MATHEMATICS

V. P. ZAROVNYI

ON THE REPRESENTATION OF THE TRANSITION FUNCTION OF A MACHINE BY ASSOCIATIVE OPERATIONS

(Presented by Academician A. I. Mal’tsev on 6 XII 1962)

1. Ginzburg \((^1)\) calls a machine a triple \(A=A(\mathfrak A,\mathfrak X,\delta)\) consisting of a set of states \(\mathfrak A\), a semigroup of input signals \(\mathfrak X\), and a transition function \(\delta\), defined on \(\mathfrak A\times\mathfrak X\) with values in \(\mathfrak A\) and satisfying the condition
\[ \delta(a,x_1x_2)=\delta[\delta(a,x_1),x_2]. \]
We shall call machines in Ginzburg’s sense right machines. Alongside them we introduce into consideration left machines, by which we shall mean the same kind of triple, with the only difference that the transition function \(\Delta\) is to be regarded as defined on \(\mathfrak X\times\mathfrak A\) and subject to the condition (writing this function from right to left)
\[ (x_1x_2,a)\Delta=[x_1,(x_2,a)\Delta]\Delta . \]

A pair \((\varphi,\psi)\) is called an isomorphism of the right machine \(P=P(\mathfrak A,\mathfrak X,\delta)\) onto the right machine \(P'=P'(\mathfrak A',\mathfrak X',\delta')\), if \(\varphi\) is a one-to-one mapping of \(\mathfrak A\) onto \(\mathfrak A'\), \(\psi\) is an isomorphism of the semigroup \(\mathfrak X\) onto \(\mathfrak X'\), and, moreover,
\[ \varphi[\delta(a,x)]=\delta'[\varphi(a),\psi(x)]. \]
The isomorphism of left machines can be defined analogously.

If, in a right (left) machine, the set of states forms a semigroup with respect to some fixed multiplication operation in it, then we shall speak of a \(p\)-machine. Speaking of an isomorphism \((\varphi,\psi)\) of \(p\)-machines, we shall require of \(\varphi\) that it be an isomorphism of the semigroups of states.

2. We shall call a triple \(M=(\bar A,\bar{\mathfrak A},\bar{\mathfrak X})\) a right (left) \(m\)-semigroup if \(\bar A\) is a semigroup, \(\bar{\mathfrak A}\) is its subset, \(\bar{\mathfrak X}\) is a subsemigroup, and moreover \(\bar{\mathfrak A}\bar{\mathfrak X}\subseteq\bar{\mathfrak A}\) (respectively \(\bar{\mathfrak X}\bar{\mathfrak A}\subseteq\bar{\mathfrak A}\)) and \(\{\bar{\mathfrak A},\bar{\mathfrak X}\}=\bar A\), where \(\{\bar{\mathfrak A},\bar{\mathfrak X}\}\) is the subsemigroup in \(\bar A\) generated by the sets \(\bar{\mathfrak A}\) and \(\bar{\mathfrak X}\). If, in particular, \(\bar{\mathfrak A}\) is a right (left) ideal, then the \(m\)-semigroup will be called an ideal right (left) \(m\)-semigroup. If \(\bar{\mathfrak A}\) is a two-sided ideal, then the \(m\)-semigroup will be called two-sided.

If a right \(m\)-semigroup \(M=(\bar A,\bar{\mathfrak A},\bar{\mathfrak X})\) is given, then, putting
\[ \delta_M(\bar a,\bar x)=\bar a\bar x,\quad \bar x\in\bar{\mathfrak X},\ \bar a\in\bar{\mathfrak A}, \]
we obtain a right machine \(P_M=P_M(\bar{\mathfrak A},\bar{\mathfrak X},\delta_M)\), of which we shall say that it is generated by the \(m\)-semigroup \(M\); analogously, if \(M\) is a left \(m\)-semigroup, then it generates a left machine \(L=L(\bar{\mathfrak A},\bar{\mathfrak X},\Delta)\), where
\[ (\bar x,\bar a)\Delta=\bar x\bar a . \]
We shall agree to say that a given right (left) \(m\)-semigroup \(M\) represents the right (left) machine \(A\), if \(A\) is isomorphically mapped onto the machine generated by the \(m\)-semigroup \(M\). In other words, a right \(m\)-semigroup \(M=(\bar A,\bar{\mathfrak A},\bar{\mathfrak X})\) represents the right machine \(P=P(\mathfrak A,\mathfrak X,\delta)\) if there exists a pair \((\varphi,\psi)\) consisting of a one-to-one mapping \(\varphi\) of \(\mathfrak A\) onto \(\bar{\mathfrak A}\) and an isomorphism \(\psi\) of \(\mathfrak X\) onto \(\bar{\mathfrak X}\), for which the equality
\[ \delta(a,x)=\varphi^{-1}[\varphi(\bar a)\psi(\bar x)] \]
holds. Analogously for left \(m\)-semigroups and machines.

An isomorphism of two \(m\)-semigroups is an isomorphism of the machines generated by them. In order for two \(m\)-semigroups to represent isomorphic machines, it is necessary and sufficient that they themselves be isomorphic. Ideal \(m\)-semigroups represent \(p\)-machines, and conversely: if a \(p\)-machine is representable by an \(m\)-semigroup, then it is representable by an ideal one.

Proposition 1. Any right machine \(A=A(\mathfrak A,\mathfrak X,\delta)\) can be isomorphically embedded in such a right \(p\)-machine \(A'=A'(\mathfrak A',\mathfrak X,\delta')\), representable by a right (of course, ideal) \(m\)-semigroup, that \(\mathfrak A'\) differs from \(\mathfrak A\) by only one element.

Proposition 2. If a right \(p\)-machine can be represented by a right (of course, ideal) \(m\)-semigroup, then it can be isomorphically embedded in such a right \(p\)-machine that is representable by a two-sided \(m\)-semigroup.

These propositions show that in studying representations of machines by \(m\)-semigroups, attention must be paid to \(p\)-machines and to their representations by ideal (in particular, two-sided) \(m\)-semigroups.

1. The condition for representability of a right \(p\)-machine by a right \(m\)-semigroup is given by

Theorem 1. In order that a right \(p\)-machine \(P=P(\mathfrak A,\mathfrak X,\delta)\) be representable by an (ideal) right \(m\)-semigroup, it is necessary and sufficient that the function \(\delta\) be left-homogeneous with respect to the states, i.e., that the equality

\[ \delta(a_1a_2,x)=a_1\delta(a_2,x). \]

hold.

1. To obtain the conditions for representability of a \(p\)-machine by a two-sided \(m\)-semigroup, several lemmas are proved. The principal one is the following (we recall the necessary definitions). If for an \(m\)-semigroup \(M=(\bar A,\bar{\mathfrak A},\bar{\mathfrak X})\) one of the conditions holds: \(\bar A=\bar{\mathfrak A}\cup\bar{\mathfrak X}\), \(\bar{\mathfrak A}\cap\bar{\mathfrak X}=\varnothing\), \(\bar a\bar x=\bar x\bar a\) \((\bar a\in\bar{\mathfrak A},\bar x\in\bar{\mathfrak X})\), then \(M\) is called respectively minimal, separated, or commutative. An isomorphism \((\varphi,\psi)\) of a minimal \(m\)-semigroup \((A,\mathfrak A,\mathfrak X)\) onto a minimal \(m\)-semigroup \((A',\mathfrak A',\mathfrak X')\) is called a strong isomorphism if, for any \(\bar x\in\bar{\mathfrak X}\), \(\bar a\in\bar{\mathfrak A}\), the following condition holds:

\[ \bar x\bar a= \begin{cases} \varphi\left[\psi^{-1}(\bar x)\varphi^{-1}(\bar a)\right], & \text{if } \psi^{-1}(\bar x)\varphi^{-1}(\bar a)\in\mathfrak A,\\ \psi\left[\psi^{-1}(\bar x)\varphi^{-1}(\bar a)\right], & \text{if } \psi^{-1}(\bar x)\varphi^{-1}(\bar a)\notin\mathfrak A \end{cases} \]
\[ \text{(but then necessarily } \psi^{-1}(\bar x)\varphi^{-1}(\bar a)\in\mathfrak X,\text{ so that }\psi\text{ is applicable).} \]

Lemma. In order that a minimal right ideal \(m\)-semigroup \(M=(A,\mathfrak A,\mathfrak X)\) be strongly isomorphically mapped onto some separated minimal right ideal \(m\)-semigroup, it is necessary and sufficient that, for the set \(I=\mathfrak X\mathfrak A\setminus(\mathfrak X\mathfrak A\cap\mathfrak A)\), the inclusions \(\mathfrak X I\subseteq I\), \(I\mathfrak X\subseteq I\) hold.

1. The condition for representability of a right machine by a two-sided \(m\)-semigroup is given by

Theorem 2. In order that a right \(p\)-machine \(P(\mathfrak A,\mathfrak X,\delta)\) be representable by a two-sided \(m\)-semigroup, it is necessary and sufficient that there exist such a left machine \(L(\mathfrak A,\mathfrak X,\Delta)\) that

\[ \delta(a_1a_2,x)=a_1\delta(a_2,x);\qquad \delta(a_1,x)a_2=a_1(x,a_2)\Delta; \]
\[ (x,a_1a_2)\Delta=(x,a_1)\Delta a_2;\qquad \delta\bigl[(x_1,a)\Delta,x_2\bigr]=[x_1,\delta(a_2,x)]\Delta. \]

In this case the representing two-sided \(m\)-semigroup can be constructed in the set \(\bar A=\bar{\mathfrak A}\cap\bar{\mathfrak X}\), where \(\bar{\mathfrak A}\) and \(\bar{\mathfrak X}\) are arbitrary isomorphic

the images of the semigroups \(\mathfrak A\) and \(\mathfrak X\): \(\overline{\mathfrak A}=\varphi(\mathfrak A)\), \(\overline{\mathfrak X}=\psi(\mathfrak X)\), \(\overline{\mathfrak A}\cap\overline{\mathfrak X}=\varnothing\), with the operation defined by:

\[ \begin{aligned} \bar a_1\bar a_2&=\varphi\,[\varphi^{-1}(\bar a_1)\varphi^{-1}(\bar a_2)];& \bar x_1\bar x_2&=\psi\,[\psi^{-1}(\bar x_1)\psi^{-1}(\bar x_2)];\\ \bar a\bar x&=\varphi\,[\delta(\varphi^{-2}(\bar a),\psi^{-1}(\bar x))];& \bar x\bar a&=\varphi\,[(\psi^{-1}(\bar x),\varphi^{-1}(\bar a))\,\Delta]. \end{aligned} \]

1. From Theorem 2 there follows without difficulty

Theorem 3. In order that a right \(p\)-machine \(P(\mathfrak A,\mathfrak X,\delta)\) be representable by a commutative \(m\)-semigroup, it is necessary and sufficient that its function \(\delta\) satisfy the conditions:

\[ \delta(a_1,x)a_2=\delta(a_1a_2,x)=a_1\delta(a_2,x); \]

\[ \delta(a,x_1x_2)=\delta(a,x_2x_1). \]

In this case the representing \(m\)-semigroup may be constructed in the same way as in Theorem 2, but with \(\Delta\) replaced by \(\delta\).

1. It is not hard to verify that free automata \((^2)\), considered as machines, are representable by \(m\)-semigroups.

Received
1 XII 1962

REFERENCES

\(^{1}\) S. Ginsburg, Trans. Am. Math. Soc., 96, 400 (1960).
\(^{2}\) V. M. Glushkov, UMN, 16, no. 5 (101), 3 (1961).