SEMIGROUPS OF ONE-TO-ONE TRANSFORMATIONS

E. G. SHUTOV

(Presented by Academician A. I. Maltsev on 22 V 1961)

MATHEMATICS

1°. Let (\Sigma_1) be the class of all semigroups with left cancellation and without left identities ((^3)), and let (\Sigma_2) be the class of all semigroups with left cancellation, with left invertibility ((^2)), and without idempotents. Every semigroup of the class (\Sigma_1) is isomorphic to a subsemigroup of the semigroup (H_\Omega) of all one-to-one mappings of some infinite set (\Omega) into itself, and every semigroup of the class (\Sigma_2) is isomorphic to a subsemigroup of the semigroup (P_\Omega) of all (a \in H_\Omega) whose defect ((^6)) is equal to the cardinality of (\Omega). Moreover, (H_\Omega) belongs to the class (\Sigma_1), and (P_\Omega) to the class (\Sigma_2). It follows from this that the study of the semigroups (H_\Omega) and (P_\Omega) is important not only from the point of view of transformation theory, but also from the point of view of the general theory of semigroups. In the present note all stable equivalences and normal complexes of the semigroups (H_\Omega) and (P_\Omega) are described. Owing to these results, a certain sufficient condition has been found for the possibility of embedding semigroups with left cancellation in semigroups that are simple with respect to homomorphisms. A number of properties of stable equivalences of the semigroups (H_\Omega) and (P_\Omega) are considered; in particular, the dependence of these equivalences on the stable equivalences of the symmetric groupoid ((^1)) and of the symmetric generalized group ((^5)) is established.

Terms of semigroup theory that are not defined are used in the usual sense ((^2)). We shall denote the cardinality of a set (\Delta) by (\tau\Delta). Let (a) and (b) be mappings of the set (\Omega) into itself, and let (\Delta \subset \Omega). Denote by (a\Delta) the image of (\Delta) under (a), and by (\Pi(a,b)) the set of all such (\alpha \in \Omega) that (a\alpha \ne b\alpha), and let (\tau\Pi(a,b)=\omega(a,b)).

2°. In what follows, by (\Omega) we shall mean an infinite set, and by (q) an infinite cardinal number (\leq \lambda), where (\lambda) is the least cardinal number (>\tau\Omega). Let (A_\Omega) be an arbitrary semigroup of transformations of the set (\Omega). Denote by (\sigma[q]) the binary relation of the semigroup (A_\Omega) such that elements (x,y \in A_\Omega) are in the relation (\sigma[q]) if and only if (\omega(x,y)<q).

It is obvious that in any semigroup (A_\Omega) the relation (\sigma[q]) is a left-stable equivalence.

Theorem 1. In order that in the semigroup (A_\Omega) all relations (\sigma[q]) be stable equivalences, it is necessary and sufficient that, for any (a,b,c \in A_\Omega), (\Delta \subset \Omega), from (c\Delta=\alpha \in \Pi(a,b)) it follow that (\tau\Delta\omega(a,b)).

Theorem 2. All relations (\sigma[q]) are stable equivalences of the semigroup (P_\Omega) ((^{10})), and these relations exhaust all stable equivalences of the semigroup (P_\Omega) distinct from equality.

3°. Let (a \in P_\Omega), (q \leq \lambda) ((^{20})). Denote by (M(a,q)) the set of all such (b \in P_\Omega) that (\omega(a,b)<q). From Theorem 2 the following theorems are derived:

Theorem 3. Every set (M(a,q)) is a normal complex of the semigroup (P_\Omega), and all normal complexes of the semigroup (P_\Omega) consisting of more than one element are exhausted by the sets (M(a,q)).

Theorem 4. The semigroup (P_{\Omega}) has no normal subsemigroups distinct from (P_{\Omega}).

(4^\circ). Let (R) be such a subset of (P_{\Omega}) that if (a \in R), (x \in P_{\Omega}), (a\Omega \supset x\Omega), (\tau(a\Omega \setminus x\Omega)=\tau\Omega), then (x \in R).

Theorem 5. Every subset of the form (R) of the semigroup (P_{\Omega}) is its right ideal, and these ideals exhaust all ideals of the semigroup (P_{\Omega}).

(5^\circ). Let (\xi \leqslant \tau\Omega), (\Delta \subset \Omega), (\tau\Omega=\tau\Delta). Denote by (H_{\Omega}^{\xi}) and (H_{\Omega,\Delta}), respectively, the sets of all such (x \in H_{\Omega}) and (z \in H_{\Omega}) ((1^\circ)) that (\delta(x)\geqslant \xi) and (z\Omega \subset \Omega), where (\delta(x)=\tau(\Omega \setminus x\Omega)).

Theorem 6. The only two-sided and right ideals of the semigroup (H_{\Omega}) are, respectively, all the sets (H_{\Omega}^{\xi}) and (H_{\Omega,\Delta}). Moreover, every left ideal of (H_{\Omega}) is its two-sided ideal.

(6^\circ). Let (a,b \in H_{\Omega}), (a\Omega=b\Omega). Denote by ([a,b]) such a (c \in H_{\Omega}) that (ca=b^{-1}aa) for every (a \in \Omega). Let (S_{\Omega}) be the group of all (a \in H_{\Omega}) for which (a\Omega=\Omega) and (q\leqslant \lambda) ((2^\circ)).

Denote by (R_q) the set of all such (x \in S_{\Omega}) that (\omega(x,e)<q), where (e) is the identity in (S_{\Omega}), and by (R_1) the alternating subgroup of the group (S_{\Omega}).

Definition. Let
[
0 \leqslant \xi_0 \leqslant \xi_1 \leqslant \xi_2 < \cdots < \xi_r=\lambda
]
be such a sequence of cardinal numbers (finite or infinite) that if (\xi_1) is infinite, then (\xi_0=\xi_1), and if (\xi_k) ((k\geqslant 1)) is finite, then (\xi_k-\xi_{k-1}\leqslant 1). To each (\xi_k) assign such an infinite cardinal number (\eta_k\leqslant \lambda) that if (i\lambda_0) and (q=1,\lambda_0) when (\eta_0=\lambda_0). Denote by (\sigma[\xi,\eta]) such a binary relation of the semigroup (H_{\Omega}) that transformations (a,b \in H_{\Omega}) are in the relation (\sigma[\xi,\eta]) if and only if one of the conditions is satisfied: 1) the transformations (a) and (b) are equal; 2) (a\Omega=b\Omega), (\delta(a)=\xi_0), ([a,b]\in R_q), (\omega(a,b)<\eta_0); 3) (\xi_{n_0}\leqslant \delta(a),\delta(b)<\lambda_0), (\omega(a,b)<\eta_{n_0}), (\delta(a)-\delta(b)=m_{n_0}p) ((p=0,\pm1,\pm2,\ldots)); 4) (\xi_k\leqslant \delta(a),\delta(b)<\xi_{k+1}), (\omega(a,b)<\eta_k) ((k\ne0,\ k\ne n_0)).

Theorem 7. Every relation (\sigma[\xi,\eta]) is a stable equivalence of the semigroup (H_{\Omega}), and these equivalences exhaust all stable equivalences of the semigroup (H_{\Omega}).

(7^\circ). Let (\xi_1<\xi_2\leqslant \lambda) be arbitrary cardinal numbers, (\eta\leqslant \lambda) infinite, (a \in H_{\Omega}), (\xi_1\leqslant \delta(a)<\xi_2). Denote by (M_1(a,\xi_1,\xi_2,\eta)) the set of all such (x \in H_{\Omega}) that (\xi_1\leqslant \delta(x)<\xi_2), (\omega(a,x)<\eta). Let (n\geqslant 0), (m\geqslant 1) be finite, (n\leqslant \delta(a)<\lambda_0). Denote by (M_2(a,n,m,\eta)) the set of all such (x \in H_{\Omega}) that (n\leqslant \delta(x)<\lambda_0), (\omega(a,x)<\eta), (\delta(a)-\delta(x)=mp) ((p=0,\pm1,\pm2,\ldots)). Further, let (\delta(a)=n), (R_q) ((6^\circ)) be such that (q=\eta) if (\eta>\lambda_0), and (q=1,\eta) if (\eta=\lambda_0). Denote by (M_3(a,n,R_q,\eta)) the set of all (x \in H_{\Omega}) for which (x\Omega=a\Omega), ([a,x]\in R_q), (\omega(a,x)<\eta).

Theorem 8. Every set (M_1(a,\xi_1,\xi_2,\eta)), (M_2(a,n,m,\eta)), (M_3(a,n,R_q,\eta)) is a normal complex of the semigroup (H_{\Omega}), and these sets exhaust all normal complexes of the semigroup (H_{\Omega}) containing more than one element.

(8^\circ). Let (\xi,\eta\leqslant \lambda) be infinite. Denote by (R(\xi,\eta)) the set of all such (a \in H_{\Omega}) that (\delta(a)<\xi), (\omega(a,e)<\eta), and if (\xi=n\geqslant 0) is finite, then by (R_1(n,\eta)) the set of all (b \in H_{\Omega}) for which (\delta(b)=kn), (\omega(b,e)<\eta), where (k=0,\pm1,\pm2,\ldots).

Theorem 9. The only normal subsemigroups of the semigroup (H_{\Omega}) are the unit subsemigroup, the alternating subgroup of the group (S_{\Omega}), and all subsets of the form (R(\xi,\eta)) and (R_1(n,\eta)).

From Theorem 9 there follows the following known ((^8)) corollary:

Corollary. The only normal divisors of the group (S_{\Omega}) are the unit subgroup and all (R_{\xi}), where (\xi=1), (\xi_0\leqslant \xi\leqslant \lambda).

9°. A semigroup is called simple if each of its stable equivalences is either equality or contains all pairs of elements of this semigroup. In (⁴, ⁷) three examples were first constructed of simple semigroups containing elements of infinite order and distinct from groups. The following theorem gives one more example of such semigroups.

Theorem 10. The factor semigroup of the semigroup (P_{\Omega}) by the equivalence (\sigma[\tau\Omega]) (2°) is a simple semigroup with left reversibility and without idempotents.

10°. Let (A) be an infinite semigroup with left cancellation, let (a, b \in A), and let (\Delta) be the set of all such (x \in A) that (ax \ne bx). If for any (a) and (b) one always has (\tau\Delta = \tau(A \setminus aA)), then the semigroup (A) will be called an (l)-semigroup. From Theorem 10 the following theorem is derived:

Theorem 11. Every (l)-semigroup is a subsemigroup of some simple semigroup with left reversibility and without idempotents.

Let (F) be the semigroup of such continuous strictly monotone functions (f(x)), defined on the interval ([a,b]), (a < b), that (f[a,b] \ne [a,b]). From Theorem 11 the following corollary follows:

Corollary. Each of the semigroups (P_{\Omega}), (F) is a subsemigroup of some simple semigroup with left reversibility and without idempotents.

Since every semigroup of class (\Sigma_2) is isomorphic to a subsemigroup of some semigroup (P_{\Omega}), it follows from this:

Corollary. Every semigroup with left reversibility, with left cancellation and without idempotents, is a subsemigroup of some simple semigroup with left reversibility and without idempotents.

11°. Let (A_1) be a subsemigroup of the semigroup (A_2). An equivalence (\sigma_1) of the semigroup (A_1) is called the restriction of an equivalence (\sigma_2) of the semigroup (A_2) if elements (a) and (b) of the semigroup (A_1) are in the relation (\sigma_1) if and only if they are in (A_2) in the relation (\sigma_2). Let (C_{\Omega}) be the symmetric groupoid (¹) and (W_{\Omega}) the symmetric generalized group (⁵). From (¹, ⁵), thanks to Theorems 2 and 7, the following theorems are derived:

Theorem 12. Every stable equivalence of the semigroup (P_{\Omega}) is the restriction of some stable equivalence of each of the semigroups (C_{\Omega}), (W_{\Omega}), and (H_{\Omega}).

Theorem 13. All stable equivalences of the semigroup (H_{\Omega}) that are restrictions of stable equivalences of each of the semigroups (C_{\Omega}), (W_{\Omega}), are exhausted by all equivalences of the form (\sigma[q]) (2°).

Udmurt State Pedagogical Institute
named after the Tenth Anniversary of the Udmurt Autonomous Region

Received
14 V 1961

CITED LITERATURE

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