and nontrivial defining relations

\[ \varphi_i(b_1, b_2, \ldots, b_s) = 1 \quad (i = 1, 2, \ldots, k_1). \tag{8} \]

By the hypothesis of Theorem 1, the identity problem is decidable in \(F\), since the lengths of the left-hand sides of the relations (8) are \(\leq l\).

If \(E\) is an integral word with respect to the system of relations \(\mathfrak M\), then by Lemma 1 it has a unique decomposition into elementary two-sided factors. The image \(f(E)\) of the integral word \(E\) is the result of replacing, in this decomposition, each elementary two-sided word from the set \(\mathfrak M_j\) by the letter \(b_j\). The image of any integral word is a word in the positive alphabet of the group \(F\). If \(T\) is a word in the positive alphabet of the group \(F\), then by a preimage \(f^{-1}(T)\) of the word \(T\) we shall mean any integral word whose image is \(T\).

Let \(A\) be an arbitrary word of the system \(\mathfrak B\), and let
\[ A \equiv XEY, \]
where \(E\) is an integral word with respect to the system of relations \(\mathfrak M\), and \(X,Y\) are certain words, possibly empty. Denote by \(T_1\) the image of the word \(E\). Let \(\rho\) be the length of the word \(A\). Using the algorithm that solves the identity problem in the group \(F\), we find all possible words in the positive alphabet of the group \(F\)
\[ T_1,\ T_2,\ \ldots,\ T_m, \tag{9} \]
which are equal in \(F\) to the word \(T_1\) and have length \(\leq \rho\). Every word
\[ B \equiv X f^{-1}(T_i)Y, \]
obtained for some choice of an integral subword \(E\) of the word \(A\), a word \(T_i\) from the set (9) corresponding to this \(E\), and a preimage \(f^{-1}(T_i)\), will be called a word directly generated from \(A\), if the length of \(B\) does not exceed \(\rho\).

We shall say that a word \(B\) is generated from \(A\) if one can indicate such a finite sequence of words
\[ A_1,\ A_2,\ \ldots,\ A_j,\ \ldots,\ A_r \]
that
\[ A_1 \equiv A,\qquad A_r \equiv B, \]
and \(A_i\) is directly generated from \(A_{i-1}\) for
\[ i=2,3,\ldots,r. \]
Since every word generated from a fixed word \(A\) of length \(\rho\) has length \(\leq \rho\), the set \(\pi(A)\) of all words generated from \(A\) is finite, and we have an algorithm for finding the set \(\pi(A)\) from the given word \(A\).

4. The second operation of extension of a system of defining relations will be applied to systems of relations satisfying all six conditions indicated in item 3. Using the operation of generation, for each \(\mathfrak M_j\) we find the set \(\mathfrak M'_j\) of all those words that are generated from words of the set \(\mathfrak M_j\). Then we merge those \(\mathfrak M'_j\) that have a nonempty intersection. In doing so, if we merge two sets \(\mathfrak M'_i\) and \(\mathfrak M'_j\), then in all functions
\[ \varphi_i(x_1,x_2,\ldots,x_s) \]
we identify the variables \(x_i\) and \(x_j\). Having performed a finite number of such mergings and the corresponding identifications, we finally obtain a collection of sets
\[ \mathfrak M^*_1,\ \mathfrak M^*_2,\ \ldots,\ \mathfrak M^*_{s_1}, \tag{10} \]
where \(s_1 \leq s\), and the sets \(\mathfrak M^*_j\) are pairwise disjoint. At the same time the functions
\[ \varphi_i(x_1,x_2,\ldots,x_s) \]
are transformed into
\[ \psi_i(x_1,x_2,\ldots,x_{s_1}). \]
We define the system of relations \(\mathfrak M^*\) as the collection of relations obtained from the schemes of relations
\[ \varphi_i^*(x_1,x_2,\ldots,x_{s_1})=1 \qquad (i=1,2,\ldots,k_1) \]
as a result of replacing, independently of one another, all occurrences of each \(x_j\) by arbitrary elements of the corresponding set \(\mathfrak M^*_j\). The system of relations \(\mathfrak M^*\) and the collection of sets (10) will be called the result of applying the second operation of extension to the system of relations \(\mathfrak M\) and the collection of sets (6). If \(s_1=s\) and every \(\mathfrak M^*_j\) coincides with \(\mathfrak M_j\), then \(\mathfrak M^*\) will also coincide with \(\mathfrak M\). In this case we shall say that the system of relations \(\mathfrak M\) is closed with respect to the second operation of extension.

5. The third operation of extension is applied to the result \(\mathfrak M^*\) of applying the second operation of extension, although it uses

the generation operation defined on the basis of the system of relations \(\mathfrak M\) and the corresponding group \(F\). For each proper left (right) wing \(C\) of each relation of the system \(\mathfrak M^*\), we find the set \(\pi(C)\), if \(C\) is not a word two-sided with respect to the system \(\mathfrak M\). We run through all possible pairs \([\pi(C_1), \pi(C_2)]\), where \(C_1\) is the left wing of some relation and \(C_2\) is the right wing of some relation. If \(\pi(C_1)\) and \(\pi(C_2)\) have a nonempty intersection, then we declare the elements \(C_1\) and \(C_2\) two-sided. To the system of relations \(\mathfrak M^*\) we then add all relations obtained by a cyclic permutation of \(C_1\) (\(C_2\)) in all those relations in which \(C_1\) (\(C_2\)) is the left (right) wing. The system of relations \(\mathfrak M^*\), together with the totality of the sets (10), is called closed with respect to the third extension operation if, for none of the indicated pairs, the sets \(\pi(C_1)\) and \(\pi(C_2)\) intersect.

1. The three extension operations described are applied in the following order: first the first is applied until a system of relations closed with respect to the first extension operation is obtained; then the second is applied, and immediately after it the third extension operation; after this everything begins again from the beginning, and so on. It is established that after a finite number of steps a system of relations will be obtained that is closed with respect to the application of each of these three operations and satisfies all six conditions of item 3. For such a system of relations the fundamental lemma is proved, one of whose assertions states:

Two words \(X\) and \(Y\) are equal in the system \(\mathfrak B\) if and only if the corresponding sets of words \(\pi(X)\) and \(\pi(Y)\) have a nonempty intersection.

From this we obtain an algorithm solving the identity problem in the associative system \(\mathfrak B\).

1. In \((^2)\) the solvability of the identity problem was established for an associative system given by a single irreducible defining relation \(A = B\), where \(A, B\) are nonempty words. From Theorem 1 with \(k = 1\), on the basis of B. Magnus’s result \((^1)\) on the solvability of the identity problem in any group with one defining relation, it follows:

Theorem 2. In an associative system with one defining relation of the form \(A = 1\), the identity problem is solvable.

In connection with Theorem 1 the question arises: can there exist associative systems of deficiency \(l\) with an unsolvable identity problem? The following theorem gives an answer to this question.

Theorem 3. For every natural \(l \ge 3\) one can construct an associative system of deficiency \(l\) in which the identity problem is unsolvable. There exists an algorithm solving the identity problem in every associative system of deficiency 2.

An associative system given by defining relations
\[ A_i = B_i \quad (i = 1, 2, \ldots, m) \]
is called homogeneous if the length of each \(A_i\) is equal to the length of \(B_i\). In homogeneous associative systems the identity problem is solved very simply. From Theorem 3 it follows:

Theorem 4. Whatever nonempty word \(A\) may be, one can construct a homogeneous associative system \(\mathfrak A\) such that in the system \(\mathfrak A_1\), obtained from \(\mathfrak A\) by adding the new relation \(A = 1\), the identity problem is unsolvable.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
4 VII 1960

CITED LITERATURE

\(^{1}\) W. Magnus, UMN, vol. 8, 365 (1940).
\(^{2}\) S. I. Adian, DAN, 138, No. 2, 6 (1960).