UDC 519.50

MATHEMATICS

I. D. STUPINA

ON SOME PROPERTIES OF \(R\)-, \(R^c\)-OPERATIONS AND PROJECTIVE OPERATIONS ON UNCOUNTABLE SYSTEMS OF SETS IN CONNECTION WITH BRANCHING HYPOTHESES

(Presented by Academician M. A. Lavrent'ev, 12 I 1970)

In the paper \((^1)\) we considered certain properties of \(R\)- and \(R^c\)-operations on countable systems of sets. Analogous properties of certain operations were considered in \((^2)\). In the present note analogous questions are considered for the same operations on uncountable systems of sets. We use the notation introduced in note \((^1)\).

1. Let \(T=\langle \mathcal E,<\rangle\) be a branching table, \(U\subset \mathcal E\), \(T^*=\langle U(T),<\rangle\). If \(\rho(T^*)=\omega_\lambda\), \(\omega_\tau\) is the final character of the number \(\omega_\lambda\), and \(F=(x_\beta)_{\beta<\omega_\lambda}\) is a monotone dissection of the table \(T'\), then \((\forall \beta<\omega_\lambda)\cdot [\overline{(x_\beta,\cdot)_{T^*}\cap U}\ge \aleph_\tau]\), and further either \(\overline{F}\cap \overline{U}=\aleph_\tau\), or there exists a disjunctive subset \(K\subset U\) such that \((\forall \beta<\omega_\lambda)[\overline{(x_\beta,\cdot)_{T^*}\cap K}\ge \aleph_\tau]\).

In what follows, by \(\aleph_\nu\) we shall denote a strongly inaccessible cardinal number. If \(\overline E=p\), then \(E\) will be called a \(p\)-set. The following hypotheses are used in the paper:

\((\alpha_1)\) Every branching table \(T\) of rank \(\omega_\nu\) such that \((\forall \alpha<\omega_\nu)\cdot[\overline{T_\alpha}<\aleph_\nu]\) attains its rank.

\((\alpha_2)\) If in a branching table \(T=\langle \mathcal E,<\rangle\) of rank \(\omega_\nu\) the cardinality of every disjunctive subset \(U\subset \mathcal E\) is less than \(\aleph_\nu\), then the table \(T\) attains its rank \((^3)\).

These hypotheses are respectively equivalent to the following:

\((\beta_1)\) Every branching table \(T\) of rank \(\omega_\nu\) either attains its rank or has an \(\aleph_\nu\)-node.

\((\beta_2)\) Every branching table \(T=\langle \mathcal E,<\rangle\) of rank \(\omega_\nu\) either attains its rank or satisfies the condition: there exists a disjunctive \(\aleph_\nu\)-subset \(U\subset \mathcal E\) not satisfying condition \(i_\nu\).

Obviously, \((\alpha_1)\Rightarrow(\alpha_2)\). Theorems proved with the aid of the hypotheses \((\alpha_1)\) and \((\alpha_2)\) will be marked by \((*)\) and \((**)\), respectively.

\((*)\) For every \(\aleph_\nu\)-set \(U\subset \mathcal E\) in the table \(T=\langle \mathcal E,<\rangle\) of rank \(\le \omega_\nu\), \(I_\nu\) or \(II_\nu\) is valid.

1. By the \(\Pi^\tau\)-product of discrete spaces \(\{X_\alpha:\alpha<\omega_\tau\}\), denoted by us as \(\Pi^\tau X_\alpha\) and called a generalized Baire space, we shall mean the space on the abstract product

\[ \prod_{\alpha<\omega_\tau} X_\alpha, \]

whose open-closed base \(B^\tau\) is given by the totality of generalized Baire intervals \(\delta_{(i_\beta)_{\beta<\alpha}}\), where

\[ \delta_{(i_\beta)_{\beta<\alpha}} = \left\{(j_\beta)_{\beta<\omega_\tau}\in \prod_{\beta<\omega_\tau}X_\beta: (\forall \beta<\alpha)\,[j_\beta=i_\beta]\right\}. \]

We shall assume

\[ \delta_{(i_\beta)_{\beta<0}} = \prod_{\alpha<\omega_\tau} X_\alpha. \]

If \((\forall \alpha<\omega_\tau)\,[X_\alpha=J]\) and \(J\ne \aleph_\tau\), then the space \(\Pi^\tau X_\alpha\) is denoted by \(J^{\omega_\tau}\). Put \(T^\tau=\langle B^\tau,<\rangle\), where as the relation \(<\) the relation \(\supset\) of strict inclusion is taken.

\((*)\) In order that a closed set \(E\subseteq \Pi^\tau X_\alpha\) not be \(\aleph_\nu\)-bicompact, it is necessary and sufficient that it have a disjunctive

an $\aleph_\nu$-covering $S \subset B^\nu$, satisfying condition II$_\nu$, i.e., such that the branching table $\langle S(T^\nu), <\rangle$ has an $\aleph_\nu$-node.

Remark. I. I. Parovichenko ($^4$) proved that hypothesis $(a_1)$ is equivalent to the assertion of $\aleph_\nu$-bicompactness of the $T^\nu$-product of discrete spaces $\{X_\alpha:\alpha<\omega_\nu\}$, where $(\forall \alpha<\omega_\nu)[\overline{\overline{X}}_\alpha<\aleph_\nu]$. Since the classes of open sets in this space and in the space $\prod^\nu X_\alpha$ coincide, hypothesis $(a_1)$ is equivalent to the assertion of $\aleph_\nu$-bicompactness of the space $\prod^\nu X_\alpha$, when $(\forall \alpha<\omega_\nu)[\overline{\overline{X}}_\alpha<\aleph_\nu]$.

1. The set of all tuples of the form $(i_\beta)_{\beta<\alpha}$, corresponding to all points $x\in J^\nu$, will be denoted by $W$. A set $U\subset W$ will be called a $W$-base. Unless special qualifications are made, the $R$- and $R^c$-operations are considered with the full depth of chains $\upsilon=\omega_\nu$. The rigid $W$-bases of the operations $R_{\mathfrak M}$, $R_{\mathfrak M}^c$, $R_N^\alpha$, $R_N^{\alpha c}$, $R^\alpha$, $R^{\alpha c}$, $0\leqslant\alpha<\omega_{\nu+1}$ ($^5$, $^6$) will be denoted respectively by $\theta_{\mathfrak M}$, $\theta_{\mathfrak M}^c$, $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $\theta^\alpha$, $\theta^{\alpha c}$. By $\chi_\alpha$ [$\chi_\alpha^c$] we denote the rigid base of the projective $A_\alpha$ [$CA_\alpha$]-operation, $0\leqslant\alpha<\omega_{\nu+1}$.

2. We shall say that a rigid base $N$ admits a $V$-transformation if, for every $J'\subset J$, under the condition $N^{J'}\ne\varnothing$, there exists a set $[J']$ such that, putting for an arbitrary system of sets $(E_i)$ $E_i=E_i'$, if $i\notin [J']$, and $E_i=\varnothing$, if $i\in [J']$, we obtain $\Phi_N^{J'}(E_i)=\Phi_N(E_i)$.

We have proved: if $N$ and $N^c$ are rigid bases of mutually complementary operations, then each of them admits a $V$-transformation.* Therefore the bases $\chi_\alpha$, $\chi_\alpha^c$, $0\leqslant\alpha<\omega_{\nu+1}$, and also the bases $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $0\leqslant\alpha<\omega_{\nu+1}$, under the condition that $N$ and $N^c$ are rigid bases, admit a $V$-transformation.

A rigid base $N$ is called $\aleph_\tau$-regular ($^8$) for a class of sets $\mathcal K$ if $(\forall J'\subset J)[\Phi_N^{J'}(\mathcal K)\subset\Phi_N(\mathcal K)]$ and the class of sets $\Phi_N(\mathcal K)$ is invariant with respect to the operations $\underset{\aleph_\tau}{\cup}$, $\underset{\aleph_\tau}{\cap}$. A property $H$ of chains of a rigid base $N$ is called $N$-regular ($^8$) for a class of sets $\mathcal K$ if the operations $\Phi_{HN}$, $\Phi_{(HN)^i}$ are weaker than the operation $\Phi_N$ with respect to the class of sets $\mathcal K$.

Each of the bases $\chi_\alpha$, $\chi_\alpha^c$, $0\leqslant\alpha<\omega_{\nu+1}$, and also each of the bases $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $1\leqslant\alpha<\omega_{\nu+1}$, when the conditions are fulfilled: $1^\circ$. $N$ and $N^c$ are rigid bases; $2^\circ$. The operation $\Phi_N$ is stronger than the operation $\underset{\aleph_\nu}{\cup}$, is $\aleph_\nu$-regular for the class of sets $\mathcal K\ni\varnothing,\Xi$, where $\Xi$ is the basic space.

Hence, and by virtue of Theorem 3 of I. Kozlova (($^8$), Theorem 1), for the class of sets $\mathcal K\ni\varnothing,\Xi$, for every $p\leqslant\aleph_0$ the property $H_x$ is $M$-regular, if $M$ is a base $\chi_\alpha$, $\chi_\alpha^c$, $0\leqslant\alpha<\omega_{\nu+1}$, and also a base $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $1\leqslant\alpha<\omega_{\nu+1}$ in the case when conditions $1^\circ$ and $2^\circ$ are fulfilled, and, consequently, when $M$ is a base $\theta^\alpha$, $\theta^{\alpha c}$, $1\leqslant\alpha<\omega_{\nu+1}$.

1. For a base $N$ define the $\Delta\Sigma$-operation $Q_N$ over two systems of sets $(E_u)$, $(e_v)$, by putting

\[ Q_N(E_u,e_v)= \bigcup_{u,v}\ \bigcup_{\xi\in N,\ \xi'\subset\xi,\ \overline{\overline{\xi'}}=\aleph_\nu} \left(\bigcap_{u\in\xi} E_u\cap \bigcap_{v\in\xi'} e_v\right). \]

(*) For any $0\leqslant\alpha<\omega_{\nu+1}$ the operation $Q_{\theta^\alpha}$ [$Q_{\theta^{\alpha c}}$] is weaker than the operation $\Phi_{\theta^\alpha}$ [$\Phi_{\theta^{\alpha c}}$] with respect to the class of sets $\mathcal K\ni\varnothing,\Xi$.

1. Let $\mathfrak M=(N_a)_{a\in W}$, $\mathfrak M^c=(N_a^c)_{a\in W}$ be tables of rigid bases. For the operations $\Phi_{\theta\mathfrak M}$, $\Phi_{\theta\mathfrak M^c}$, we introduce thinning conditions. We shall say that a collection of tuples $\eta_x$ satisfies condition $a_1^\nu$ [$a_3^\nu$] if in the branching table $\langle\eta_x(T_W^\nu),<\rangle$ there is a node $((a_i))_{i\in J'}$ from which one can form a $>\aleph_\nu$ [$\geqslant\aleph_\nu$] $R$-($R^c$-) covering of the tuple $a$; $a_2^\nu$, if in the collection $\eta_x(T_W^\nu)$ there is a tuple and such an $R$-($R^c$-) covering of it in which $\aleph_\nu$ tuples have the property of bisection; $a_4^\nu$, if the table $\langle\mu_x,<\rangle$ attains its rank $\omega_\nu$; $a_7^\nu$, if the collection $\mu_x$ includes a disjunctive $\aleph_\nu$-subset of tuples.

* A. D. Taimanov ($^7$) gave an example of a $\delta S$-operation with a rigid base for which the complementary $\delta S$-operation has no rigid base.

Then for the operations \(\Phi_{\theta_{\mathfrak M}}, \Phi_{\theta_{\mathfrak M}^{c}}\) the following holds:

1) for \(\nu<\omega_\nu\):
\[ \overline{M}_x>\mathfrak N_\nu \Longleftrightarrow a_1^\nu \vee a_2^\nu,\qquad \overline{M}_x\geq \mathfrak N_\nu \Longleftrightarrow a_2^\nu \vee a_3^\nu,\qquad \overline{M}_x=\mathfrak N_\nu \Longleftrightarrow \neg(a_1^\nu\vee a_2^\nu)\ \&\ a_3^\nu; \]

2) \((**)\)
\[ \overline{M}_x\geq \mathfrak N_\nu\ \&\ \neg a_3^\nu\ \&\ \neg a_7^\nu \Rightarrow a_4^\nu; \]

3) \((**)\).
\[ \overline{\overline{M}}_x\geq \mathfrak N_\nu \Longleftrightarrow a_3^\nu\vee a_4^\nu\vee a_7^\nu. \]

For the operation \(\Phi_{\theta_{\mathfrak M}^{c}}\) the following holds:

4) \((*)\)
\[ \overline{\overline{M}}_x>\mathfrak N_\nu \Longleftrightarrow a_1^\nu\vee a_2^\nu; \]

5) \((*)\)
\[ \overline{\overline{M}}_x=\mathfrak N_\nu \Longleftrightarrow \neg(a_1^\nu\vee a_2^\nu)\ \&\ a_3^\nu. \]

1. Let the \(W\)-base \(U\) coincide with the base \(\theta_{\mathfrak M}\) or \(\theta_{\mathfrak M}^{c}\). For \(k=2,4,7\) put
   \[ \Phi_{H_{a_k}U}(E_a)=\{x\in\Phi_U(E_a):\ \text{the set }\eta_x\text{ satisfies the condition }a_k^\nu\}. \]

\((*)\) For the class of sets \(\mathcal K\ni\varnothing,\ \Xi\), the operations
\[ \Phi_{H_{a_2}\theta^\alpha},\quad \Phi_{H_{a_2}\theta^{\alpha c}},\quad 0\leq \alpha<\omega_{\nu+1}, \]
are weaker than the operations \(\Phi_{\theta^\alpha}, \Phi_{\theta^{\alpha c}}\), respectively.

\((**)\) If conditions \(1^0\) and \(2^0\) are fulfilled, the operations
\[ \Phi_{H_{a_4}\theta_N^\alpha},\quad \Phi_{H_{a_7}\theta_N^\alpha},\quad 2\leq \alpha<\omega_{\nu+1}, \]
are weaker than the operation \(\Phi_{\theta_N^\alpha}\), and the operations
\[ \Phi_{H_{a_4}\theta_N^{\alpha c}},\quad \Phi_{H_{a_7}\theta_N^{\alpha c}},\quad 1\leq \alpha<\omega_{\nu+1}, \]
are weaker than the operation \(\Phi_{\theta_N^{\alpha c}}\).

\((**)\) If conditions \(1^0\) and \(2^0\) are fulfilled, the property \(H_{\mathfrak N_\nu}\) is \(M\)-regular if the base \(M\) coincides with the base \(\theta_N^\alpha,\ 2\leq\alpha<\omega_{\nu+1},\ \theta_N^{\alpha c},\ 1\leq\alpha<\omega_{\nu+1}\), in particular, with the base \(\theta^\alpha\) for \(2\leq\alpha<\omega_{\nu+1}\), and \(\theta^{\alpha c}\) for \(1\leq\alpha<\omega_{\nu+1}\).

\((*)\) The property \(H_{\widehat{\mathfrak Z}_\nu}\) is \(\theta^{\alpha c}\)-regular for the class of sets \(\mathcal K\ni\varnothing,\Xi\).

1. Let \(J=\{\alpha:\alpha<\omega_\nu\}\). If each chain \(\xi\) of the rigid base \(M\) is ordered in the order of increase of its elements, then we obtain a rigid reduced base \(\breve M\). Let \(\breve M\subset J^{\omega_\nu}\). Put
   \[ \Phi_{H_c\breve M}(E_i) = \{x\in\Phi_M(E_i):\ \text{the closure of the set }\breve M_x\text{ is not }\mathfrak N_\nu\text{-bicompact}\}, \]
   where \(\breve M_x\) is the set of all chains of the base \(\breve M\) whose kernels contain the point \(x\). Let
   \[ q=(i_\beta)_{\beta<\alpha}\in W \quad\text{and}\quad (\beta'<\beta'')\Rightarrow (i_{\beta'}<i_{\beta''}). \]
   For a system of sets \((E_i)\) put \(E_i^q=\varnothing\), if
   \[ (\forall \beta<\alpha)(i\ne i_\beta)\ \&\ (\exists\beta_0)[i<i_{\beta_0}], \]
   and \(E_i^q=E_i\) in all remaining cases. Let
   \[ \mathscr E_q=\Phi_M(E_i^q),\qquad S_\alpha'=\bigcup_{q\in W}\mathscr E_{qp}. \]

Then
\[ (*)\qquad \Phi_{H_c\breve M}(E_i)= \bigcup_{q\in W}\overline{\lim_i}^{\,\mathfrak N_\nu} S_{qi}. \]

\((*)\) If the rigid base \(M\) is \(\mathfrak N_\nu\)-proper for the class of sets \(\mathcal K\ni\varnothing\), then the property \(H_c\) is \(M\)-regular.

\((*)\) The property \(H_c\) is \(M\)-regular for the class of sets \(\mathcal K\ni\varnothing,\Xi\), if the base \(M\) coincides with the base \(\chi_\alpha,\chi_\alpha^{c},\ 0\leq\alpha<\omega_{\nu+1}\), and also, when conditions \(1^0\) and \(2^0\) are fulfilled, with the base \(\theta_N^\alpha,\theta_N^{\alpha c},\ 1\leq\alpha<\omega_{\nu+1}\), in particular, with the base \(\theta^\alpha,\theta^{\alpha c},\ 0\leq\alpha<\omega_{\nu+1}\).

Volgograd State Pedagogical
Institute
named after A. S. Serafimovich

Received
30 XII 1969

CITED LITERATURE

\({}^{1}\) I. D. Stupina, DAN, 188, No. 5, 1010 (1969).
\({}^{2}\) I. D. Stupina, DAN, 117, No. 2, 188 (1957).
\({}^{3}\) G. Kurera, Publ. math. de l’Univ. de Belgrade, 4, 1 (1935).
\({}^{4}\) I. I. Parovichenko, DAN, 174, No. 1, 30 (1967).
\({}^{5}\) A. A. Lyapunov, Algebra and Logic, Seminar, 2, No. 2, 47 (1963).
\({}^{6}\) Z. I. Kozlova, Scientific Notes, Volgograd Ped. Inst., Mathematics, issue 23 (1969).
\({}^{7}\) A. D. Taimanov, Izv. AN SSSR, ser. matem., 14, 443 (1950).
\({}^{8}\) Z. I. Kozlova, DAN, 188, No. 5, 1001 (1969).