UDC 519

MATHEMATICS

E. I. GRINDLINGER

ON THE UNSOLVABILITY OF THE WORD IDENTITY PROBLEM FOR ONE CLASS OF SEMIGROUPS WITH A SOLVABLE ISOMORPHISM PROBLEM

(Presented by Academician P. S. Novikov, February 4, 1966)

In the paper (²) the isomorphism problem was solved for the class \(K_1\) of semigroups defined by systems of defining relations such that none of the defining words occurs in any one of the defining words distinct from it. The question of the solvability of the identity problem for these semigroups remained open, although for a certain subclass of the class of semigroups \(K_1\) the identity problem was solved in (¹).

Theorem 1. The class \(K_1\) contains a semigroup \(\mathfrak A\) whose word identity problem has the highest recursively enumerable degree of unsolvability.*

The semigroup \(\mathfrak A\) is defined by 15 generating elements: \(a, b, c, d, e, f, x, y, p, r, s, q, m, n, v\), and 13 defining relations:

\[ ad = da,\qquad bc = cb,\qquad bd = db,\qquad ac = ca = vn,\qquad aef = mn,\qquad ccm = cv, \]

\[ fc = rs,\qquad sa = qf,\qquad erq = ce,\qquad fd = xy,\qquad yb = pf,\qquad exp = de. \]

The semigroup \(\mathfrak A\) contains, as a subsemigroup, the semigroup \(\mathfrak B\) of G. S. Tseitin (⁶), defined by generating elements \(a, b, c, d, e\) and defining relations:

\[ ac = ca,\qquad ad = da,\qquad bc = cb,\qquad bd = db, \]

\[ eca = ce,\qquad edb = de,\qquad cca = ccae. \]

It is easy to verify that the mapping \(\varphi\) of the semigroup \(\mathfrak B\) into the semigroup \(\mathfrak A\), which sends

\[ a \to a,\qquad b \to b,\qquad c \to c,\qquad d \to d,\qquad e \to ef, \]

is a homomorphism. The proof that \(\varphi\) is a monomorphism is omitted because of its cumbersome nature.

From the works of G. S. Tseitin (⁶) and P. S. Novikov (⁵) it is clear that the word identity problem of the semigroup \(\mathfrak B\), and consequently also of the semigroup \(\mathfrak A\), has the highest recursively enumerable degree of unsolvability.

Definition 1. By the strictly restricted isomorphism problem for a finitely defined semigroup \(\mathfrak A\), given by generating elements \(a_1, \ldots, a_n\) and defining relations \(A_i = B_i\) \((i = 1, \ldots, k)\), we shall understand the following problem: to find an algorithm which, for any semigroup \(\mathfrak B\) given by the same number \(n\) of generators and by a finite number of defining relations, makes it possible to determine whether the semigroup \(\mathfrak B\) is isomorphic to the semigroup \(\mathfrak A\), or to prove that no such algorithm exists.

Theorem 2. Let the semigroup \(\mathfrak A\) belong to the class \(K_1\). Then the degree of unsolvability of the word identity problem of the semigroup \(\mathfrak A\) is \(\leq\) the degree of unsolvability of the strictly restricted isomorphism problem for \(\mathfrak A\).

* For the definition of the notion of the degree of unsolvability, see (³).

Theorem 2 easily follows from the theorem that the semigroup \(\mathfrak A\) is not isomorphic to any of its proper factor-semigroups (the latter theorem is stated without proof in paper \((^2)\), Remark 3).

Theorem 3. Let the semigroup \(\mathfrak A\) belong to the class \(K_1\). Then the degree of unsolvability of the strictly bounded isomorphism problem for the semigroup \(\mathfrak A\) is \(\leq\) the degree of unsolvability of the word identity problem for the semigroup \(\mathfrak A\).

In the proof of Theorem 3 the following lemma is used:

Lemma 1. Let \(\mathfrak A\) be a semigroup of the class \(K_1\), given by generators \(a_1,\ldots,a_n\) and defining relations \(A_i = B_i\) \((i = 1,\ldots,k)\). Let \(\mathfrak B\) be any semigroup isomorphic to the semigroup \(\mathfrak A\) and given by generators \(b_1,\ldots,b_m\) and defining relations \(C_j = D_j\) \((j = 1,\ldots,s)\), where \(C_j \ne D_j\) (for all \(j\)). Then any isomorphism \(\varphi\) of the semigroup \(\mathfrak B\) onto the semigroup \(\mathfrak A\) maps the set \(\{b_k\}\) one-to-one onto \(\{a_k\}\) \((k = 1,\ldots,m)\); \(\varphi^{-1}(A_i)\) and \(\varphi^{-1}(B_i)\) are defining words of the semigroup \(\mathfrak B\), and all defining words of the semigroup \(\mathfrak B\) are nonempty. Moreover, if some defining word \(C_j\) or \(D_j\) of the semigroup \(\mathfrak B\) does not contain another defining word of the semigroup \(\mathfrak B\) as a proper subword, then the equality \(\varphi(C_j)=\varphi(D_j)\) is a defining relation in the semigroup \(\mathfrak A\).

From Theorems 2 and 3 follows Theorem 4:

Theorem 4. Let the semigroup \(\mathfrak A\) belong to the class \(K_1\). Then the degree of unsolvability of the strictly bounded isomorphism problem for the semigroup \(\mathfrak A\) is equal to the degree of unsolvability of the word identity problem for the semigroup \(\mathfrak A\).

Theorem 5. The class \(K_1\) contains both an infinite number of pairwise nonisomorphic semigroups with solvable strictly bounded isomorphism problem and an infinite number of pairwise nonisomorphic semigroups, each of which is not isomorphic to any semigroup with a smaller number of generators and has the highest recursively enumerable degree of unsolvability of this problem.

It is interesting to compare Theorem 5 with the result of A. A. Markov \((^4)\) that, for every semigroup with \(n\) generators, the problem of recognizing the isomorphism of this semigroup to any previously given semigroup with \(n+4\) generators is unsolvable. In an unpublished work, G. S. Tseitin improved A. A. Markov’s result by replacing the number of generators \(n+4\) by the number of generators \(n+2\). Well-known examples of semigroups for which the strictly bounded isomorphism problem is unsolvable are, however, isomorphic to semigroups with a smaller number of generators.

In conclusion I express my gratitude for advice and attention to the work to M. Grindlinger, S. I. Adian, A. A. Markov, and P. S. Novikov.

Ivanovo Institute of Chemical Technology

Received
19 I 1966

CITED LITERATURE

\(^1\) E. I. Grindlinger, Sibirsk. matem. zhurn., 5, No. 1, 77 (1964).
\(^2\) E. I. Grindlinger, Sibirsk. matem. zhurn., 5, No. 4, 788 (1964).
\(^3\) S. C. Kleene, E. L. Post, An. Math., 59, 379 (1954).
\(^4\) A. A. Markov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 42 (1954).
\(^5\) P. S. Novikov, ibid., 44 (1955).
\(^6\) G. S. Tseitin, ibid., 52, 173 (1958).