MATHEMATICS

E. G. SHUTOV

ON SEMIGROUPS OF ALMOST IDENTITY TRANSFORMATIONS

(Presented by Academician A. N. Kolmogorov, 3 V 1960)

1°. In the present note all stable equivalences are found for the semigroup of all almost identity transformations of an infinite set (\Omega). Thanks to these results, all endomorphisms of this semigroup distinct from isomorphisms are determined. Further, certain properties of the semigroup of all almost identity partial transformations of the set (\Omega) are investigated. An abstract characterization of this semigroup is given, and all its ideals, normal subsemigroups, and automorphisms are described. In finding the stable equivalences of the semigroup of all almost identity transformations, results of A. I. Mal'tsev are used ((^{1})). For finite sets (\Omega), all the results indicated above have already been obtained ((^{1-3})).

2°. All notions which are not specially defined in the note are used in their usual sense (see, for example, ((^{1-4}))).

An equivalence of a semigroup is called stable if it is compatible with the multiplication of this semigroup. Let (A_1) be a subsemigroup of the semigroup (A). We shall say that an equivalence (\Sigma_1) of the semigroup (A_1) is a restriction of an equivalence (\Sigma) of the semigroup (A), if elements of the semigroup (A) are comparable with respect to the equivalence (\Sigma_1) if and only if they are comparable with respect to the equivalence (\Sigma). The identity equivalence of a semigroup and the equivalence under which all elements of the semigroup are comparable with one another are called trivial equivalences.

3°. Let (\Omega) be an infinite set. Denote by (H_\Omega) the semigroup of all almost identity transformations of the set (\Omega), i.e., the totality of all such transformations of the set (\Omega), each of which moves no more than a finite number of elements, and by (S_\Omega) the group of all one-to-one transformations from (H_\Omega). If under (a \in H_\Omega) the set (\Delta) is mapped onto (\Delta_1), then we shall write (a\Delta=\Delta_1). The set of all (\alpha \in \Omega) which the transformation (a) maps into a single element is called the contracting complex of (a). Transformations (a) and (b) from (H_\Omega) will be called related if their contracting complexes are the same and (a\Omega=b\Omega).

4°. Let (a,b \in H_\Omega), and let (\Delta) be the set of all those (\alpha) from (\Omega), each of which is mapped into itself by the transformations (a) and (b) and is the contracting complex of (a) and (b). Denote by (\Pi(a,b)) the set of all (\alpha\in\Omega) which are not contained in (\Delta). Let (a) and (b) be related. Then denote by ([a,b]=c) such a transformation from (S_\Omega) that for any
(\alpha\in\Pi(a,b)), (\beta\in a\Pi(a,b)) one has

[
ca\alpha=b\alpha,\qquad c\beta=\beta.
]

For any (a,b,c\in H_\Omega) the following properties hold:

1. If (a) is related to (b), and (b) is related to (c), then (a) is related to (c) and
   [
   [b,c][a,b]=[a,c].
   ]

2. If (a,b) are related and the defects of (ca), (a) are the same, then (cb) is related to (ca) and the transformations ([a,b]), ([ca,cb]) are conjugate.

1. If (a) and (b) are related and the defects (ac, a) are the same, then (ac) is related to (bc) and ([ac,bc]=[a,b]).

(5^0). Definition. Let (n) be a finite nonnegative integer, and let (R) be a normal divisor of the group (S_\Omega). Transformations (a) and (b) from (H_\Omega) will be called comparable modulo (R) relative to (n) if one of the following conditions is satisfied: 1) (a) and (b) are equal; 2) the defects of (a) and (b) are greater than (n); 3) the defects of (a) and (b) are equal to (n), (a) and (b) are related, and ([a,b]\in R).

By virtue of the properties (4^0), the comparison just defined is a stable equivalence of the semigroup (H_\Omega). These equivalences, for all possible (R) and (n), will be called equivalences modulo normal divisors.

(6^0). Let (\Delta) be a finite subset of (\Omega) containing more than four elements. The aggregate (P_\Delta) of all such (a\in H_\Omega) that (a\Delta\subset\Delta) and (a\alpha=\alpha) for every (\alpha\in\Delta), forms a semigroup isomorphic to the semigroup of all substitutions of the set (\Delta). Let (\Delta\subset\Delta_1), where (\Delta_1) is finite, and let (\Sigma) be a stable equivalence of the semigroup (P_\Delta) under which at least two transformations from (P_\Delta) are comparable, the rank of each of which is greater than four. Then, using results of A. I. Mal’cev ((^1)), one can show that the equivalence (\Sigma) has the following properties:

1. There exists an equivalence (\Sigma_1) of the semigroup (H_\Omega) modulo a normal divisor ((5^0)) such that the equivalence (\Sigma) is the restriction of the equivalence (\Sigma_1) ((2^0)).

2. If the equivalence (\Sigma) is the restriction of each of the stationary equivalences (\Sigma_1,\Sigma_2) of the semigroup (P_{\Delta_1}), then the equivalences (\Sigma_1) and (\Sigma_2) are the same.

(7^0). With the aid of properties 1 and 2 ((6^0)) the following is proved.

Theorem 1. The only nontrivial stable equivalences of the semigroup (H_\Omega) are the equivalences modulo normal divisors ((5^0)).

From the theorem follows

Corollary. The only normal subsemigroups of the semigroup (H_\Omega) are the semigroup (H_\Omega) itself and the normal divisors of the group (S_\Omega).

This result was obtained by N. N. Vorob’ev ((^4)).

(8^0). A homomorphism of a semigroup into itself is called an endomorphism of this semigroup. We shall denote by (\varphi_1) the endomorphism of the semigroup (H_\Omega) onto the subsemigroup consisting of one element.

The set (P_2), consisting of two such transformations (a) and (b) from (H_\Omega) that

[
ab=ba=a^2=c,\qquad b^2=b,
]

forms a subsemigroup of (H_\Omega). The set (P_3), consisting of three such transformations (a,b,c) from (H_\Omega) that

[
ab=ba=ac=ca=a^2=a,\qquad bc=cb=c,\qquad b^2=c^2=b.
]

also forms a subsemigroup of (H_\Omega). Let (S^1) be the set of all even (a) from (S_\Omega), and let (H^1) be the set of all (a) from (H_\Omega) of positive defects. Then denote by (\varphi_2) such a mapping of (H_\Omega) onto the subsemigroup (P_2) that for any (d_1\in S_\Omega,\ d_2\in H^1) one has

[
\varphi_2 d_1=b,\qquad \varphi_2 d_2=a.
]

Next denote by (\varphi_3) such a mapping of (H_\Omega) onto its subsemigroup (P_3) that for any (d_1\in S_\Omega\setminus S^1,\ d_2\in S^1,\ d_3\in H^1) one has

[
\varphi_3 d_1=b,\qquad \varphi_3 d_2=c,\qquad \varphi_2 d_3=a.
]

The mappings (\varphi_2) and (\varphi_3) are endomorphisms of the semigroup (H_\Omega), and there exist no other endomorphisms of (H_\Omega) onto (P_2) and (P_3) distinct from (\varphi_2) and (\varphi_3).

Thanks to Theorem 1, the following theorem is proved.

Theorem 2. The only endomorphisms of the semigroup (H_{\Omega}) distinct from isomorphisms are endomorphisms of the form (\varphi_1, \varphi_2, \varphi_3).

9°. Let (\Delta_1, \Delta_2) be subsets of (\Omega), which may also be empty. A mapping (a) of the set (\Delta_1) into (\Delta_2) is called a partial transformation of the set (\Omega). We shall denote each of (\Delta_1, \Delta_2), respectively, by (\Pi_1(a)) and (\Pi_2(a)). The cardinality of (\Pi_2(a)) is called the rank of (a). With respect to the usual multiplication of partial transformations, the totality (W_{\Omega}) of all partial transformations of the set (\Omega) is a semigroup. The set (V_{\Omega}) of all (a \in W_{\Omega}), each of which moves no more than a finite number (\alpha) from (\Omega), forms a subsemigroup of (W_{\Omega}), which is called the semigroup of all almost identical partial transformations. The totality (V_{\Omega}^{1}) of all (a) from (V_{\Omega}) of rank (\leqslant 1) is the minimal densely embedded ideal of (V_{\Omega}).

10°. Let (A) be a semigroup with zero (0). Denote by (A^) the set of all nonzero (a \in A) such that, for any (b, c \in A), from (bc = 0) it follows that (bac = 0). Let (a \in A). The set of all (b \in A^) for which (ab \neq 0) will be denoted by (A_a).

Theorem 3. In order that a semigroup (A) be isomorphic to the semigroup (V_{\Omega}^{1}), it is necessary and sufficient that (A) have the following properties:

1. (A) contains a zero (0), and the sets (\Omega), (A^) are equipotent.*
2. For every nonzero (a \in A) there exists in (A^) a unique left identity (e_a), and the set (A_a) is nonempty and finite.*
3. For every (a \in A^) and every nonempty finite subset (M) of the set (A^), there exists a unique (b \in A) such that

[
A_b = M, \qquad e_b = a.
]

1. If (e_b \in A_a) ((a, b \in A)), then (ab \neq 0).
2. If (ab \neq 0) ((a, b \in A)), then (A_{ab} = A_b).

11°. Thanks to results obtained by L. M. Gluskin ((^3)), the following theorem is valid.

Theorem 4. In order that a semigroup (A) be isomorphic to the semigroup (V_{\Omega}), it is necessary and sufficient that (A) have a minimal densely embedded ideal isomorphic to the semigroup (V_{\Omega}^{1}) (9°).

12°. If all elements of the set (\Delta_1), with the exception, perhaps, of a finite number of them, are contained in the set (\Delta_2), then we shall write (\Delta_1 \sigma \Delta_2).

13°. Let (a \in V_{\Omega}), let (\Delta) be a subset of (\Omega), and let (\Delta_1 = \Delta \cap \Delta_2), where (\Delta_2) is the set of all (\alpha \in \Pi_1(a)) such that (a\alpha = \alpha), and if (a\beta = \alpha), then (\beta = \alpha). We shall denote each of the sets (\Delta \setminus \Delta_1) and (\Pi_2(a) \setminus \Delta_1), respectively, by (\Gamma(\Delta,a)) and (\Gamma(a,\Delta)).

Definition. Let (\Delta) be a subset of (\Omega), and let (n) be a finite nonnegative integer not exceeding the cardinality of the set (\Omega \setminus \Delta). By the set (V(\Delta,n)) we shall mean the set of all (a \in V_{\Omega}) such that:

1. (\Pi_1(a) \sigma \Delta) (12°),
2. If the sets (\Gamma(\Delta,a)) and (\Gamma(a,\Delta)) have cardinalities (p) and (q), respectively, then (q \leqslant p+n).

Obviously, if (\Delta) contains a finite number (m) of elements, then (V(\Delta,n)) is the totality of all such (a) from (V_{\Omega}) whose ranks do not exceed (n+m).

Each set (V(\Delta,n)), for arbitrary (\Delta) and (n), is an ideal of the semigroup (V_{\Omega}).

Theorem 5. All ideals of the semigroup (V_{\Omega}) are exhausted by set-theoretic sums of ideals of the form (V(\Delta,n)).

14° Definition. Let (\Omega^*) be any set whose elements are distinct subsets (\Delta_i) of the set (\Omega), and let (\Omega_a = \Omega \setminus \Pi_1(a)),

where (a \in V_\Omega). By a set (V(\Omega^)) we shall mean the collection of all such (a) from (V_\Omega) for each of which there exists in (\Omega^) a finite number of such (\Delta_i), whose union is (\Delta), such that (\Omega_a \subset \Delta) ((12^0)).

In order to exhaust all possible sets (V(\Omega^)), it is enough to assume that (\Omega^) either consists of one (\Delta_1) (finite or infinite), or of an infinite number of such infinite (\Delta_i) that the union of any finite number of them differs from the union of all sets from (\Omega^*) by an infinite number of elements.

Every normal divisor of the group (S_\Omega) ((3^0)) and every set of the form (V(\Omega^*)) are normal subsemigroups of the semigroup (V_\Omega).

Theorem 6. All normal subsemigroups of the semigroup (V_\Omega) are exhausted by the normal divisors of the group (S_\Omega) and by all possible subsemigroups of the form (V(\Omega^)).*

(15^0). Theorem 7. Every automorphism of the semigroup (V_\Omega) is induced by a unique automorphism of the semigroup (W_\Omega) ((9^0)).

The only automorphisms of the semigroup (W_\Omega) are inner automorphisms ((^3)); therefore Theorem 7 gives a complete description of all automorphisms of the semigroup (V_\Omega).

Since not all invertible elements of the semigroup (W_\Omega) belong to (V_\Omega), it follows, according to Theorem 7, that the semigroup (V_\Omega) has automorphisms which are not inner.

Udmurt State
Pedagogical Institute
named after the Decade of the UASSR

Received
24 IV 1960

CITED LITERATURE

(^1) A. I. Mal’tsev, Matem. sborn., 31 (73), 1, 136 (1952).
(^2) E. S. Lyapin, DAN, 88, No. 1, 13 (1953).
(^3) L. M. Gluskin, Matem. sborn., 47 (89), 1, 111 (1959).
(^4) N. N. Vorob’ev, Scientific Notes of the Leningrad State Pedagogical Institute named after Herzen, 89, 161 (1953).