V. I. Frenkel

ON ALGORITHMIC PROBLEMS IN PARTIALLY ORDERED GROUPS

(Presented by Academician P. S. Novikov on 22 III 1963)

The article considers questions connected with the solution of algorithmic problems arising in the partial ordering of finitely presented groups (I), and questions of the solvability of algorithmic problems arising when partially ordered groups are specified by a finite number of generators and a finite number of defining inequalities (II) (an exact description of such a specification of partially ordered groups will be given below). In accordance with this, the article is divided into two parts.

I. Let \(\mathfrak A\) be a group given by a finite number of generators and defining relations. A set of elements of \(\mathfrak A\) closed with respect to the group operation is a subsemigroup of the group \(\mathfrak A\). A subsemigroup of the group \(\mathfrak A\) will be called invariant if, for every word \(P\) of this subsemigroup, the word \(X^{-1}PX\), where \(X\) is any word from \(\mathfrak A\), also belongs to this subsemigroup. An invariant subsemigroup \(\widetilde{\mathfrak A}^{+}\) of the group \(\mathfrak A\) will be called proper if among its elements there is no \(1\) of the group \(\mathfrak A\). The subsemigroup \(\mathfrak A^{+}=\widetilde{\mathfrak A}^{+}\cup 1\) of the group \(\mathfrak A\) will be called positive. If in the group \(\mathfrak A\) there is at least one positive subsemigroup, then, as is known \((^{2})\), the group \(\mathfrak A\) can be partially ordered in the following way: \(a\leqslant b\) (\(a,b\) are elements of the group \(\mathfrak A\)) if and only if the element \(ba^{-1}\) or \(a^{-1}b\) belongs to the given positive subsemigroup. If for any elements \(a,b\in\mathfrak A\) there exists an element \(c\) such that \(a\leqslant c\) and \(b\leqslant c\), then the group \(\mathfrak A\) is called directed.

In the author’s paper \((^{3})\) the notion of an effectively partially orderable (e.p.o.) group was introduced: a group \(\mathfrak A\) is called e.p.o. with respect to a given positive subsemigroup \(\mathfrak A^{+}\) if the problem of membership of words from \(\mathfrak A\) in \(\mathfrak A^{+}\) is decidable, i.e. there exists an algorithm determining, for every word from \(\mathfrak A\), whether it belongs to \(\mathfrak A^{+}\) or not. A directed e.p.o. group will mean an e.p.o. group for which, for any two words \(A\) and \(B\), one can effectively construct a word \(C\) greater than the words \(A\) and \(B\). It is not difficult to see that for e.p.o. groups the word identity problem is decidable.

In the work of P. S. Novikov \((^{1})\) finitely presented groups with the so-called regular system of passing letters are considered; we shall denote them by \(\mathfrak A_p\). The subsemigroup of the group \(\mathfrak A_p\) generated by the set of words \(X^{-1}Ap_iBX\), where \(p_i\) are passing letters of the group \(\mathfrak A_p\), \(X\) are arbitrary words from \(\mathfrak A_p\), and \(A,B\) are all possible words from \(\mathfrak A_p\) not containing passing letters, is a proper subsemigroup. We shall denote this subsemigroup by \(\widetilde{\mathfrak A}^{+}_{p}\), and the positive subsemigroup \(\widetilde{\mathfrak A}^{+}_{p}\cup 1\) by \(\mathfrak A^{+}_{p}\).

Consider the equation in the group \(\mathfrak A_p\)

\[ Z_0Y_1Z_1\ldots Y_kZ_k=1, \tag{1} \]

where \(Z_i\) are arbitrary words from \(\mathfrak A_p\), for which solutions are considered only in words without passing letters.

If there exists an algorithm deciding, for every equation (1), whether it has a solution or not, then we shall say that the solvability problem for equations (1) is decidable.

Theorem 1. If in the group \(\mathfrak A_p\) the solvability problem for equations (1) is decidable, then in this group the word membership problem for words of the group \(\mathfrak A_p\) in the pure subsemigroup \(\mathfrak A_p^{+}\) of this group is decidable.

The proof of Theorem 1 and of the other assertions of this paper cannot be given within the limits of the present article. However, we shall describe the algorithm for membership of words of the group \(\mathfrak A_p\) in the pure subsemigroup \(\mathfrak A_p^{+}\).

The number equal to the sum of all exponents for each passing letter \(p_i\) is an invariant of the transformation of a word from \(\mathfrak A_p\) into any word equal to it.

Let, for some word \(\gamma\), \(k_1, k_2, \ldots, k_t\) be the sums of the exponents at the passing letters \(p_i\) \((i=1,2,\ldots,t)\) of the group \(\mathfrak A_p\). If for at least one \(p_i\), \(k_i<0\), or all \(k_i=0\), then it is obvious that \(\gamma \notin \mathfrak A_p^{+}\).

Suppose that for the word \(\gamma\),
\[ k_1+k_2+\cdots+k_t=k>0 \]
and \(k_i \ge 0\) \((i=1,2,\ldots,t)\). Consider all possible representations of the word \(\gamma\) in the form
\[ Z_0 p Z_1 p \ldots p Z_k, \tag{2} \]
where the passing letters, for simplicity, are denoted without indices. In the representation (2) of the word \(\gamma\), \(k\) passing letters are chosen so that, if the sum of the exponents at the passing letter \(p_i\) is equal to \(k_i\), then \(k_i\) letters \(p_i\) are written out. The segments between adjacent passing letters written in (2) in the word \(\gamma\) are denoted by \(Z_i\).

To each representation (2) of the word \(\gamma\) we assign an equation of the form (1):
\[ Z_0Y_1Z_1Y_2\ldots Y_kZ_k=1. \tag{3} \]

If for at least one representation (2) of the word \(\gamma\) equation (3) is solvable, then \(\gamma \in \mathfrak A_p^{+}\). If, however, for no representation (2) of the word \(\gamma\) does equation (3) have a solution, then \(\gamma \notin \mathfrak A_p^{+}\).

Theorem 2. If in the group \(\mathfrak A_p\): a) the solvability problem for equations (1) is decidable and b) the identity problem is decidable, then the group \(\mathfrak A_p\) can be effectively partially ordered by its proper subsemigroup \(\mathfrak A_p^{+}=\mathfrak A_p^{+}\cup 1\). In this case the group \(\mathfrak A_p\) is a directed e.i.u. group with respect to the proper subsemigroup \(\mathfrak A_p^{+}\).

Consider the free product
\[ \mathfrak A'_p=\mathfrak A_p * \{\bar p\}, \]
where \(\{\bar p\}\) is an infinite cyclic group. To the system of passing letters of the group \(\mathfrak A'_p\), besides those contained in the group \(\mathfrak A_p\), we also adjoin the letter \(\bar p\).

Theorem 3. In order that the group \(\mathfrak A'_p=\mathfrak A_p * \{\bar p\}\) can be effectively partially ordered by the proper subsemigroup \(\mathfrak A_p^{\prime +}=\mathfrak A_p^{+}\cup 1\), it is necessary and sufficient that in the group \(\mathfrak A_p\): a) the solvability problem for equations (1) be decidable and b) the identity problem be decidable. Moreover, if for the group \(\mathfrak A_p\) conditions a) and b) are satisfied, then the group \(\mathfrak A'_p\) is a directed e.i.u. group with respect to the subsemigroup \(\mathfrak A_p^{\prime +}\).

From Theorem 3 it follows:

Theorem 4. There exists a finitely presented directed e.i.u. group with an undecidable conjugacy problem.

In Theorem 4 one considers the group
\[ \mathfrak A'_{p_1}=\mathfrak A_{p_1} * \{\bar p\}, \]
where \(\mathfrak A_{p_1}\) is a group with an undecidable conjugacy problem, constructed by P. S. Novikov [1]. As the proper subsemigroup of the group \(\mathfrak A'_{p_1} * \{\bar p\}\) one takes the subsemigroup
\[ \mathfrak A_{p_1}^{\prime +}\cup 1, \]
the pure subsemigroup of which \(\mathfrak A_{p_1}^{\prime +}\) is generated by the words
\[ X^{-1}A\bar pBX, \]
where the words \(A\) and \(B\) do not contain the letter \(\bar p\).

Theorem 5. There exists a finitely presented group \(\mathfrak A_0\) with decidable identity problem and with such a proper subsemigroup \(\mathfrak A_0^{+}\),

that the group \(\mathfrak A_0\) cannot be effectively partially ordered by this semigroup \(\mathfrak A_0^+\).

The proof of Theorem 5 is based on Theorem 3 and on the properties of the group of P. S. Novikov with unsolvable conjugacy problem \((^1)\).

II. Let a finite alphabet \(a_1,\ldots,a_n,a_1^{-1},\ldots,a_n^{-1}\) and a finite number of pairs of words in this alphabet \((A_i,B_i)\), \(i=1,2,\ldots,m\), be given.

Between the words of the given alphabet we introduce a binary relation: \(X \leqslant Y\) if and only if this inequality is derivable by a finite number of applications of the following rules of inference: 1) \(A_i \leqslant B_i\) \((i=1,2,\ldots,m)\); 2) \(a_j a_j^{-1} \leqslant 1,\ a_j^{-1}a_j \leqslant 1,\ 1 \leqslant a_j a_j^{-1},\ 1 \leqslant a_j^{-1}a_j\) \((j=1,2,\ldots,n)\); 3) \(A \leqslant A\); 4) from \(A \leqslant B\) and \(B \leqslant C\) it follows that \(A \leqslant C\); 5) from \(A \leqslant B\) it follows that \(XAY \leqslant XBY\), where \(A,B,X\), and \(Y\) are arbitrary words.

We also define a relation of “equality” between words: 6) \(A=B\) if and only if \(A \leqslant B\) and \(B \leqslant A\).

The algebraic system \(\Gamma\) so defined is, as is not hard to see, a partially ordered group. We shall say that the group \(\Gamma\) is given by generators \(a_1,\ldots,a_n,a_1^{-1},\ldots,a_n^{-1}\) and defining inequalities \(A_i \leqslant B_i\) \((i=1,2,\ldots,m)\).

When groups are given by generators and defining inequalities, it may happen that in the group \(\Gamma\) the reverse inequalities \(B_i \leqslant A_i\) hold for all \(i\). In this case the defining inequalities of the group \(\Gamma\), according to rule 6), turn into equalities. Moreover, any group given by generators and defining equalities can in a trivial way be given by defining inequalities. Thus, specifying groups by generators and defining inequalities is a more universal way of specifying groups than specifying groups by means of generators and defining equalities.

For groups \(\Gamma\), along with the algorithmic problems of identity and conjugacy, new algorithmic problems arise.

Word inequality problem: construct an algorithm which, for any pair of words \(A\) and \(B\) of the group \(\Gamma\), determines whether the inequality \(A \leqslant B\) holds.

Weak conjugacy problem: construct an algorithm which, for any pair of words \(A\) and \(B\) of the group \(\Gamma\), determines whether there exists a word \(C\) such that the inequality \(A \leqslant C^{-1}BC\) holds.

Theorem 6. There exists a group \(\Gamma_1\), given by a finite number of generators and a finite number of defining inequalities, for which the word inequality problem is unsolvable, while the identity problem is solvable.

The construction of the group \(\Gamma_1\) is based on the group of P. S. Novikov with unsolvable conjugacy problem \((^1)\).

We note that Theorem 6 strengthens Theorem 5, proved by other methods.

Theorem 7. There exists a group \(\Gamma_2\), with a finite number of generators and a finite number of defining inequalities, for which the weak conjugacy problem is unsolvable, while the word inequality problem and the conjugacy problem are solvable.

Consider a calculus \(I\) over the alphabet \(a_1,a_2,\ldots,a_n\), in which word transformations are carried out according to the scheme: \(X \to X,\ XA_iY \to XB_iY,\ aX \to Xa\), for arbitrary words \(X,Y\) and a letter \(a\). (We agree to regard the words \(A_i\) and \(B_i\) as nonempty.)

Lemma 1. There exists a calculus \(I_2\) for which the word derivability problem is unsolvable.

From the calculus \(I\) we construct a group \(\Gamma\). The group \(\Gamma\) is given by the alphabet \(a_1,\ldots,a_n,a_1^{-1},\ldots,a_n^{-1}\) and by the defining inequalities

\[ \begin{aligned} A_i p &\leqslant pB_i, \qquad && i=1,2,\ldots,m;\\ a_j p &\leqslant p a_j, \qquad && j=1,2,\ldots,n. \end{aligned} \]

Theorem 7 follows from Lemma 1 and the following principal lemma.

Principal Lemma. In order that a word \(X\) be weakly conjugate to a word \(Y\) in the group \(\Gamma\), where \(X\) and \(Y\) are words consisting of positive letters of the alphabet \(a_1, a_2, \ldots, a_n\), it is necessary and sufficient that \(Y\) be derivable from \(X\) in the calculus \(\Gamma\).

From Theorems 6 and 7 there follows the scheme:

\[ \boxed{\begin{gathered} \text{Weak}\\ \text{conjugacy}\\ \text{problem} \end{gathered}} \;\longrightarrow\; \boxed{\begin{gathered} \text{Inequality}\\ \text{problem} \end{gathered}} \;\longrightarrow\; \boxed{\begin{gathered} \text{Identity}\\ \text{problem} \end{gathered}} \]

where the dependence between algorithmic problems indicated by an arrow means that from the solvability of the given algorithmic problem for an arbitrary group \(\Gamma\) there follows the solvability in this group of the algorithmic problem following from it, but not conversely.

REFERENCES

\(^{1}\) P. S. Novikov, Izv. Akad. Nauk SSSR, Ser. Mat., 18, 485 (1954).
\(^{2}\) G. Birkhoff, Lattice Theory, IL, 1952.
\(^{3}\) V. I. Frenkel, Uspekhi Mat. Nauk, 17, no. 4, 173 (1962).