UDC 519.50

MATHEMATICS

I. D. STUPINA

ON SOME PROPERTIES OF \(R\)- AND \(R^c\)-OPERATIONS

(Presented by Academician L. V. Kantorovich on 8 January 1969)

The article uses the terminology and notation introduced in papers \((1–5)\), and applies the apparatus of D. Kurepa’s ramified tables.

1. A ramified table is a partially ordered system \(T=\langle \mathcal E,<\rangle\) such that, for every \(x\in\mathcal E\), the set \(\mathcal P_x=\{y:y<x\}\) is well ordered. The order type of the set \(\mathcal P_x\) is called the order of the element \(x\). The rank \(\rho(T)\) of the table \(T\) is called \(\sup\limits_{x\in\mathcal E}\rho(x)\). A node of order \(\alpha\) is a set \(\{x\in\mathcal E:\mathcal P_x=\mathcal P_y\}\) for some \(y\in\mathcal E\) of order \(\alpha\). The set \(T_\alpha=\{x\in\mathcal E:\rho(x)=\alpha\}\) is called the layer of order \(\alpha\). Elements \(a,b\in\mathcal E\) are called comparable if \(a<b\) or \(b<a\); otherwise they are called disjunctive. A set \(U\subset\mathcal E\) is called a section of the table \(T\) if it has a unique common element with each layer \(T_\alpha\). One says that the table \(T\) attains its rank if it contains a well-ordered subset of type \(\rho(T)\).

Remark. A ramified table attains its rank if and only if it admits a monotone section.

Put \((x,\cdot)_T=\{y:y\in\mathcal E \text{ and } x<y\}\). A set \(U\subset\mathcal E\) is called full if, for every \(x\in U\), one has \(\mathcal P_x\subset U\). The completion \(U(T)\) of a set \(U\) in the table \(T\) is the least full set containing \(U\). Then
\[ U(T)=U\cup\left(\bigcup_{x\in U}\mathcal P_x\right). \]

Thickening conditions. We shall say that a set \(U\), where \(\overline{\overline U}=\mathfrak N_\tau\), satisfies the conditions: \(\mathrm I_\tau\), if the table \(\langle U(T),<\rangle\) attains its rank \(\omega_\tau\); \(\mathrm{II}_\tau\), if the table \(\langle U(T),<\rangle\) has a node of cardinality \(\mathfrak N_\tau\).

1. In all that follows we shall consider the set \(\mathcal E\) countable and \(\rho(T)\leq \omega\). The table \(T=\langle\mathcal E,<\rangle\) will be called countable. We note some properties of countable tables.

Theorem 1. Let \(U\subset\mathcal E\), \(T^*=\langle U(T),<\rangle\). If \(\rho(T^*)=\omega\) and \(F=(x_i)_{i<\omega}\) is a monotone section of the table \(T^*\), then, first,
\[ (\forall i)\,[\,\overline{(x_i,\cdot)_{T^*}\cap U}\,]=\mathfrak N_0, \]
and, second, either
\[ \overline{F\cap U}=\mathfrak N_0, \]
or there exists a countable disjunctive subset \(B\subset U\) such that
\[ (\forall i)\,[\,\overline{(x_i,\cdot)_{T^*}\cap B}=\mathfrak N_0\,]. \]

Theorem 2. For every countable set \(U\subset\mathcal E\) in the table \(T=\langle\mathcal E,<\rangle\), either \(\mathrm I_0\) or \(\mathrm{II}_0\) holds (here \(\omega_0=\omega\)).

Denote by \(I\) the set of all natural numbers and by \(W\) the set of all finite tuples of natural numbers, including the empty tuple \(\{\ \}\). Let \(T_W=\langle W,<\rangle\), where as the relation \(<\) we take the relation of subordination of tuples. Obviously, \(\rho(T_W)=\omega\).

1. Let \(H\) be a property of sets of chains of a given rigid base \(N\). We shall also denote by \(H\) the set of all subsets \(H_\xi\subset N\) possessing the property \(H\). Let
   \[ HN=\left(\bigcup_{\xi\in H_\xi}\xi\right)_{H_\xi\in H}' . \]
   The following notation is introduced: \(H_pN\) (\(p\) natural), \(H_{\mathfrak N_0}N\), \(H_{\hat{\mathfrak N}_0}N\), if \(H_\xi\) contains, respectively, not fewer than \(p\) chains, not fewer than \(\mathfrak N_0\) chains, more than \(\mathfrak N_0\) chains of the base \(N\). Let \((E_i)\) be a sequence of sets of the basic space \(\Xi\). The kernel of a chain \(\xi\in N\)

call \(\bigcap E_i\). If \(x \in \Phi_N(E_i)\), then by \(M_x\) we denote the set of all chains \(\xi \in N\) to whose kernels the point \(x\) belongs. If \(I^* \subset I\), then the truncated base \(N^{I^*}\) of the operation \(\Phi_N\) is called \(\{\xi \in N: I^* \subset \xi\}\).

1. Consider an \(R^c_{\mathfrak M^c}\)-operation, where \(\mathfrak M^c = (N^c_{n_1\ldots n_k})_{\{n_1\ldots n_k\}\in W}\) is a table of bases. A. A. Lyapunov \((^4)\) showed that, in the case when all bases \(N^c_{n_1\ldots n_k}\) are rigid, the \(R^c_{\mathfrak M^c}\)-operation also has a rigid base \(\theta^c_{\mathfrak M}\). We shall call the \(\{n_1\ldots n_k\}\)-section of an \(R^c_{\mathfrak M^c}\)-chain \(\vartheta\), and denote by \(\vartheta_{n_1\ldots n_k}\), the totality of all tuples from the chain \(\vartheta\) either coinciding with the tuple \(\{n_1\ldots n_k\}\) or subordinate to it. Denote by \(\Phi_{\vartheta^c_{\mathfrak M};\{n_1\ldots n_k\}}\) the \(\delta s\)-operation whose base is \(\{\vartheta_{n_1\ldots n_k}:\vartheta \in \theta^c_{\mathfrak M}\}\). Obviously, \(\Phi_{\vartheta^c_{\mathfrak M};\{\}}=\Phi_{\theta^c_{\mathfrak M}}\). The totality of tuples \(((n_1\ldots n_k n^i_{k+1}))_i\) will be called an \(R^c\)-covering of the tuple \(\{n_1\ldots n_k\}\), if \(\xi=(n^i_{k+1})_i \in N^c_{n_1\ldots n_k}\).

Theorem 3. Every rigid \(R^c_{\mathfrak M^c}\)-chain consists of pairwise non-subordinate tuples.

Theorem 4. If \(\vartheta\) is a rigid \(R^c_{\mathfrak M^c}\)-chain, and \(\vartheta(T_W)\) is its completion in the table \(T_W\), then every node of the table \(\langle \vartheta(T_W), < \rangle\) of the form \((\{n_1,\ldots,n_k n^i_{k+1}\})_i\) is an \(R^c\)-covering of the tuple \(\{n_1\ldots n_k\}\).

Theorem 5. No \(R^c_{\mathfrak M^c}\)-chain \(\vartheta\) satisfies condition \(I_0\), i.e., either the rank of the table \(T^*=\langle \vartheta(T_W), < \rangle\) is less than \(\omega\), or \(\rho(T^*)=\omega\), but the table \(T^*\) does not attain it.

Theorem 6. For any tuple \(\{n_1\ldots n_k\}\in W\), the operation \(\Phi_{\vartheta^c_{\mathfrak M};\{n_1\ldots n_k\}}\) is weaker than the operation \(\Phi_{\theta^c_{\mathfrak M}}\).

Theorem 7. Whatever the tuple \(\{n_1\ldots n_k\}\in W\) may be, the operation whose base is an arbitrary truncated base of the operation \(\Phi_{\vartheta^c_{\mathfrak M};\{n_1\ldots n_k\}}\) is weaker than the operation \(\Phi_{\theta^c_{\mathfrak M}}\) relative to the class of sets \(\mathcal K \ni \varnothing\).

A consequence of Theorem 7 and of a theorem of I. Kozlova \((^6)\) is

Theorem 8. If the class of sets \(\mathcal K \ni \varnothing\) and the class \(\Phi_{\theta^c_{\mathfrak M}}(\mathcal K)\) are invariant with respect to the operations \(\Sigma\) and \(\Pi\), then for any tuple \(\{n_1\ldots n_k\}\in W\) and any natural \(p\) one has

\[ \Phi_{H_p(\vartheta^c_{\mathfrak M};\{n_1\ldots n_k\})}(\mathcal K)\subset \Phi_{\theta^c_{\mathfrak M}}(\mathcal K). \]

1. Let \(x\in \Phi_{\theta^c_{\mathfrak M}}(E_{n_1\ldots n_k})\). Denote by \(\eta_x\) the totality of all tuples entering chains that belong to the set \(M_x\).

We introduce the duality property. We shall say that a tuple \(\{n_1\ldots n_k\}\in \eta_x(T_W)\) has the duality property if from the tuples of the collection \(\eta_x\) one can form at least two chains of the base of the operation \(\Phi_{\vartheta^c_{\mathfrak M};\{n_1\ldots n_k\}}\). Denote by \(\mu_x\) the totality of all tuples of the collection \(\eta_x(T_W)\) that have the duality property.

Theorem 9. Two distinct \(R^c_{\mathfrak M^c}\)-chains \(\vartheta\) and \(\vartheta'\) differ either in that their completions \(\vartheta(T_W)\) and \(\vartheta'(T_W)\) include subsets \(\mu_x\) that are distinct from one another, or in that there is a tuple \(\{n_1\ldots n_k\}\in \vartheta(T_W)\cap \vartheta'(T_W)\) such that \(\vartheta(T_W)\) and \(\vartheta'(T_W)\) include \(R^c\)-coverings of this tuple that are distinct from one another.

For the \(R^c_{\mathfrak M^c}\)-operation we introduce condensation conditions. We shall say that a set of tuples \(\eta_x\) satisfies condition: \(a_1\), if in the table \(\langle \eta_x(T_W), < \rangle\) there exists a node \((\{n_1\ldots n_k n^i_{k+1}\})\), from which one can form \(>\mathfrak N_0\) \(R^c\)-coverings of the tuple \(\{n_1\ldots n_k\}\); \(a_2\), if in the collection \(\eta_x(T_W)\) there are a tuple \(\{n_1\ldots n_k\}\) and such an \(R^c\)-covering of it \((\{n_1\ldots n_k n^i_{k+1}\})_i\), in which \(\mathfrak N_0\)

tuples have the property of duality; \(a_3\), if in the table \(\langle \eta_x(T_W), <\rangle\) there exists a node \(\bigl(\{n_1\ldots n_k n_{k+1}^{\,i}\}\bigr)_i\), from which an \(\geqslant \aleph_0\) \(R^c\)-covering of the tuple \(\{n_1\ldots n_k\}\) can be formed; \(a_4\), if the table \(\langle \mu_x, <\rangle\) attains its rank \(\omega\).

Theorem 10. For every set \(\eta_x\) one has
\[ \overline{\overline{M}}_x \geqslant \aleph_0 \ \&\ \neg a_2 \ \&\ \neg a_3 \Rightarrow a_4 . \]

Theorem 11. For every set \(\eta_x\) one has
\[ a_1 \vee a_2 \Longleftrightarrow \overline{\overline{M}}_x > \aleph_0 . \]

Theorem 12. For every set \(\eta_x\) one has
\[ \neg(a_1 \vee a_2)\ \&\ (a_3 \vee a_4)\Longleftrightarrow \overline{\overline{M}}_x=\aleph_0 . \]

Theorem 13. For every set \(\eta_x\) one has
\[ a_2 \vee a_3 \vee a_4 \Longleftrightarrow \overline{\overline{M}}_x \geqslant \aleph_0 . \]

6. Let a \(\delta_s\)-operation \(\Phi_N\) and two sequences of sets \((E_i)\), \((e_i)\) be given. Put
\[ Q(N;E_i;e_i)=\bigcup_{\xi,\xi'}\left(\bigcap_{i\in \xi}E_i\cap \bigcap_{i\in \xi'}e_i\right), \]
where the union is taken over all pairs \((\xi,\xi')\) of chains such that \(\xi\in N\) and \(\xi'\) is an infinite subchain of the chain \(\xi\). If \(N\) is a rigid base of the operation of countable union, then for any ordinal number \(\alpha<\omega_1\) by \(\theta^\alpha,\theta^{\alpha c}\) we denote the rigid bases of the operations \(R_N^\alpha, R_N^{\alpha c}\), respectively.

Theorem 14. If the class of sets \(\mathcal K\ni \varnothing\) and for every tuple \(\{m_1\ldots m_t\}\) the sets \(E_{m_1\ldots m_t}, e_{m_1\ldots m_t}\) belong to the class \(\mathcal K\), then for any ordinal number \(\alpha<\omega_1\) the sets
\[ Q_{\{m_1\ldots m_t\}}\bigl(\theta^\alpha;E_{m_1\ldots m_t};e_{m_1\ldots m_t}\bigr), \]
\[ Q_{\{m_1\ldots m_t\}}\bigl(\theta^{\alpha c};E_{m_1\ldots m_t};e_{m_1\ldots m_t}\bigr) \]
belong, respectively, to the classes \(\Phi_{\theta^\alpha}(\mathcal K)\), \(\Phi_{\theta^{\alpha c}}(\mathcal K)\).

7. Let \((E_{m_1}\ldots E_{m_t})\) be a table of sets of the class \(\mathcal K\). We construct the sets
\[ \Phi_{H\aleph_0,\mathfrak M}^{\theta c}(E_{m_1\ldots m_t}),\quad \Phi_{H\aleph_0,\mathfrak M}^{\theta_{\mathfrak M}^{c}}(E_{m_1\ldots m_t}). \]
Put
\[ \mathcal E_{n_1\ldots n_k} = \Phi_{\theta_{\mathfrak M}^{c;\{n_1\ldots n_k\}}}(E_{m_1\ldots m_t}), \]
\[ \mathcal E^{*}_{n_1\ldots n_k} = \Phi_{H\hat{\ }\,\aleph_0 N^{c}_{n_1\ldots n_k}}\bigl(\mathcal E_{n_1\ldots n_k i}\bigr), \quad \mathcal E^{**}_{n_1\ldots n_k} = \Phi_{H_2(\theta_{\mathfrak M}^{c};\{n_1,\ldots,n_k\})}(E_{m_1\ldots m_t}), \]
\[ \mathcal E^{***}_{n_1\ldots n_k} = \Phi_{i\,H\aleph_0 N^{c}_{n_1\ldots n_k}}\bigl(\mathcal E_{n_1\ldots n_k i}\bigr), \quad Y_{n_1\ldots n_k} = Q_i\bigl(N^{c}_{n_1\ldots n_k};\mathcal E_{n_1\ldots n_k i};\mathcal E^{**}_{n_1\ldots n_k i}\bigr). \]

Let
\[ \theta_{\mathfrak M}^{c\{n_1\ldots n_k\}} = \{\vartheta\in\theta_{\mathfrak M}^{c}:\{n_1\ldots n_k\}\in\vartheta\}. \]
Put
\[ E^{n_1\ldots n_k}_{m_1\ldots m_t} = \begin{cases} E_{m_1\ldots m_t}, & \text{if } \{m_1\ldots m_t\}\ne \{n_1\ldots n_k\},\\ \Xi, & \text{if } \{m_1\ldots m_t\}= \{n_1\ldots n_k\}. \end{cases} \]

Let
\[ \mathcal E^{n_1\ldots n_k} = \Phi_{\theta_{\mathfrak M}^{c\{n_1\ldots n_k\}}}(E_{m_1\ldots m_t}), \quad \widetilde{\mathcal E}^{\,n_1\ldots n_k} = \Phi_{\theta_{\mathfrak M}^{c\{n_1\ldots n_k\}}}\bigl(E^{n_1\ldots n_k}_{m_1\ldots m_t}\bigr), \]
\[ Z^{*}_{n_1\ldots n_k} = \widetilde{\mathcal E}^{\,n_1\ldots n_k}\cap \mathcal E^{*}_{n_1\ldots n_k}, \quad Z^{**}_{n_1\ldots n_k} = \widetilde{\mathcal E}^{\,n_1\ldots n_k}\cap Y_{n_1\ldots n_k}, \quad Z^{***}_{n_1\ldots n_k} = \widetilde{\mathcal E}^{\,n_1\ldots n_k}\cap \mathcal E^{***}_{n_1\ldots n_k}, \]
\[ Z^{(1)}=\bigcup_{\{n_1\ldots n_k\}\in W} Z^{*}_{n_1\ldots n_k}, \quad Z^{(2)}=\bigcup_{\{n_1\ldots n_k\}\in W} Z^{**}_{n_1\ldots n_k}, \quad Z^{(3)} = \bigcup_{\{n_1\ldots n_k\}\in W} Z^{***}_{n_1\ldots n_k}, \]
\[ B_{n_1\ldots n_k} = \mathcal E^{n_1\ldots n_{k-1}n_k} \cup \bigcup_{\substack{n'_k\ne n_k\\ \{m_1\ldots m_t\}\in W}} \mathcal E^{n_1\ldots n_{k-1}n'_k m_1\ldots m_t}. \]

By \(\Phi_{\mathfrak A}\) denote the \(\delta_s\)-operation whose base \(\mathfrak A\) consists of all chains that are a countable totality of tuples of increasing ranks, each succeeding one being subordinate to the preceding one. It is easy to show that the operation \(\Phi_{\mathfrak A}\) is weaker than the \(A\)-operation. Put
\[ Z^{(4)}=\Phi_{\mathfrak A}(B_{n_1\ldots n_k}). \]

Theorem 15. \(\Phi_{H_{\aleph_0}^{\theta^c_{\mathfrak M}}}(E_{m_1\ldots m_t})=Z^{(1)}\cup Z^{(2)}\).

Theorem 16. \(\Phi_{H_{\aleph_0}^{\theta^c_{\mathfrak M}}}(E_{m_1\ldots m_t})=\displaystyle\bigcup_{k=2}^{4} Z^{(k)}\).

Theorem 17. If 1) for any tuple \(\{n_1\ldots n_k\}\in W\), for a class of sets \(\mathcal K\supset \varnothing,\ \Xi\), the conditions are satisfied: a) the operation \(\Phi_{N^c_{n_1\ldots n_k}}\) is stronger than the \(\Sigma\)-, \(\Pi\)-operations, b) the operation \(\Phi_{H_{\aleph_0}^{\,}N^c_{n_1\ldots n_k}}\) is weaker than the operation \(\Phi_{N^c_{n_1\ldots n_k}}\), c) the \((\theta^c_{\mathfrak M};\,N^c_{n_1\ldots n_k})\)-type is weaker than the \(\theta^c_{\mathfrak M}\)-type; 2) for an arbitrary class of sets \(\mathcal K'\supset \varnothing\), the set \(Q(N^c_{n_1\ldots n_k};\,E_i;\,e_i)\) belongs to the class \(\Phi_{N^c_{n_1\ldots n_k}}(\mathcal K')\), whenever the sets \(E_i,\ e_i\) belong to the class \(\mathcal K'\), then

\[ \Phi_{H_{\aleph_0}^{\theta^c_{\mathfrak M}}}(\mathcal K)\subset \Phi_{\theta^c_{\mathfrak M}}(\mathcal K). \]

Theorem 18. If conditions 1c), 2) of Theorem 17 are preserved and conditions 1a)—1b) are replaced by the conditions: 1a′) the operation \(\Phi_{N^c_{n_1\ldots n_k}}\) is stronger than the \(A\)-operation, 1b′) the operation \(\Phi_{H_{\aleph_0}^{\,}N^c_{n_1\ldots n_k}}\) is weaker than the operation \(\Phi_{N^c_{n_1\ldots n_k}}\), we have:

\[ \Phi_{H_{\aleph_0}^{\theta^c_{\mathfrak M}}}(\mathcal K)\subset \Phi_{\theta^c_{\mathfrak M}}(\mathcal K). \]

Corollary. For every ordinal number \(1\le \alpha<\omega_1\), the operations \(\Phi_{H_{\aleph_0}^{\theta\alpha c}}\), \(\Phi_{H_{\aleph_0}^{\theta^a c}}\) are weaker than the operation \(\Phi_{\theta\alpha c}\), and the operation \(\Phi_{H_{\aleph_0}^{\theta^a}}\) is weaker than the operation \(\Phi_{\theta^a}\) relative to a class of sets \(\mathcal K\supset \varnothing,\ \Xi\).

Volgograd Pedagogical Institute
named after A. S. Serafimovich

Received
7 VII 1969

REFERENCES

1. G. Kurepa, Ensembles ordonnés et ramifiés, Publications Mathématiques de l’Université de Belgrade, 4, 1935, 1.
2. Yu. S. Ochan, UMN, 10, no. 3, 71 (1955).
3. A. A. Lyapunov, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, 40, 3 (1953).
4. A. A. Lyapunov, Izv. AN SSSR, ser. matem., 17, 563 (1953).
5. A. A. Lyapunov, Tr. Mosk. matem. obshch., 6, 195 (1957).
6. E. I. Kozlova, Izv. AN SSSR, ser. matem., 19, 125 (1955).