UDC 519.40

MATHEMATICS

M. A. TAITSLIN

ON ALGORITHMIC PROBLEMS

FOR COMMUTATIVE SEMIGROUPS

(Presented by Academician A. I. Mal’tsev on IV 8, 1967)

1. Algorithms for solving various problems in the theory of commutative semigroups were proposed by A. I. Mal’tsev \((^{1})\), Tseitlin, Biryukov \((^{2-5})\), Emelichev \((^{6})\), and Khalezov \((^{7})\). In papers \((^{8,9})\) the author described a universal algorithm which solves any problem, provided only that it is written in the language of first-order predicate calculus of a definite signature. For a more precise formulation of these results, with each commutative semigroup \(L(\mathfrak A,\mathfrak B)\), given by a finite set of generators \(\mathfrak A\) and a set of defining relations \(\mathfrak B\), associate the signature
\(\sigma(\mathfrak A)=\langle a,G_a\mid a\in\mathfrak A\rangle\), where \(G_a\) for \(a\in\mathfrak A\) is a unary predicate symbol. We shall say that an algorithm solves problem A if, from \(\mathfrak A,\mathfrak B\) and from an arbitrary closed formula \(\Phi\) of first-order predicate calculus of signature \(\sigma(\mathfrak A)\), this algorithm determines whether the formula \(\Phi\) is true in the semigroup \(L(\mathfrak A,\mathfrak B)\) in the case when the predicate \(G_a(x)\) for each \(a\in\mathfrak A\) is interpreted in \(L(\mathfrak A,\mathfrak B)\) as “\(x\) is equal to some natural multiple of \(a\).” The principal content of papers \((^{8,9})\) is the description of an algorithm solving problem A.

The algorithm described consisted in successively obtaining consequences from a certain system of axioms until either the formula \(\Phi\) or the formula \(\neg\Phi\) was derived. A certain shortcoming of papers \((^{8,9})\) was the use of model-theoretic considerations in proving completeness of the system of axioms used. This distinguished the description of the universal algorithm from constructive descriptions of algorithms for solving individual problems in the cited papers.

In the present note another algorithm is described, also solving problem A. The description given here is constructive and simpler. The following result of this note may also be of independent interest. If by the elements of a commutative semigroup with \(n\) generators one understands equivalence classes defined on \(n\)-tuples of natural numbers, then the corresponding equivalence relation is always definable on the natural numbers by some formula of first-order predicate calculus containing no nonlogical symbols other than the symbol of the addition operation.

2. In what follows, without explanation, the terminology adopted in item 1 of § 1 of the survey \((^{10})\) is used. Let an algebraic system \(\mathfrak M\) of signature \(\sigma\) have a decidable theory. Let
\(\mathfrak G(x_1,\ldots,x_k,y_1,\ldots,y_k)\) be a formula of signature \(\sigma\), containing no free variables other than \(x_1,\ldots,x_k,y_1,\ldots,y_k\). Let the relation \(\sim\), defined in the system \(\mathfrak M^k\) by the condition: for \((x_1,\ldots,x_k)\), \((y_1,\ldots,y_k)\) from \(\mathfrak M^k\) if and only if \((x_1,\ldots,x_k)\sim (y_1,\ldots,y_k)\), when \(\mathfrak G(x_1,\ldots,x_k,y_1,\ldots,y_k)\) is true in \(\mathfrak M\), be a congruence relation in \(\mathfrak M^k\). Then the factor system \(\mathfrak M^k/\sim\) also has a decidable theory. If the relation \(\sim\) and the formula
\(\mathfrak G(x_1,\ldots,x_k,y_1,y_2,\ldots,y_k)\) are connected by the condition considered above, we shall say that the relation \(\sim\) is elementary in \(\mathfrak M\) and that the formula
\(\mathfrak G(x_1,\ldots,x_k,y_1,\ldots,y_k)\) defines the relation \(\sim\).

It is known that the system \(\mathfrak N=\langle N,+\rangle\), where \(N=\{0,1,2,\ldots\}\) is the set of natural numbers and \(+\) is the usual operation of addition of natural numbers,

numbers, has a decidable theory. Therefore every factor system of the system \(\mathfrak N^k\) with respect to an elementary equivalence relation also has a decidable theory.

Each semigroup \(L(\mathfrak A,\mathfrak L)\) can be regarded as a factor semigroup of the semigroup \(\mathfrak N^k\), where \(k\) is the number of elements of \(\mathfrak A\). To solve problem A it is therefore enough to note that this factor semigroup is obtained by factoring \(\mathfrak N^k\) with respect to an elementary equivalence relation and that the formula defining this equivalence relation is effectively constructed from \(\mathfrak A\) and \(\mathfrak B\).

In the case where \(L(\mathfrak A,\mathfrak L)\) is a semigroup with cancellation, this is observed trivially. Indeed, let
\(\mathfrak A=\{a_1,\ldots,a_k\}\), \(\mathfrak L=\{(c_i,d_i)\mid i=1,\ldots,q\}\), where

\[ c_i=\sum_{j=1}^{k}\alpha_j^{(i)}a_j;\qquad d_i=\sum_{j=1}^{k}\beta_j^{(i)}a_j;\qquad \alpha_j^{(i)},\beta_j^{(i)}\in N . \]

In this case \((x_1,\ldots,x_k)\sim(y_1,\ldots,y_k)\) is equivalent to the condition

\[ (\exists t_1)\ldots(\exists t_q)(\exists t'_1)\ldots(\exists t'_q) \left(\bigwedge_{i=1}^{k} \left[ x_i+\sum_{j=1}^{q}t_j(\alpha_i^{(j)}-\beta_i^{(j)}) = y_i+\sum_{j=1}^{q}t'_j(\alpha_i^{(j)}-\beta_i^{(j)}) \right]\right). \]

This condition can obviously be represented in the form of the required formula.

1. In this paragraph we recall some definitions and results from § 1 of [9]. By \(L(\mathfrak A)\) we denote the semigroup, free in the class of commutative semigroups with zero, with set \(\mathfrak A\) of free generators. We think of the set \(L(\mathfrak A)\) as the set of all possible linear forms in the letters \(a_1,\ldots,a_k\), where \(\mathfrak A=\{a_1,\ldots,a_k\}\), with natural coefficients. A semigroup \(L(\mathfrak A,\mathfrak L)\), given in the class of commutative semigroups with zero by finite sets of generators \(\mathfrak A\) and defining relations \(\mathfrak B\), is regarded by us as a factor semigroup of the semigroup \(L(\mathfrak A)\). For \(x\in L(\mathfrak A)\), by \(\bar{x}\) we denote the image of \(x\) under the canonical mapping \(L(\mathfrak A)\to L(\mathfrak A,\mathfrak L)\). By \(M(\mathfrak A,\mathfrak L)\) we denote the semigroup, given in the class of commutative semigroups with cancellation and with zero, by finite sets of generators \(\mathfrak A\) and defining relations \(\mathfrak B\). We also regard the semigroup \(M(\mathfrak A,\mathfrak L)\) as a factor semigroup of the semigroup \(L(\mathfrak A)\). For \(x\in L(\mathfrak A)\), by \([x]\) we denote the image of \(x\) under the canonical mapping \(L(\mathfrak A)\to M(\mathfrak A,\mathfrak L)\).

For \(a\in\mathfrak A\) and \(f\in L(\mathfrak A)\), by \((f)_a\) we denote the coefficient of the letter \(a\) in the form \(f\). For \(f,g\in L(\mathfrak A)\) we write \(f\geq g\) if \((f)_a\geq(g)_a\) for all \(a\in\mathfrak A\). We write \(\mathfrak A_1\subset\mathfrak A\) if \(\mathfrak A_1\ne\mathfrak A\) and \(\mathfrak A_1\) is a subset of the set \(\mathfrak A\). The symbol \(\varnothing\) denotes the empty set. By \(L(\varnothing)\) we denote the zero semigroup. If \(\mathfrak A_1\subset\mathfrak A\), the semigroup \(L(\mathfrak A_1)\) is regarded as a subsemigroup of the semigroup \(L(\mathfrak A)\). We assume that \(\mathfrak A\subseteq L(\mathfrak A)\). For \(\mathfrak A_1\subset\mathfrak A\) and \(f\in L(\mathfrak A)\), by \(\mathfrak A_1(f)\) we denote such a form from \(L(\mathfrak A\setminus\mathfrak A_1)\) that \((\mathfrak A_1(f))_a=(f)_a\) for all \(a\in\mathfrak A\setminus\mathfrak A_1\). Let \(q(\mathfrak L)\) be the number of relations in \(\mathfrak B\) and \(\lambda(\mathfrak L)\) the greatest coefficient in these relations. By \(h(\mathfrak A,\mathfrak L)\) we denote the form

\[ \sum_{a\in\mathfrak A}\lambda(\mathfrak L)(q(\mathfrak L)+1)a \]

from \(L(\mathfrak A)\).

Let \(\mathfrak A_1\subset\mathfrak A\). By \(\mathfrak N(\mathfrak A_1,\mathfrak L)\) we denote the set of all \(x\in L(\mathfrak A\setminus\mathfrak A_1)\) such that the inequality \(\overline{x+y}\ne\overline{h(\mathfrak A,\mathfrak L)+z}\) holds in \(L(\mathfrak A,\mathfrak L)\) for all \(y\in L(\mathfrak A_1)\), \(z\in L(\mathfrak A)\). By \(\mathfrak M(\mathfrak A_1,\mathfrak L)\) we denote the set of elements of \(\mathfrak N(\mathfrak A_1,\mathfrak L)\) that are maximal in the sense of the relation \(\geq\). By \(\mathfrak M_1(\mathfrak A,\mathfrak L)\) we denote the set

\[ \{x\in L(\mathfrak A\setminus\mathfrak A_1)\mid(\exists y)(y\in\mathfrak M(\mathfrak A_1,\mathfrak L)\ \&\ y\geq x)\}. \]

The sets \(\mathfrak M(\mathfrak A_1,\mathfrak L)\) and \(\mathfrak M_1(\mathfrak A_1,\mathfrak L)\) are finite.

We divide the set \(\mathfrak M_1(\mathfrak A_1,\mathfrak L)\) into reducibility classes, assigning \(x,y\in \mathfrak M_1(\mathfrak A_1,\mathfrak L)\) to one class if there exist \(z,u\in L(\mathfrak A_1)\) such that \(\overline{x+z}=y+u\) in \(L(\mathfrak A,\mathfrak L)\). By \(i(\mathfrak A_1,\mathfrak L)\) we denote the number of reducibility classes, and by
\[ \mathfrak M_{2,1}(\mathfrak A,\mathfrak L),\ldots,\mathfrak M_{2,i(\mathfrak A_1,\mathfrak L)}(\mathfrak A,\mathfrak L) \]
all these distinct classes.

For \(x,y\in \mathfrak M_{2,j}(\mathfrak A,\mathfrak L)\), \(j\in\{1,\ldots,i(\mathfrak A_1,\mathfrak L)\}\), \(\mathfrak A_3\subset \mathfrak A_1\), by \(\mathfrak N(\mathfrak A_1,\mathfrak A_3,x,y,\mathfrak L)\) we denote the set of all such \(z\in L(\mathfrak A_1\setminus \mathfrak A_3)\) that for all \(u\in L(\mathfrak A_3)\), \(v\in L(\mathfrak A_1)\) in \(L(\mathfrak A,\mathfrak L)\) we have the inequality \(\overline{x+z+u}\ne y+v\). By \(\mathfrak M(\mathfrak A_1,\mathfrak A_3,x,y,\mathfrak L)\) we denote the set of maximal elements of \(\mathfrak N(\mathfrak A_1,\mathfrak A_3,x,y,\mathfrak L)\). The set \(\mathfrak M(\mathfrak A_1,\mathfrak A_3,x,y,\mathfrak L)\) is finite. Recall that:

(1) the sets \(\mathfrak M(\mathfrak A_1,\mathfrak L)\), \(\mathfrak M_1(\mathfrak A_1,\mathfrak L)\), \(\mathfrak M(\mathfrak A_1,\mathfrak A_3,x,y,\mathfrak L)\) are effectively constructed from \(\mathfrak A,\mathfrak B,\mathfrak A_1,\mathfrak A_3,x,y\) \((^9)\);

(2) the set \(\mathfrak M_1(\mathfrak A_1,\mathfrak L)\) is effectively, from \(\mathfrak A,\mathfrak B,\mathfrak A_1\), divided into reducibility classes \((^9)\);

(3) there exists an effective procedure which, from \(\mathfrak A,\mathfrak B,\mathfrak A_1,x\) such that \(\mathfrak A_1\subset \mathfrak A\) and \(x\in L(\mathfrak A\setminus \mathfrak A_1)\), constructs such a finite set \(\mathfrak L(\mathfrak A_1,x)\) of defining relations for \(\mathfrak A_1\) that for \(u,v\in L(\mathfrak A_1)\) then and only then \(\bar u=\bar v\) in \(L(\mathfrak A_1,\mathfrak L(\mathfrak A_1,x))\), when \(\overline{x+u}=\overline{x+v}\) in \(L(\mathfrak A,\mathfrak L)\) \((^9)\);

(4) for \(x,y\in L(\mathfrak A)\) then and only then \(\overline{x+h(\mathfrak A,\mathfrak L)}=\overline{y+h(\mathfrak A,\mathfrak L)}\) in \(L(\mathfrak A,\mathfrak L)\), when \([x]=[y]\) in \(M(\mathfrak A,\mathfrak L)\) (Zeitlin’s lemma, (7));

(5) for \(x\in L(\mathfrak A)\) either there exists \(z\in L(\mathfrak A)\) such that \(\bar x=\overline{h(\mathfrak A,\mathfrak L)+z}\) in \(L(\mathfrak A,\mathfrak L)\), or there exists \(\mathfrak A_1\subset \mathfrak A\) such that \(\mathfrak A_1(x)\in \mathfrak M_1(\mathfrak A_1,\mathfrak L)\) \((^9)\);

(6) for \(\mathfrak A_1\subset \mathfrak A\), \(x,y\in \mathfrak M_{2,j}(\mathfrak A_1,\mathfrak L)\), \(j\in\{1,\ldots,i(\mathfrak A_1,\mathfrak L)\}\), \(u\in L(\mathfrak A_1)\), either there exists \(v\in L(\mathfrak A_1)\) such that \(\overline{x+u}=\overline{y+v}\) in \(L(\mathfrak A,\mathfrak L)\), or there exist \(\mathfrak A_3\subset \mathfrak A_1\) and \(v\in \mathfrak M(\mathfrak A_1,\mathfrak A_3,x,y,\mathfrak L)\) such that \(\mathfrak A_3(u)\le v\) \((^9)\).

4. Theorem. Let \(\mathfrak A=\{a_1,\ldots,a_k\}\). The equivalence relation in \(\mathfrak N^k\), defined by the condition
\[ (\alpha_1,\ldots,\alpha_k)\sim(\beta_1,\ldots,\beta_k) \]
if and only if
\[ \sum_{i=1}^{k}\alpha_i\bar a_i=\sum_{i=1}^{k}\beta_i\bar a_i \]
in \(L(\mathfrak A,\mathfrak L)\), is elementary in \(\mathfrak N\). There exists an effective procedure which, from \(\mathfrak A\) and \(\mathfrak B\), constructs a formula
\[ \mathfrak C(\mathfrak A,\mathfrak L;\alpha_a,\beta_a\mid a\in\mathfrak A) \]
defining the relation \(\sim\).

Proof is carried out by induction on the number of elements of \(\mathfrak A\). For a one-element \(\mathfrak A\) it is obvious. Suppose that we can construct
\[ \mathfrak C(\mathfrak A,\mathfrak B;\alpha_a,\beta_a\mid a\in\mathfrak A) \]
in the case when \(\mathfrak A\) contains fewer than \(k\) elements. Let \(\mathfrak A\) contain \(k\) elements. Let
\[ \mathfrak C^*(\mathfrak A,\mathfrak L;\alpha_a,\beta_a\mid a\in\mathfrak A) \]
denote a formula which defines such a congruence relation \(\sim_1\) in \(\mathfrak N^k\) that the semigroups \(M(\mathfrak A,\mathfrak L)\) and \(\mathfrak N^k/\sim_1\) are isomorphic.

For \(\mathfrak A_1\subset \mathfrak A\), \(j\in\{1,\ldots,i(\mathfrak A_1,\mathfrak L)\}\), \(u,v\in \mathfrak M_{2,j}(\mathfrak A,\mathfrak L)\), by \(\Phi(\mathfrak A_1,u,v)\) we denote the set of all such functions \(\varphi\) which to each \(\mathfrak A_3\subset \mathfrak A_1\) and each \(e\in \mathfrak M(\mathfrak A_1,\mathfrak A_3,u,v,\mathfrak L)\) assign an element \(\varphi(\mathfrak A_3,e)\in \mathfrak A_1\setminus \mathfrak A_3\). For \(\varphi\in \Phi(\mathfrak A_1,u,v)\) consider the set
\[ \mathfrak A(\varphi)=\{\varphi(\mathfrak A_3,e)\mid \mathfrak A_3\subset \mathfrak A_1,\ e\in \mathfrak M(\mathfrak A_1,\mathfrak A_3,u,v,\mathfrak L)\}. \]
For \(a\in \mathfrak A(\varphi)\) by \(\mu_a\) we denote the natural number
\[ \max \{(e)_a\mid \mathfrak A_3\subset \mathfrak A_1,\ e\in \mathfrak M(\mathfrak A_1,\mathfrak A_3,u,v,\mathfrak L),\ \varphi(\mathfrak A_3,e)=a\}. \]
By \(F_\varphi(u,v,\mathfrak A_1)\) we denote the conjunction of all formulas
\[ \alpha_a=(u)_a,\qquad \beta_a=(v)_a,\qquad \alpha_c=\mu_c+z_c+1 \]
for all possible \(a\in \mathfrak A\setminus \mathfrak A_1\), \(c\in \mathfrak A(\varphi)\), where \(z_c,\alpha_c,\alpha_a,\beta_a\) are symbols of object variables.

Let
\[ d=\sum_{a\in \mathfrak A\setminus \mathfrak A_1}\alpha_a a+\sum_{a\in \mathfrak A(\varphi)}(\mu_a+1)a. \]

From (6) it follows that \(\bar d=\bar v+\bar d_1\) in \(L(\mathfrak A,\mathfrak L)\) for some \(d_1\in L(\mathfrak A_1)\). Now in \(\mathfrak C(\mathfrak A,\mathfrak L(\mathfrak A_1,v);\,\alpha_a,\beta_a\mid a\in\mathfrak A_1)\), in place of \(\alpha_a\) substitute \((d_1)_a+z_a\) if \(a\in\mathfrak A(\varphi)\), and substitute \(\alpha_a+(d_1)_a\) if \(a\notin\mathfrak A(\varphi)\). We obtain the formula \(G_\varphi(u,v,\mathfrak A_1)\). By \(D_\varphi(u,v,\mathfrak A_1)\) we denote the formula that is obtained if, before the formula \(F_\varphi(u,v,\mathfrak A_1)\ \&\ G_\varphi(u,v,\mathfrak A_1)\), one prefixes existential quantifiers over \(z_c\) for all \(c\in\mathfrak A(\varphi)\). By \(D(\mathfrak A_1,u,v)\) we denote the disjunction of the formulas \(D_\varphi(u,v,\mathfrak A_1)\) for all \(\varphi\in\Phi(\mathfrak A_1,u,v)\).

By \(\Psi\) we denote the set of all such functions \(\psi\) that to each \(\mathfrak A_1\subset\mathfrak A\) and to each \(c\in\mathfrak M(\mathfrak A_1,\mathfrak L)\) assign an element \(\psi(\mathfrak A_1,c)\in\mathfrak A\setminus\mathfrak A_1\). For \(\psi\in\Psi\) consider the set \(\mathfrak A_\psi=\{\psi(\mathfrak A_1,c)\mid \mathfrak A_1\subset\mathfrak A,\ c\in\mathfrak M(\mathfrak A_1,\mathfrak L)\}\). For \(a\in\mathfrak A_\psi\), by \(\nu_a\) we denote the natural number

\[ \max\{(c)_a\mid \mathfrak A_1\subset\mathfrak A,\ c\in\mathfrak M(\mathfrak A_1,\mathfrak L),\ \psi(\mathfrak A_1,c)=a\}. \]

By \(H_\psi(\alpha_c,z_c\mid c\in\mathfrak A_\psi)\) we denote the conjunction of all formulas \(\alpha_c=\nu_c+z_c+1\) for all possible \(c\in\mathfrak A_\psi\), where \(\alpha_c,z_c\) are symbols for object variables.

Let

\[ f_\psi=\sum_{a\in\mathfrak A_\psi}(\nu_a+1)a. \]

From (5) it follows that \(\bar f_\psi=\overline{h(\mathfrak A,\mathfrak L)}+g_\psi\) for some \(g_\psi\in L(\mathfrak A)\). By \(B(\psi,\tau)\) we denote the formula obtained if, in \(\mathfrak C^*(\mathfrak A,\mathfrak L;\,\alpha_a,\beta_a\mid a\in\mathfrak A)\), one substitutes \(z_c+(g_\psi)_c\) for \(\alpha_c\), for \(c\in\mathfrak A_\psi\); substitutes \(\alpha_a+(g_\psi)_a\) for \(\alpha_a\), for all \(a\in\mathfrak A\setminus\mathfrak A_\psi\); substitutes \(y_d+(g_\tau)_d\) for \(\beta_d\), for all \(d\in\mathfrak A_\tau\); and substitutes \(\beta_a+(g_\tau)_a\) for \(\beta_a\), for all \(a\in\mathfrak A\setminus\mathfrak A_\tau\). By \(E(\psi,\tau)\) we denote the formula obtained if, before the conjunction

\[ H_\psi(\alpha_c,z_c\mid c\in\mathfrak A_\psi)\ \&\ H_\tau(\beta_d,y_d\mid d\in\mathfrak A_\tau)\ \&\ B(\psi,\tau) \]

one prefixes existential quantifiers over all \(z_c\) and \(y_d\), for \(c\in\mathfrak A_\psi\) and \(d\in\mathfrak A_\tau\).

From (4), (5), (6) it follows that as \(\mathfrak C(\mathfrak A,\mathfrak L;\,\alpha_a,\beta_a\mid a\in\mathfrak A)\) one may take the disjunction of all formulas \(E(\psi,\tau)\), \(D(\mathfrak A_1,u,v)\) for all possible \(\psi,\tau\in\Psi\), all possible \(\mathfrak A_1\subset\mathfrak A\), all possible \(j\in\{1,\ldots,i(\mathfrak A_1,\mathfrak L)\}\), and all possible \(u,v\in\mathfrak M_{2,j}(\mathfrak A_1,\mathfrak L)\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
31 III 1967.

REFERENCES

1. A. I. Mal'tsev, Scientific Notes of the Ivanovo Pedagogical Institute, 6, 227 (1958).
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