A. Ya. Aizenshtat

On the Semisimplicity of the Semigroup of Endomorphisms of Ordered Sets

(Presented by Academician A. I. Maltsev on 8 VIII 1961)

Let (\Omega) be an ordered set (an equivalent term is a partially ordered set). According to the general definition of an endomorphism of a set in which certain relations are specified, a transformation (S) of the set (\Omega) is called an endomorphism of (\Omega) if from (\alpha \leq \beta) ((\alpha, \beta \in \Omega)) it always follows that (S\alpha \leq S\beta). It is known that the set of all endomorphisms of (\Omega) is a semigroup with respect to the usual multiplication of transformations ((S_1S_2(\alpha)=S_1(S_2(\alpha)))) ((^3)). In what follows, by (\Omega) we shall denote an ordered set, and by (\Sigma) the semigroup of all its endomorphisms.

It is of interest to establish a connection between the properties of the set (\Omega) and those of the semigroup (\Sigma). Some results in this direction have already been obtained. Thus, L. M. Gluskin proved that the order relation in (\Omega) is completely determined by the semigroup (\Sigma), apart from one trivial case ((^1)). The present note is also devoted to this direction.

Let (\mathfrak A) be an arbitrary semigroup, and (I) its ideal (two-sided). We say that (a,b \in \mathfrak A) are comparable modulo the ideal (I) if (a=b) or (a,b \in I). Comparison modulo the ideal (I) is a two-sided stable equivalence; it is usually called an ideal equivalence. A semigroup (\mathfrak A) is called semisimple if all its two-sided stable equivalences distinct from equality are ideal ((^2)). It is known that finding two-sided stable equivalences in a semigroup is equivalent to finding its homomorphisms up to isomorphism.

In the article a characterization of a finite linearly ordered set is indicated with the aid of properties of the semigroup (\Sigma).

We shall call the elements of the semigroup (\Sigma) transformations. The rank of a transformation (S \in \Sigma), as usual, will mean the cardinality of the set (S\Omega), and will be denoted by (rS).

The simplest notions of semigroup theory may be found in the book ((^3)).

§ 1. In this section we shall assume (\Omega) to be a finite linearly ordered set consisting of (n) elements. We shall describe all two-sided stable equivalences of the semigroup (\Sigma), using the approach to the description of homomorphisms from ((^4)).

Let (k \leq n). Denote by (I_k) the set of all transformations in (\Sigma) whose ranks do not exceed (k).

Theorem 1. The subsets (I_k), and only they, are ideals of the semigroup (\Sigma).

In the proof of the main theorem, Lemmas 1, 2, and 3 are used.

Lemma 1. Under any two-sided stable equivalence distinct from equality, all transformations of unit rank are equivalent to one another.

Lemma 2. Under any two-sided stable equivalence, substitutions equivalent to substitutions of unit rank form an ideal of the semigroup (\Sigma).

From Theorem 1 and Lemma 2 there follows

Corollary. If, under some two-sided stable equivalence, there are two equivalent substitutions (A, B) such that (rA = 1,\ rB = k > 1), then all substitutions of rank not exceeding (k) will be equivalent to one another.

Lemma 3. If, under some two-sided stable equivalence, a substitution (A) is not equivalent to substitutions of unit rank and a substitution (B) is equivalent to (A), then (rA = rB).

Theorem 2. If (\Omega) is a finite linearly ordered set, then the semigroup (\Sigma) of all endomorphisms of (\Omega) is semisimple.

§ 2. In the semigroup (\Sigma) we define a relation (\mathfrak R). (A \sim B) ((\mathfrak R)), if one of the following conditions is satisfied: 1) (A, B) are substitutions of finite rank; 2) (A, B) are of infinite rank, and if (\Omega_0) is the set of those elements of (\Omega) which the substitutions (A, B) transform differently, then the sets (A\Omega_0,\ B\Omega_0) are finite. The relation (\mathfrak R) is a two-sided stable equivalence on (\Sigma) ((^4)).

Let (\Omega) be an infinite linearly ordered set. We shall write the elements of the semigroup (\Sigma) in the form of substitutions. An element of (\Omega) to which a substitution sends only itself will sometimes be omitted in writing the substitution.

We shall prove that the equivalence (\mathfrak R) is not ideal. For this it is enough to prove that there exist at least two (\mathfrak R)-classes containing more than one element. One such class is formed by all substitutions of finite rank.

Let (m_1 < m_2) be elements of (\Omega), chosen so that the number of elements (\alpha < m_1) or the number of elements (\beta > m_2) is infinite. It is easy to show that such elements (m_1, m_2) can be found in any infinite linearly ordered set.

The substitutions

[
A =
\left(
\frac{
\overbrace{m_1 \ldots m_2}
}{
m_1
}
\right),
\qquad
B =
\left(
\frac{
\overbrace{m_1 \ldots m_2}
}{
m_2
}
\right),
]

where the dots are understood to denote all (\gamma \in \Omega) such that (m_1 < \gamma < m_2), have infinite ranks and, consequently, belong to a second (\mathfrak R)-class containing more than one element.

Thus, it has been proved:

Theorem 3. If (\Omega) is an infinite linearly ordered set, then the semigroup (\Sigma) of all endomorphisms of (\Omega) is not semisimple.

Let the ordering in (\Omega) not be linear. To prove that the semigroup (\Sigma) is not semisimple, we shall need one auxiliary theorem.

Let

[
S =
\begin{pmatrix}
\Omega_1 & \Omega_2 & \ldots & \Omega_k \
a_1 & a_2 & \ldots & a_k
\end{pmatrix}
\in \Sigma,
]

where (\Omega_i) is the set of elements of (\Omega) transformed by the substitution (S) into (a_i), all (a_1, a_2, \ldots, a_k) are distinct, and (\Omega = \Omega_1 \cup \Omega_2 \cup \ldots \cup \Omega_k). We shall say that the substitution (S) effects a partition of (\Omega) into the subsets (\Omega_1, \Omega_2, \ldots, \Omega_k).

Theorem 4. In order that the ordering in the set (\Omega) be non-linear, it is necessary and sufficient that in the semigroup (\Sigma) there exist two distinct substitutions (A, B) satisfying the conditions: 1) (A, B) effect the same partition of (\Omega) into subsets; 2) (A\Omega = B\Omega).

Define in the semigroup $\Sigma$ the relation $L_k$ for any natural number $k$. Let $A \sim B\ (L_k)$ if one of the following conditions is satisfied: 1) $A = B$; 2) the ranks of the substitutions $A, B$ are less than $k$; 3) $rA = rB = k$, the substitutions $A, B$ effect the same partition $\Omega$ into subsets, and $A\Omega = B\Omega$. It is easy to show that the relation $L_k$ is a two-sided stable equivalence.

From Theorem 4 we obtain that, for some $k$, the two-sided stable equivalence $L_k$ is not ideal. Hence it follows:

Theorem 5. If the ordering on $\Omega$ is not linear, then the semigroup $\Sigma$ is not semisimple.

Combining the results given above, we obtain:

Theorem 6. In order that the ordered set $\Omega$ be finite linearly ordered, it is necessary and sufficient that the semigroup of all endomorphisms of $\Omega$ be semisimple.

Received
1 VIII 1961

REFERENCES CITED

¹ L. M. Gluskin, DAN, 129, No. 1, 16 (1959).
² E. S. Lyapin, Izv. AN SSSR, ser. matem., 14, 367 (1950).
³ E. S. Lyapin, Semigroups, Moscow, 1960.
⁴ A. I. Maltsev, Matem. sborn., 31 (73), 1, 136 (1952).