UDC 519

MATHEMATICS

V. A. OSIPOVA

ON THE IDENTITY PROBLEM FOR FINITELY PRESENTED SEMIGROUPS

(Presented by Academician P. S. Novikov, 13 IV 1967)

Let the semigroup (\Pi) be given by generators

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

and defining relations

[
A_i = B_i \quad (\text{the words } A_i \text{ and } B_i \text{ are nonempty, } i=1,2,\ldots,k).
\tag{2}
]

The set (M={A_i}\cup{B_i}) ((i=1,2,\ldots,k)) is called the basis of the semigroup (\Pi). The beginning (end) (P) of a word (X) will be called a proper beginning (proper end) of the word (X), if (\partial(P)\geq \frac12\partial(X)).

We shall say that the semigroup (\Pi) with generators (1) and defining relations (2) belongs to the class (K_{1/2}), if its basis (M) satisfies the following conditions: a) if (A,B\in M), (B \stackrel{\circ}{=} RPQ), and (P) is a proper beginning of the word (A), then (R) is empty; b) if (A,B\in M), (B \stackrel{\circ}{=} RPQ), and (Q) is a proper end of the word (A), then (Q) is empty.

Theorem 1. If (\Pi\in K_{1/2}), then for every word (Z) in the alphabet (1) there exists only a finite number of words equal to it in (\Pi).

From Theorem 1 it follows easily that

Theorem 2. There exists an algorithm that solves the identity problem for all semigroups of the class (K_{1/2}).

The same result for a substantially narrower class of semigroups was obtained by E. I. Grindlinger (see ((^2))).

By (\overline{K}{1/2}) we denote the class of semigroups obtained analogously to the class (K), if in the definition of proper beginnings and proper ends one considers, instead of the non-strict inequality, the strict inequality (\partial(P) > \frac12\partial(X)).

It is interesting to note that in the class (\overline{K}_{1/2}) one can indicate an example of a finitely presented semigroup with an unsolvable identity problem. Such examples are easily constructed on the basis of the known result of G. S. Tseitin ((^4)). In particular, the semigroup given by the generators (a,b,c,d,e,m,n,p,q,m',n',y',q',\alpha,\beta,\gamma,\delta,\eta,\theta) and defining relations

[
\begin{gathered}
ac=ca,\quad ad=da,\quad bc=cb,\quad bd=db,\quad e=mn,\quad nc=cp,\quad mc=cq,\
qpa=e,\quad e=m'n',\quad n'd=dp',\quad m'd=dq',\quad q'p'b=e,\quad ca=\alpha\beta\gamma,\
c\alpha=\delta\theta,\quad cc=\delta\eta,\quad e=\alpha\gamma,\quad \eta aa=\theta\beta,
\end{gathered}
\tag{3}
]

belongs to the class (\overline{K}_{1/2}), and in it the identity problem is unsolvable. An analogous example is indicated in the paper of E. I. Grindlinger ((^3)).

We proceed to the proof of Theorem 1. Let (\Pi\in K_{1/2}). A word

[
E \stackrel{\circ}{=} S_1S_2\ldots S_t
\tag{4}
]

in the alphabet (1) will be called normal if the factors (S_1,S_2,\ldots,S_t) satisfy the conditions:

I. Each (S_i) is contained in some (C_i\in M).

II. (S_1) is a proper beginning of (C_1); (S_t) is a proper end of (C_t).

III. For each (i<t), either (S_i) is a proper end (C_i), or (S_{i+1}) is a proper beginning (C_{i+1}).

Every decomposition (4) with properties I, II, and III will be called a normal decomposition of the word (E).

Lemma 1. If the word (PAQ), where (A\in M), has a normal decomposition consisting of (t) factors, then one can indicate such a normal decomposition (4) of the word (PAQ) that (P\overset{\circ}{=}S_1S_2\ldots S_{j-1}), (A\overset{\circ}{=}S_j).

Proof. Let (PAQ\overset{\circ}{=}R_1R_2\ldots R_t) be a normal decomposition, where (C_\xi\overset{\circ}{=}u_\xi R_\xi q_\xi), (C_\xi\in M) ((\xi=1,2,\ldots,t)). Obviously, one can indicate such an (i\le t) that (P\overset{\circ}{=}R_1R_2\ldots R_i'), (R_i\overset{\circ}{=}R_i'A'), (A\overset{\circ}{=}A'A''), where (A') is nonempty. Two cases are possible: 1) (\partial(A')\ge \frac12\partial(A)); 2) (\partial(A')<\frac12\partial(A)).

Case 1. In this case (A') is a proper beginning of the word (A); (C_i\overset{\circ}{=}u_iR_i'A'q_i), and, by condition a), (u_iR_i') is empty and (R_i\overset{\circ}{=}A'). If in addition (A'') is empty, then (R_i\overset{\circ}{=}A). Then put (S_1\rightleftarrows R_1,\ldots,S_{i-1}\rightleftarrows R_{i-1}), (S_i\rightleftarrows A), (S_{i+1}\rightleftarrows R_{i+1},\ldots,S_t\rightleftarrows R_t) ((\rightleftarrows) denotes equality by definition). Obviously, the required decomposition (4) has been found.

Let (A'') be nonempty. Then (R_i) is not a proper end; indeed, suppose (R_i) is a proper end. Then (q_i) is empty and (C_i\overset{\circ}{=}A'), i.e. (A') is a proper end (C_i), and then (A') is an end of (A), i.e. (A'') is empty, contrary to the assumption. From condition III it follows that (R_{i+1}) is a proper beginning. If (A''\overset{\circ}{=}R_{i+1}Z_1) for some (Z_1), then (R_{i+1}) must be a beginning of (A), which is impossible, since (A') is nonempty. Therefore (R_{i+1}\overset{\circ}{=}A''R_{i+1}''), where (R_{i+1}'') is nonempty. Note that if (R_{i+1}) is at the same time also a proper end, then (R_{i+1}'') is a proper end (C_{i+1}). Put (S_1\rightleftarrows R_1,\ldots,S_{i-1}\rightleftarrows R_{i-1}), (S_i\rightleftarrows A), (S_{i+1}\rightleftarrows R_{i+1}'', S_{i+2}\rightleftarrows R_{i+2},\ldots,S_t\rightleftarrows R_t), and the assertion of Lemma 1 is fulfilled for (j=i).

Case 2. We have (\partial(A'')>\frac12\partial(A)), consequently (A'') is a proper end of (A). If for some (Z_2) we have (R_{i+1}\overset{\circ}{=}A''Z_2), then (Z_2) is empty, since (A'') must be an end of (C_{i+1}). Therefore one may suppose that (A''\overset{\circ}{=}R_{i+1}Z_3) for some (Z_3).

Since (A') is nonempty, (R_{i+1}) is not a proper beginning. Indeed, suppose (R_{i+1}) is a proper beginning. Then, since (A\overset{\circ}{=}A'R_{i+1}Z_3), (R_{i+1}) is a beginning of (A), i.e. (A') is empty, contrary to the assumption. From condition III it follows that (R_i) is a proper end. If at the same time (R_i') is empty, then (R_i\overset{\circ}{=}A'), and (A') must be an end of (A). But this is impossible, since (A'') is nonempty. Hence (R_i') is nonempty. At the same time, if (R_i) was also a proper beginning, then (R_i\overset{\circ}{=}C_i). Then (R_i') is a proper beginning for (C_i), since (\partial(A)<\frac12\partial(C_i)).

If (R_{i+1}) is a proper end, then (Z_3) is empty. Then put (S_1\rightleftarrows R_1,\ldots,S_{i-1}\rightleftarrows R_{i-1}), (S_i\rightleftarrows R_i'), (\ldots), (S_{i+1}\rightleftarrows A), (S_{i+2}\rightleftarrows R_{i+2},\ldots,S_t\rightleftarrows R_t), and for (j=i+1) the assertion of Lemma 1 is fulfilled.

Let (R_{i+1}) not be a proper end. Then (R_{i+2}) is a proper beginning. If (Z_3\overset{\circ}{=}R_{i+2}Z_4) for some (Z_4), then (R_{i+2}) must be a beginning of (A), but this is impossible. Consequently, we have (R_{i+2}\overset{\circ}{=}Z_3R_{i+2}'), where (R_{i+2}') is nonempty. Note that if (R_{i+2}) is at the same time also a proper end, then (R_{i+2}\overset{\circ}{=}C_{i+2}). Then (R_{i+2}') is a proper end (C_{i+2}), since (\partial(Z_3)<\frac12\partial(C_{i+2})). The latter follows from the fact that in the word (A\overset{\circ}{=}A'R_{i+1}Z_3), (Z_3) is not a beginning. Put (S_1\rightleftarrows R_1,\ldots,S_{i-1}\rightleftarrows R_{i-1}), (S_i\rightleftarrows R_i'), (S_{i+1}\rightleftarrows A), (S_{i+2}\rightleftarrows R_{i+2}'), (S_{i+3}\rightleftarrows R_{i+3},\ldots,S_t\rightleftarrows R_t), (j=i+1). Lemma 1 is proved.

From Lemma 1 it follows immediately that

Lemma 2. If (PAQ) has a normal decomposition into (t) factors, and (PAQ\to PBQ) is an elementary transformation of the semigroup (\Pi) (see (1), p. 7), then (PBQ) also has a normal decomposition into (t) factors.

Let us call a normal word (V) the maximal normal beginning of the word (VQ), if no word (VQ'), with nonempty (Q_1) and (Q\overset{\circ}{=}Q_1Q_2)

is not a normal word. The following lemma follows directly from Lemma 2.

Lemma 3. Let (V) be the maximal normal beginning of the word (VQ), and let (V\to V_1) be an elementary transformation of (\Pi). Then (V_1) is the maximal normal beginning of the word (V_1Q).

Lemma 4. If (V) is the maximal normal beginning of the word (VY) and (VY \stackrel{\circ}{=} PAQ), where (A\in M), then either (\partial(V)<\partial(P)), or (\partial(V)\geq \partial(PA)).

Proof. Suppose that (\partial(P)\leq \partial(V)<\partial(PA)). Then we have (V\stackrel{\circ}{=}PQ'), (A\stackrel{\circ}{=}A'A''), (Y\stackrel{\circ}{=}A''Q), where (A'') is nonempty. We shall prove that (VA'') is a normal word. Let (R_1\ldots R_t) be the normal decomposition of the word (V). It is enough to prove that (A'') is a proper end. Suppose this is not so. Then (A') is a proper beginning.

Either (R_t) is an end of the word (A'), or (A') is an end of (R_t). Arguing as in the proof of Lemma 1, it is easy to verify that both these cases are impossible. Consequently, (VA'') is a normal beginning of (VY). This contradicts the fact that (V) is the maximal normal beginning of (VY), since (A'') is nonempty. Lemma 4 is proved.

Lemma 5. If (\partial(X)=l) and (X=Y) in (\Pi), then (\partial(Y)\leq lm^l), where (m=\max{\partial(A_i),\partial(B_i)}).

We shall prove this assertion by induction on the length of the word (X).

Let (\partial(X)=0), i.e., (X) is the empty word. If (X=Y) in (\Pi), then (Y) is the empty word, since the defining relations (2) consist of nonempty words. Therefore, (\partial(Y)=0).

Let (\partial(X)=l>0). Then (X\stackrel{\circ}{=}aW), where (a) is one of the generators (1). By the induction hypothesis, the length of any word (W_j) equal to (W) in (\Pi) does not exceed ((l-1)m^{l-1}). Let (Y\stackrel{\circ}{=}aW'). Among all sequences of elementary transformations (aW_j\stackrel{\circ}{=}F_0\to\cdots\to F_q\stackrel{\circ}{=}Y), for various words (W_j) equal to (W), choose a sequence

[
aW_k\stackrel{\circ}{=}E_0\to E_1\to\cdots\to E'i\to EY,}\to\cdots\to E_2\stackrel{\circ}{=
\tag{5}
]

containing the smallest number of transformations. Let (aW_k\stackrel{\circ}{=}VQ), where (V) is the maximal normal beginning of (VQ).

If (V) is empty, then the first elementary transformation in (5) has the form (P_1AP_2\to P_1BP_2), where (P_1) is nonempty. Then (P_1\stackrel{\circ}{=}aP'_1). Put (W_s\stackrel{\circ}{=}P'_1AP_2). Obviously (W_s=W_k=W) in (\Pi), and the sequence (aW_s\stackrel{\circ}{=}E_1\to E_2\to\cdots\to E_r\stackrel{\circ}{=}Y) is shorter than (5), which contradicts the condition by which the sequence (5) was chosen. Consequently, if (V) is empty, then (Y\stackrel{\circ}{=}aW_k), and

[
\partial(Y)\leq 1+(l-1)m^{l-1}\leq lm^l.
]

If (V) is nonempty, then (V\stackrel{\circ}{=}aV') and (W_k\stackrel{\circ}{=}V'Z). By induction on the number (i) of the word (E_i), we shall prove that (E_i\stackrel{\circ}{=}V_iZ), where (V_i) is the maximal normal beginning of (E_i), and that there exists a sequence

[
aV'\stackrel{\circ}{=}V_0\to V_1\to\cdots\to V_i,
\tag{6}
]

containing exactly (i) elementary transformations. The word (E_0) has the indicated form. Suppose that our assertion is true for (E_t), and let us prove it for (E_{t+1}). Let the elementary transformation (E_t\to E_{t+1}) have the form (PAQ\to PBQ), where (A,B\in M). By Lemma 4, either (\partial(V_t)<\partial(P)), or (\partial(V_t)\geq \partial(PA)). We shall prove that the first case is impossible. Indeed, in this case we have (P\stackrel{\circ}{=}V_tP') and (Z\stackrel{\circ}{=}P'AQ). Then (W_k\stackrel{\circ}{=}V'P'BQ) in (\Pi), whence (V'P'BQ=W) in (\Pi). With the aid of the sequence (6), it is easy to construct the sequence of elementary transformations
[
aV'P'BQ\to V_1P'BQ\to\cdots\to V_tP'BQ\stackrel{\circ}{=}E_{t+1}\to E_{t+2}\to\cdots\to E_r,
]
which is shorter than the sequence (5).

In the second case we have (V_t\stackrel{\circ}{=}PAV't) and (Q\stackrel{\circ}{=}V'_tZ). Then (EPBV'_tZ), and the transition (PAV'_t\to PBV'_t) is an elementary transformation of (\Pi). By Lem-}\stackrel{\circ}{=

3, (PBV_t') is the maximal normal beginning of the word (PBV_t'Z), i.e. (PBV_t') is the required maximal normal beginning of (V_{t+1}). Consequently, (Y \circ V_rZ), where (V_r = V) in (\Pi). Then

[
\partial(Y)=\partial(V_r)+\partial(Z)\leq m[1+(l-1)m^{l-1}-\partial(Z)]+\partial(Z)=
]
[
=\partial(Z)(1-m)+lm^l+m(1-m)^{l-1}\leq lm^l .
]

Lemma 5 is proved.

The assertion of Theorem 1 follows directly from Lemma 5.

Moscow State University
named after M. V. Lomonosov

Received
12 IV 1967

References Cited

(^{1}) S. I. Adyan, Tr. Mat. Inst. im. V. A. Steklova, Academy of Sciences of the USSR, 85 (1966).
(^{2}) E. I. Grinlinger, Siberian Math. J., 5, No. 1, 77—85 (1964).
(^{3}) E. I. Grinlinger, DAN, 171, No. 3, 519 (1966).
(^{4}) G. S. Tseitlin, Tr. Mat. Inst. im. V. A. Steklova, Academy of Sciences of the USSR, 52, 172 (1958).