UDC 519.47+517.11

MATHEMATICS

R. A. BAIRAMOV

CRITERIA FOR THE FUNDAMENTALITY OF PERMUTATION GROUPS AND TRANSFORMATION SEMIGROUPS

(Presented by Academician S. L. Sobolev on 17 IV 1969)

Let (k \ge 3); (E_k={0,\ldots,k-1}); (\mathscr P_k) be the set of all functions of (k)-valued logic; (P_k=\langle \mathscr P_k;\Phi={\eta,\tau,\Delta,\nabla,*}\rangle) be the Post algebra of rank (k) (for the definition of the signature operations from (\Phi), see ((^1))); (\Omega_k) be the symmetric semigroup of degree (k); (\sigma_k) the symmetric group of degree (k); (A_k) the alternating group of degree (k) (the subgroup of all even permutations in (\sigma_k)). Let us denote by (\mathscr T), and call the Słupecki algebra, the subalgebra in (P_k) generated by all functions depending essentially on only one argument and by all functions omitting at least one value from (E_k). Following A. Salomaa ((^2)), we shall call a subset (\mathfrak M\subset \mathscr P_k) fundamental if the condition is satisfied (prior to the reference to ((^3)) we do not assume the maximality of the subalgebra (\mathscr T) in (P_k) to be known):

[
\bigl([\mathfrak M]\ne \mathscr P_k\bigr)\ \&\ \bigl(\forall f\in \mathscr P_k\setminus(\mathscr T\cup[\mathfrak M])\bigl([\mathfrak M+{f}]=\mathscr P_k\bigr)\bigr).
\tag{1}
]

Since, when testing a set for fundamentality, it must first be closed, we are interested only in the fundamentality of closed sets (subalgebras in (P_k)). For well-known reasons, special interest attaches to the study of the fundamentality of subalgebras of unary functions (strictly speaking, one studies the fundamentality of subsemigroups of the symmetric semigroup (\Omega_k)); for them condition (1) is transformed into the form

[
\forall f\in \mathscr P_k\setminus \mathscr T\bigl([\mathfrak M+{f}]=\mathscr P_k\bigr).
]

It should be noted that the existence of fundamental sets not contained in (\mathscr T) is an obvious fact (such are, for example, all maximal subalgebras in (P_k) distinct from (\mathscr T)); whereas the existence of fundamental semigroups of unary functions is a non-obvious fact. In ((^3)) the fundamentality of the semigroup (\Omega_k) was proved (and thereby the existence of fundamental semigroups), and in ((^4)) the fundamentality of its ideal (\Omega_{k,k-1}) (by (\Omega_{k,t}) we denote the ideal of the semigroup (\Omega_k) consisting of all mappings of rank (\le t), (1\le t\le k-1)). After it had been proved in ((^5)) that the symmetric group (\sigma_k=\Omega_k\setminus\Omega_{k,k-1}) is fundamental ((k\ge 5)), the study of fundamental semigroups naturally split into three directions: the study of the fundamentality of permutation groups (subgroups of (\sigma_k)), the study of the fundamentality of subsemigroups of the ideal (\Omega_{k,k-1}) (semigroups having no (\sigma_k)-component), and the study of the fundamentality of semigroups having both a (\sigma_k)-component and an (\Omega_{k,k-1})-component (we shall call such semigroups mixed). From ((^6)) it follows that the ideal (\Omega_{k,k-2}) (and consequently any of its subsemigroups) is no longer a fundamental semigroup, and among the semigroups (X) satisfying the condition (\Omega_{k,k-2}\subset X\subset \Omega_{k,k-1}) there are both fundamental and nonfundamental ones. From ((^4,^5,^7)) it follows that for (k\ge 5) all maximal subgroups in (\Omega_k) are fundamental, and that the unique nonfundamental semigroup among the maximal subsemigroups in (\Omega_q) ((q=3,4)) is (\Omega_{q,q-2}+\sigma_q). In ((^8)) the fundamentality was proved

groups (A_k) ((k\geqslant 5)), in [2]—for (k\geqslant 5) the fundamentality of every 4-fold transitive permutation group, and for (5\leqslant k\ne 2^i) the fundamentality of every 3-fold transitive one. A. I. Mal’cev in [9] proved the fundamentality of every ((k-1))-fold transitive subsemigroup in (\Omega_k) ((k\geqslant 5)), which absorbs the results of [3, 4, 5], but does not absorb the results of [8, 2]. The author has not encountered other investigations on fundamental sets in the literature.

In the present note, for “almost all” (k), a necessary and sufficient criterion is given for the fundamentality of permutation groups (as a consequence of the latter, a sufficient criterion is derived for the fundamentality of mixed semigroups of mappings), and the possibility of extending to all (k) the method of its proof is also analyzed. Before turning to the formulations of the assertions, we introduce the necessary notation.

By (N) we denote the natural series, by (N_i) the set of (i)-th powers of all numbers, by (P) the set of all prime numbers. Let

[
\widetilde N=N\setminus (P\cup N_2\cup N_3\cup\ldots).
]

For every algebra (\mathfrak A), by (\operatorname{Sub}(\mathfrak A)) we denote the set of all its subalgebras, and by (\operatorname{Sub}^{\max}(\mathfrak A)) the set of all its maximal subalgebras (the latter, by definition, are regarded as proper). If (\mathfrak R\subseteq \operatorname{Sub}(\mathfrak A)), then by (\mathfrak R^{\max}) we denote the collection of all maximal elements of the set (\mathfrak R), partially ordered by set-theoretic inclusion, and by (\mathfrak R^{\operatorname{Sub}}) the collection of all subalgebras of all algebras from (\mathfrak R). The fundament of a subalgebra (\mathfrak A\in \operatorname{Sub}(P_k)) is called its (\sigma_k)-component; denote it by (\operatorname{Fund}(\mathfrak A)). If (\mathfrak M\subseteq \operatorname{Sub}(P_k)), then

[
\mathfrak M^{\operatorname{Fund}}={X:X\in \operatorname{Sub}(\sigma_k),\ \exists Y\in \mathfrak M\ (X=\operatorname{Fund}(Y))}.
]

If (\tau) is some operator mapping (\mathfrak R\subseteq \operatorname{Sub}(\sigma_k)) onto some subset of it, then by (\mathfrak R^\tau) we denote the range of the operator (\tau). In particular, the set of all imprimitive subgroups in (\sigma_k) will be denoted by (\operatorname{Sub}^{\operatorname{impr}}(\sigma_k)), the set of all linear subgroups in (\sigma_k) by (\operatorname{Sub}^{\operatorname{lin}}(\sigma_k)) (a subgroup (X\in \operatorname{Sub}(\sigma_k)) is called linear if either (X=L_k), or (X) is conjugate to (L_k), where (L_k) is the subgroup of all permutations from (\sigma_k) of the form (\psi(x)=\alpha+\beta x \pmod k)). Finally, by (\operatorname{Sub}^{\operatorname{bas}}(\sigma_k)) we denote the set of all fundamental subgroups in (\sigma_k). Everywhere below (k\geqslant 5), since for (k\leqslant 4) no fundamental groups exist.

Theorem 1. If (k\in \widetilde N),

[
(\operatorname{Sub}^{\max}(P_k)\setminus {\mathscr T})^{\operatorname{Fund}}\subseteq
\operatorname{Sub}^{\operatorname{impr}}(\sigma_k),
]

while if (k\in P),

[
(\operatorname{Sub}^{\max}(P_k)\setminus {\mathscr T})^{\operatorname{Fund}}\subseteq
\operatorname{Sub}^{\operatorname{impr}}(\sigma_k)\cup
(\operatorname{Sub}^{\operatorname{lin}}(\sigma_k))^{\operatorname{Sub}}.
]

Theorem 2.

[
(\operatorname{Sub}^{\operatorname{impr}}(\sigma_k))^{\max}
=
(\operatorname{Sub}^{\max}(\sigma_k))^{\operatorname{impr}}
\quad \text{for all } k,
]

while if (k\in P),

[
((\operatorname{Sub}^{\operatorname{lin}}(\sigma_k))^{\operatorname{Sub}})^{\max}
=
(\operatorname{Sub}^{\max}(\sigma_k))^{\operatorname{lin}\cdot\operatorname{Sub}}
=
(\operatorname{Sub}^{\max}(\sigma_k))^{\operatorname{lin}}
=
\operatorname{Sub}^{\operatorname{lin}}(\sigma_k).
]

Corollary. If (k\in \widetilde N),

[
(\operatorname{Sub}(\sigma_k)\setminus \operatorname{Sub}^{\operatorname{bas}}(\sigma_k))^{\max}
=
\operatorname{Sub}^{\max}(\sigma_k)\setminus \operatorname{Sub}^{\operatorname{bas}}(\sigma_k).
]

Theorem 3. If (k\in \widetilde N), the fundamental subgroups in (\sigma_k) are exhausted by its primitive subgroups:

[
\operatorname{Sub}^{\operatorname{bas}}(\sigma_k)=
\operatorname{Sub}(\sigma_k)\setminus \operatorname{Sub}^{\operatorname{impr}}(\sigma_k);
]

if (k\in P), they are the primitive subgroups that are not contained in any of the linear ones:

[
\operatorname{Sub}^{\operatorname{bas}}(\sigma_k)=
\operatorname{Sub}(\sigma_k)\setminus
(\operatorname{Sub}^{\operatorname{impr}}(\sigma_k)\cup
(\operatorname{Sub}^{\operatorname{lin}}(\sigma_k))^{\operatorname{Sub}}).
]

Corollary. If (k\in \widetilde N), every mixed subsemigroup in (\Omega_k) with primitive (\sigma_k)-component is fundamental. If (k\in P), fundamental will be every mixed subsemigroup in (\Omega_k) whose (\sigma_k)-component is a primitive subgroup not contained in any of the linear ones.

Theorem 4.

[
(\operatorname{Sub}(\sigma_k)\setminus \operatorname{Sub}^{\operatorname{bas}}(\sigma_k))^{\max}
=
\operatorname{Sub}^{\max}(\sigma_k)\setminus \operatorname{Sub}^{\operatorname{bas}}(\sigma_k)
\Longleftrightarrow
]

[
\Longleftrightarrow
(\operatorname{Sub}^{\max}(P_k)\setminus {\mathscr T})^{\operatorname{Fund}}
=
\bigcup_{\tau\in \mathfrak M\subseteq Z_{\mathfrak A}(\max)+\mathfrak B}
\operatorname{Sub}^{\tau}(\sigma_k),
]

where

[
\mathfrak A={\tau:\operatorname{Sub}(\sigma_k)\xrightarrow{\tau}
\operatorname{Sub}(\sigma_k)\setminus \operatorname{Sub}^{\operatorname{bas}}(\sigma_k)},
]

[
Z_{\mathfrak A}(\max)=
{\tau:\tau\in \mathfrak A,\ (\operatorname{Sub}^{\tau}(\sigma_k))^{\max}
=
(\operatorname{Sub}^{\max}(\sigma_k))^{\tau}},
]

[
\mathfrak B=
{\tau:\tau\in \mathfrak A\setminus Z_{\mathfrak A}(\max),\
\operatorname{Sub}^{\tau}(\sigma_k)\subseteq
\operatorname{Sub}^{\max}(\sigma_k)\setminus \operatorname{Sub}^{\operatorname{bas}}(\sigma_k)}.
]

It is obvious that, for those

[
k \in N \setminus (\widetilde N \cup P)=\bigcup_{i\ge 2} N_i,
]

for which ((\operatorname{Sub}_{\max}(P_k)\setminus{\mathcal T})^{\mathrm{Fund}}) is representable in the form of the indicated union, Theorem 4 gives a method for obtaining a necessary and sufficient criterion for the fundamentality of subgroups in (\sigma_k). Let us note that the set (\widetilde N \cup P) for which this has been done (Theorem 3) constitutes “almost all” of the natural numbers.

In conclusion we note that it is easy to formulate theorems dual to Theorems 3 and 4 for the corresponding predicate algebras (the set of all predicates defined on (E_k) can be turned into an algebra so that the lattice of its subalgebras is anti-isomorphic to the lattice of subgroups of (\sigma_k) (see ((^{10})))).

The author takes the opportunity to express his gratitude to S. V. Yablonskii for his attention to the work.

Institute of Cybernetics
Academy of Sciences of the Azerbaijan SSR
Baku

Received
17 IV 1969

CITED LITERATURE

¹ A. I. Maltsev, Algebra i logika, 5, no. 2, 5, Novosibirsk (1966).
² A. Salomaa, Ann. Acad. Sci. Fennicae, Ser. A I, 338 (1963).
³ J. Słupecki, C. R. séances Soc. Sci. de Varsovie, Cl. III, 32, 102 (1939).
⁴ S. V. Yablonskii, Tr. Matem. inst. im. V. A. Steklova, AN SSSR, 5 (1958).
⁵ A. Salomaa, J. Symb. Logic, 25, 203 (1960).
⁶ E. Yu. Zakharova, Problemy kibernetiki, vol. 18, 5 (1967).
⁷ R. A. Bairamov, Diskretnyi analiz, no. 8, Novosibirsk, 1966, p. 3.
⁸ A. Salomaa, Ann. Univ. Turkuensis, Ser. A I, 53 (1962).
⁹ A. I. Maltsev, Algebra i logika, 6, no. 2, 61, Novosibirsk (1967).
¹⁰ V. G. Bodnarchuk, L. A. Kaluzhnin et al., Kibernetika, no. 3, 1 (1969).