I. I. VALUTSƏ

LEFT IDEALS OF THE SEMIGROUP OF ENDOMORPHISMS OF A FREE UNIVERSAL ALGEBRA

(Presented by Academician P. S. Novikov on 6 XII 1962)

Let \(G\) be a free algebra in some primitive class of universal algebras \(\Lambda\). In the present paper a description is given of the structures of all left and all two-sided ideals of the semigroup of endomorphisms of the algebra \(G\), by means of subalgebras and certain other subsets of the algebra \(G\) itself. The results obtained can naturally be applied to various concrete examples. We note only that, if the set of operations is empty, then from the results of § 5 one easily obtains the description, given in the work of A. I. Mal’cev \((^{2})\), of the two-sided ideals of the semigroup of all mappings of a certain set into itself. Basic information on the theory of universal algebras and semigroups can be found in the books \((^{1,3})\).

The set of all subalgebras of a universal algebra, generally speaking, is not a lattice, because the intersection of subalgebras may turn out to be empty. However, this set becomes a complete lattice if it is supplemented by the empty set. The same situation may occur also for certain other kinds of subalgebras or subsets of a universal algebra (for example, for left and two-sided ideals of some semigroup). In what follows we shall assume that, whenever it is necessary, such a supplementation has been made.

Let \(x_i\), where \(i\) ranges over an index set \(I\) of cardinality \(n\), be some system of free generators of the algebra \(G\). Denote by \(A(G)\) the semigroup of all endomorphisms of the algebra \(G\).

1. We shall call a subalgebra \(F\) of the algebra \(G\) an \(n\)-subalgebra if it has at least one system of generators of cardinality not exceeding \(n\).

Denote by \(Q\) the set of all collections of \(n\)-subalgebras of the algebra \(G\) that have the following property:

a) If the \(n\)-subalgebra \(F\) belongs to the collection \(q\), and \(H\) is an \(n\)-subalgebra such that \(H \subset F\), then \(H\) also belongs to the collection \(q\).

The set \(Q\) is a complete lattice with respect to the set-theoretic operations of union and intersection.

If \(M\) is some union of subalgebras of the algebra \(G\), then by \(\mathfrak{A}_M\) we denote the left ideal consisting of all endomorphisms that map the algebra \(G\) into \(M\). Left ideals of the form \(\mathfrak{A}_M\) will be called saturated left ideals of the semigroup \(A(G)\).

The complete lattice \(\Sigma\) of all left ideals of the semigroup \(A(G)\) is isomorphic to the complete lattice \(Q\).

There exists an isomorphism between the lattices \(\Sigma\) and \(Q\) such that, if under this isomorphism the collection \(q \in Q\) corresponds to the left ideal \(\mathfrak{A} \in \Sigma\), then

\[ \mathfrak{A}=\bigcup_{F \in q} \mathfrak{A}_F . \]

1. Denote by \(U\) the set of all unions of certain subalgebras of the algebra \(G\). It is easy to see that \(U\) is a complete lattice with respect to the set-theoretic operations of intersection and union.

The set \(\Pi\) of all saturated left ideals of the semigroup \(A(G)\) is a complete lattice isomorphic to the complete lattice \(U\).

The lattice \(\Pi\) is a sublattice of the lattice \(\Sigma\) if and only if the algebra \(G\) is cyclic (i.e., an algebra with one generator); moreover, in this case \(\Pi\) and \(\Sigma\) coincide.

1. In what follows, by \(M\) we shall denote only unions of certain subalgebras of the algebra \(G\), i.e. some element of the lattice \(U\). For each \(M\), denote by \(\mathfrak{A}_M\) the set of all endomor-

morphisms from \(A(G)\) mapping \(G\) into cyclic subalgebras of \(M\). We shall say that the left ideal \(\mathfrak A \in \Sigma\) maps \(G\) onto \(M\), and shall write \(M_{\mathfrak A}=M\), if

\[ M=\bigcup_{\alpha\in\mathfrak A} G\alpha . \]

\(\overline{\mathfrak A}_{M}\) is a left ideal of the semigroup \(A(G)\), and the least among all those left ideals which map \(G\) onto \(M\).

The set \(\pi\) of all left ideals of the form \(\overline{\mathfrak A}_{M}\) is a complete lattice, isomorphic to the complete lattice \(U\), and consequently also to the complete lattice of all saturated left ideals \(\Pi\).

The lattice \(\pi\) is a sublattice of the lattice \(\Sigma\) if and only if, for every \(n\)-subalgebra contained in some cyclic subalgebra, there exists a unique least cyclic subalgebra containing the given \(n\)-subalgebra.

For each \(M\in U\), denote by \(\Sigma_M\) the set of all left ideals \(\mathfrak A\in\Sigma\) which map \(G\) onto \(M\), i.e., for which \(M_{\mathfrak A}=M\). Let \(\sigma\) be the partition of the lattice \(\Sigma\) into the classes \(\Sigma_M\).

The partition \(\sigma\) is a congruence of the complete lattice \(\Sigma\).

The quotient lattice \(\Sigma/\sigma\) is isomorphic to the complete lattice \(U\).

1. Each class \(\Sigma_M\) of the congruence \(\sigma\) is a complete lattice whose zero and unit are, respectively, the left ideals \(\overline{\mathfrak A}_M\) and \(\mathfrak A_M\).

Denote by \(Q_M\) the set of all families of \(n\)-subalgebras possessing property a) of item 1 and the property:

b) the union of all \(n\)-subalgebras of any family in \(Q_M\) is equal to \(M\).

The set \(Q_M\) is a complete lattice with respect to the set-theoretic operations of union and intersection.

It follows from item 1 that the complete lattices \(\Sigma_M\) and \(Q_M\) are isomorphic.

1. A family of subsets \(N_i\subseteq G\) is called fully characteristic if, for every subset \(N_i\) of this family and every endomorphism \(\alpha\in A(G)\), there exists a subset \(N_{i'}\) of the same family such that \(N_i\alpha\subseteq N_{i'}\). If \(N_i=N\) for all \(i\), then the set \(N\) itself is fully characteristic.

Denote by \(\mathfrak N\) the set of all fully characteristic families of \(n\)-subalgebras \(q\in Q\). The set \(\mathfrak N\) is a complete lattice with respect to the set-theoretic operations of intersection and union.

The complete lattice \(\Delta\) of all two-sided ideals of the semigroup \(A(G)\) is isomorphic to the complete lattice \(\mathfrak N\).

The left ideals \(\mathfrak A_M\) and \(\overline{\mathfrak A}_M\) are two-sided ideals of the semigroup \(A(G)\) if and only if the subset \(M\in U\) is fully characteristic.

It follows from this that the two-sided ideals of the semigroup \(A(G)\) are distributed among the classes \(\Sigma_M\) for fully characteristic \(M\), and each such class contains two-sided ideals; namely, in every case the ideals \(\overline{\mathfrak A}_M\) and \(\mathfrak A_M\) are such.

1. A partial endomorphism of the algebra \(G\) is any homomorphism \(\alpha\) of some subalgebra \(\Gamma_\alpha^1\) of the algebra \(G\) onto a subalgebra \(\Gamma_\alpha^2\) of \(G\). Multiplication of partial endomorphisms is defined in the same way as multiplication of partial transformations of some set \((^3)\). Denote by \(W(G)\) the semigroup of all partial endomorphisms of the algebra \(G\).

As in item 1, denote by \(\mathfrak A_M^W\), \(M\in U\), the set of all partial endomorphisms \(\alpha\in W(G)\) for which \(\Gamma_\alpha^2\subseteq M\). \(\mathfrak A_M^W\) is a left ideal of the semigroup \(W(G)\). The left ideals \(\mathfrak A_M^W\) are called saturated left ideals of the semigroup \(W(G)\).

The set \(\Pi^W\) of all saturated left ideals of the semigroup \(W(G)\) is a complete lattice, isomorphic to the complete lattice of all unions of subalgebras of the algebra \(G\).

1. We shall say that a left ideal \(\mathfrak A\) of the semigroup \(W(G)\) partially maps \(G\) onto \(M\), and shall write \(M_{\mathfrak A}=M\), if
   \[ M=\bigcup_{\alpha\in\mathfrak A}\Gamma_\alpha^2 . \]
   Let \(\Sigma^W\) be the structure of all left ideals of the semigroup \(W(G)\). Denote by \(\Sigma_M^W\) the set of all left ideals \(\mathfrak A\in\Sigma^W\) for which \(M_{\mathfrak A}=M\). If \(\sigma\) is the partition of the structure \(\Sigma^W\) into the classes \(\Sigma_M^W\), then:

The partition \(\sigma\) is a congruence of the complete structure \(\Sigma^W\). The factor-structure \(\Sigma^W/\sigma\) is isomorphic to the complete structure \(U\) of all unions of subalgebras of the algebra \(G\).

It is obvious that each class \(\Sigma_M^W\) contains a unique maximal left ideal, namely \(\mathfrak A_M^W\). Each such class also contains a unique minimal left ideal \(\overline{\mathfrak A}_M^W\). The set \(\pi^W\) of all left ideals \(\overline{\mathfrak A}_M^W\) is also a complete structure, isomorphic to the complete structure \(U\). Always
\[ \overline{\mathfrak A}_M^W \supset \mathfrak A_M^W, \]
and, generally speaking, the inclusion is proper.

1. In what follows only those free algebras \(G\) are considered for which the theorem on the freeness of subalgebras holds, i.e. every subalgebra of the free algebra \(G\) is itself a free algebra in the same primitive class \(\Lambda\). Denote by \(Q'\) the set of all collections of subalgebras of the algebra \(G\) possessing the property:

a′) together with some subalgebra \(F\), the given collection contains all subalgebras contained in \(F\).

The set \(Q'\) is a complete structure with respect to the set-theoretic operations of union and intersection.

The complete structures \(\Sigma^W\) and \(Q'\) are isomorphic.

Let us note that if the set \(I\) is infinite and its cardinality \(n\) is not less than the cardinality of the set of fundamental operations of the primitive class \(\Lambda\), which, as usual, are assumed to be finitary, then every subalgebra of the algebra \(G\) is an \(n\)-subalgebra. Consequently, in this case the sets \(Q\) and \(Q'\) coincide, whence we obtain that the complete structures of all left ideals of the semigroup \(A(G)\) and of all left ideals of the semigroup \(W(G)\) are isomorphic. In particular, these structures are isomorphic if \(G\) is a free group with a countable number of generators.

1. We shall call a cardinal number \(m\) admissible if: 1) for every \(m'<m\) one can indicate such an \(m''\), \(m'\leq m''<m\), that the algebra \(G\) contains a subalgebra which has a system of generators of cardinality \(m''\), but has no system of generators of smaller cardinality; 2) if a subalgebra \(H\subseteq G\) has a system of generators of cardinality less than \(m\), then every subalgebra of the algebra \(H\) also has a system of generators whose cardinality is less than \(m\).

For example, if \(G\) is a noncyclic free group of finite or countable rank, then only the numbers \(1,2\), and \(\aleph_1\) are admissible. It follows from the definition that an admissible cardinal number \(m\) does not exceed, in any case, \(\aleph_{\nu+1}\), where \(\aleph_\nu\) is the cardinality of the algebra \(G\).

Denote by \(\mathfrak A_m\) the set of all partial endomorphisms \(\alpha\in W(G)\) for which \(\Gamma_\alpha^2\) has a system of generators whose cardinality is strictly less than the admissible cardinal number \(m\). Then:

\(\mathfrak A_m\) is a two-sided ideal of the semigroup \(W(G)\). If \(m_1\) and \(m_2\) are two admissible cardinal numbers and \(m_1<m_2\), then \(\mathfrak A_{m_1}\subset \mathfrak A_{m_2}\). Every two-sided ideal \(\mathfrak A\) of the semigroup \(W(G)\) coincides with one of the ideals \(\mathfrak A_m\).

I take this opportunity to express my deep gratitude to Prof. A. G. Kurosh for his constant attention to the work and valuable advice.

CITED LITERATURE

\(^{1}\) A. G. Kurosh, Lectures on General Algebra, Moscow, 1962.
\(^{2}\) A. I. Maltsev, Matem. sborn., 31 (73), 136 (1952).
\(^{3}\) E. S. Lyapin, Semigroups, Moscow, 1960.