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Reports of the Academy of Sciences of the USSR
- Volume 142, No. 4
MATHEMATICS
B. M. SHAIN
REPRESENTATION OF SEMIGROUPS BY MEANS OF BINARY RELATIONS
(Presented by Academician A. I. Mal'cev on 2 X 1961)
In the present note the construction of an arbitrary representation of a given abstract semigroup by means of binary relations is described. The problem of finding all representations of a given semigroup by binary relations of a special kind (transformations, partial transformations) was solved by V. V. Wagner (¹) and, in another way, by E. S. Lyapin (²).
We use the notation of mathematical logic: \(\wedge\) is the symbol of conjunction, \(\to\) the symbol of implication, \(\leftrightarrow\) the symbol of logical equivalence, \(\bigwedge\) the universal quantifier, \(\bigvee\) the existential quantifier.
A binary relation between elements of a set \(A\) is any subset \(\rho\) of the set \(A \times A\). For binary relations between elements of the set \(A\) all set-theoretic operations are defined.
To binary relations \(\rho_1 \subset A \times A\) and \(\rho_2 \subset A \times A\) there corresponds their product—the binary relation \(\rho_2 \circ \rho_1 \subset A \times A\), defined by the formula
\[ (a_1,a_2)\in \rho_2\circ\rho_1 \leftrightarrow \bigvee(a)\,[(a_1,a)\in\rho_1 \wedge (a,a_2)\in\rho_2]. \tag{1} \]
The set \(\mathfrak P(A \times A)\) of all binary relations between elements of the set \(A\) is a semigroup with respect to the operation of multiplication of binary relations. The zero of this semigroup is the empty binary relation \(\varnothing\), and the identity is the identity binary relation \(\Delta_A\).
A section of a binary relation \(\rho \subset A \times A\) at an element \(a_0 \in A\) is the subset \(\rho\langle a_0\rangle\) of the set \(A\) defined by the formula
\[ a \in \rho\langle a_0\rangle \leftrightarrow (a_0,a)\in\rho. \tag{2} \]
In the case when each section of a binary relation \(\rho\) by elements of the set \(A\) contains no more than one element, the binary relation \(\rho\) is called single-valued, or a transformation of the set \(A\). If in this case for all \(a\in A\), \(\rho\langle a\rangle\ne\varnothing\), then \(\rho\) is called a complete transformation of the set \(A\)*. In the case when \(\rho\) is a transformation of the set \(A\) and \((a_1,a_2)\in\rho\), one writes: \(\rho(a_1)=a_2\).
To every binary relation \(\rho\subset A\times A\) one can put in correspondence a single-valued binary relation \(\breve\rho\subset \mathfrak P(A)\times\mathfrak P(A)\), called the extension of the binary relation \(\rho\) to the set of subsets** and defined by the formula
\[ \breve\rho(a)=\bigcup_{a\in a}\rho\langle a\rangle \quad \text{for } a\subset A. \tag{3} \]
To every binary relation \(\rho\subset A\times A\) we put in correspondence a binary relation \(\hat\rho\subset \mathfrak P(A)\times\mathfrak P(A)\), called the majorant extension of the binary relation \(\rho\) to the set of subsets—
* Often transformations are called partial transformations, and complete transformations are called (simply) transformations.
** \(\mathfrak P(A)\) is the set of subsets of the set \(A\).
the set of subsets and defined by the formula
\[ (a_1,a_2)\in \hat{\rho}\longleftrightarrow \bar{\rho}(a_1)\subset a_2 . \tag{4} \]
The majorant extension of binary relations defines a mapping of the semigroup \(\mathfrak{P}(A\times A)\) into the semigroup \(\mathfrak{P}(\mathfrak{P}(A)\times\mathfrak{P}(A))\). It turns out that this mapping is an isomorphism, i.e.
\[ \hat{\rho}_2\circ \hat{\rho}_1=\widehat{\rho_2\circ\rho_1}, \qquad \hat{\rho}_1=\hat{\rho}_2\to \rho_1=\rho_2 . \tag{5} \]
The restriction of a binary relation \(\rho\subset A\times A\) to a subset \(a\subset A\) is a binary relation \(\rho_a\subset a\times a\) such that \(\rho_a=\rho\cap a\times a\).
A representation of a semigroup \(G\) by means of binary relations between elements of a set \(A\) is a homomorphism of the semigroup \(G\) into the semigroup \(\mathfrak{P}(A\times A)\). In the case of an isomorphism, the representation is called faithful.
Thus, a mapping \(\mathrm P\) of the semigroup \(G\) into \(\mathfrak{P}(A\times A)\) will be a representation if and only if\(^*\)
\[ \mathrm P(g_2)\circ \mathrm P(g_1)=\mathrm P(g_1g_2). \tag{6} \]
If the images of the elements of the semigroup \(G\) under the representation \(\mathrm P\) are (full) transformations, then \(\mathrm P\) is called a representation by means of (full) transformations.
The kernel of a representation \(\mathrm P\) is the stable equivalence relation \(\varepsilon_{\mathrm P}\) between elements of the semigroup \(G\), defined as follows:
\[ g_1\equiv g_2(\varepsilon_{\mathrm P})\longleftrightarrow \mathrm P(g_1)=\mathrm P(g_2). \tag{7} \]
The image of the semigroup \(G\) under the representation \(\mathrm P\) is isomorphic to the quotient semigroup \(G/\varepsilon_{\mathrm P}\).
Adjoining to the semigroup \(G\) an element \(e\) not belonging to the set \(G\), and defining \(ee=e,\ eg=ge=g\) for all \(g\in G\), we obtain a semigroup \(G^*\) with identity \(e\).
The representation \(\Lambda\) of the semigroup \(G\) by means of full transformations of the set \(G^*\), defined by the formulas
\[ (g_1,g_2)\in \Lambda(g)\longleftrightarrow g_2=g_1g,\qquad (e,g)\in \Lambda(g), \tag{8} \]
is called the canonical representation of the semigroup \(G\) by means of right shifts. As is known, the representation \(\Lambda\) is faithful.
Let \(\theta\) be a one-to-one mapping of the set \(A\) onto the set \(B\). To every binary relation \(\rho\subset A\times A\) we associate the binary relation \(\dot{\rho}\subset B\times B\), defined by the formula \((a_1,a_2)\in\rho\longleftrightarrow (\theta(a_1),\theta(a_2))\in\dot{\rho}\). If \(\mathrm P\) is a representation of the semigroup \(G\) by means of binary relations between elements of the set \(A\), then, defining \(\dot{\mathrm P}(g)=\dot{\overline{\mathrm P(g)}}\), we obtain a representation \(\dot{\mathrm P}\) of the semigroup \(G\) by means of binary relations between elements of the set \(B\). The representations \(\mathrm P\) and \(\dot{\mathrm P}\) are called similar. It is clear that similar representations differ from one another inessentially, since we are not interested in the nature of the elements of the set \(A\) between whose elements the binary relations are taken.
Let \(\{\Lambda_i\}_{i\in I}\) be a family of representations of the semigroup \(G\), each of which is similar to the canonical representation \(\Lambda\), and let \(\Lambda_i\) be a representation by means of full transformations of the set \(A_i\) and of the set
\(^*\) Sometimes formula (6) is replaced by the condition \(\mathrm P(g_1)\circ \mathrm P(g_2)=\mathrm P(g_1g_2)\); however, we consider definition (6) more acceptable, since the factors in the product of binary relations are written from right to left.
from the family \(\{A_i\}_{i\in I}\) are pairwise disjoint. Then \(\Lambda_I\), defined by the formula
\[ \Lambda_I(g)=\bigcup_{i\in I}\Lambda_i(g), \tag{9} \]
will be a representation of the semigroup \(G\) by complete transformations of the set \(\bigcup_{i\in I} A_i\). This representation is called the canonical \(w\)-fold representation of the semigroup \(G\) (here \(w\) is the cardinal number of the set \(I\)). The semigroup \(G\) has various canonical \(w\)-fold representations, but all these representations are similar.
Let \(P\) be a representation of the semigroup \(G\) by binary relations between the elements of a set \(A\). Define \(\hat P(g)=\widehat{P(g)}\). Then, by formulas (5), \(\hat P\) will be a representation of the semigroup \(G\) by binary relations between the elements of the set \(\mathfrak P(A)\), and \(\varepsilon_{\hat P}=\varepsilon_P\). The representation \(\hat P\) will be called the derivative of the representation \(P\).
Let \(\mathfrak a\) be a subset of the set \(A\). Construct a mapping \(P_{\mathfrak a}\) of the semigroup \(G\) into the semigroup \(\mathfrak P(\mathfrak a\times \mathfrak a)\), defining \(P_{\mathfrak a}(g)=(P(g))_{\mathfrak a}\). If \(P_{\mathfrak a}\) proves to be a representation of the semigroup \(G\) by binary relations between the elements of the set \(\mathfrak a\), then \(P_{\mathfrak a}\) is called a restriction of the representation \(P\) to the set \(\mathfrak a\).
Theorem 1. Every representation of a semigroup by binary relations is similar to a restriction of the derivative of a canonical \(w\)-fold representation of the given semigroup, for some \(w\).
We shall give the plan of the proof of this theorem, without going into technical details.
Denote by \(H_{a_1}^{a}\), where \(a,a_1\in A\), the subset of the set \(G^{*}\) defined by the formulas
\[ g\in H_{a_1}^{a}\leftrightarrow (a,a_1)\in P(g), \qquad e\in H_{a_1}^{a}\leftrightarrow a=a_1 . \tag{10} \]
Let \(\theta_a\) be a one-to-one mapping of the set \(G^{*}\) onto some set \(B_a\), and let the sets of the family \(\{B_a\}_{a\in A}\) be pairwise disjoint. To each \(\theta_a\) there corresponds a similarity of a representation \(\Lambda\) and of some representation \(\Lambda_a\). To the family of representations \(\{\Lambda_a\}_{a\in A}\) there corresponds a canonical \(w\)-fold representation \(\Lambda_A\), where \(w\) is the cardinal number of the set \(A\). Denote by \(B^{a_1}\) the subset of the set \(B=\bigcup_{a\in A}B_a\), defined as follows: \(B^{a_1}=\bigcup_{a\in A}\theta_a(H_{a_1}^{a})\). Let \(\mathfrak A\) be the set of all \(B^a\) (for all possible \(a\in A\)). It turns out that
\[ (B^{a_1}, B^{a_2})\in \hat\Lambda_{A\mathfrak A}(g)\leftrightarrow (a_1,a_2)\in P(g) \tag{11} \]
and from \(B^{a_1}=B^{a_2}\) it follows that \(a_1=a_2\). Therefore the representations \(P\) and \(\hat\Lambda_{A\mathfrak A}\) are similar.
The least of the cardinal numbers \(w\) such that \(P\) is similar to a restriction of the derivative of a canonical \(w\)-fold representation is called the weight of the representation \(P\).
A representation \(P\) of a semigroup \(G\) by binary relations is called a universal representation of weight \(w\) if the weight of \(P\) is equal to \(w\) and every representation of the semigroup \(G\) of weight not exceeding \(w\) is similar to a restriction of the representation \(P\). It follows from Theorem 1 that the representation \(\hat\Lambda_I\), where the cardinal number of the set \(I\) is equal to \(w\), will be a universal representation of the semigroup of weight \(w\). It would be interesting to determine which semigroups can possess several non-similar universal representations of one and the same weight.
Let \(\mathfrak H=\{H_a^i\}_{a\in A}^{i\in I}\) be a family of subsets of the semigroup \(G^{*}\), specified by means of two index sets \(I\) and \(A\). Of course, one and the same subset (in particular, the empty one) may occur in the family \(\mathfrak H\) several times.
each time with different indices. The family \(\mathfrak H\) is called right admissible for the semigroup \(G\) if, for any \(g_1, g_2 \in G\), \(a_1, a_2 \in A\),
\[ (\bigwedge i)[H^i_{a_1}\{g_1g_2\}\subset H^i_{a_2}] \Longrightarrow (\bigvee a)(\bigwedge i)[H^i_{a_1}\{g_1\}\subset H^i_a \wedge H^i_a\{g_2\}\subset H^i_{a_2}]. \tag{12} \]
Define a mapping \(\mathrm P_{\mathfrak H}\) of the semigroup \(G\) into the semigroup \(\mathfrak P(A\times A)\):
\[ (a_1,a_2)\in \mathrm P_{\mathfrak H}(g) \leftrightarrow (\bigwedge i)[H^i_{a_1}\{g\}\subset H^i_{a_2}]. \tag{13} \]
Theorem 2. The mapping \(\mathrm P_{\mathfrak H}\) will be a representation of the semigroup \(G\) by means of binary relations if and only if the family \(\mathfrak H\) is right admissible for the semigroup \(G\).
For the proof it suffices to write formula (6), substituting into it the expressions from (13).
Theorem 3. For every representation \(\mathrm P\) of the semigroup \(G\) by means of binary relations there exists a family \(\mathfrak H\), right admissible for the semigroup \(G\), such that \(\mathrm P=\mathrm P_{\mathfrak H}\).
Proof. Let \(\mathrm P\) be a representation of the semigroup \(G\) by means of binary relations between the elements of a set \(A\). Consider the family \(\mathfrak H=\{H^{a_1}_{a_2}\}_{a_2\in A}^{a_1\in A}\) of subsets of the semigroup \(G^*\), whose elements are defined by formulas (10). Let \((a_1,a_2)\in \mathrm P_{\mathfrak H}(g)\). This means that \((\bigwedge a)[H^a_{a_1}\{g\}\subset H^a_{a_2}]\). In particular, \(H^{a_1}_{a_1}\{g\}\subset H^{a_1}_{a_2}\) and, since \(e\in H^{a_1}_{a_1}\), \(g\in H^{a_1}_{a_2}\). By formulas (10), \((a_1,a_2)\in \mathrm P(g)\). Conversely, let \((a_1,a_2)\in \mathrm P(g)\). If \(g_1\in H^a_{a_1}\), then \((a,a_1)\in \mathrm P(g_1)\) and \((a,a_2)\in \mathrm P(g)\circ \mathrm P(g_1)=\mathrm P(g_1g)\), or \(g_1g\in H^a_{a_2}\), whence \(H^a_{a_1}\{g\}\subset H^a_{a_2}\), i.e. \((a_1,a_2)\in \mathrm P_{\mathfrak H}(g)\). Thus, \(\mathrm P=\mathrm P_{\mathfrak H}\). Therefore the family \(\mathfrak H\) is right admissible for the semigroup \(G\). It turns out that the weight of the representation \(\mathrm P\) is equal to the least of the cardinal numbers of the upper index sets of families \(\mathfrak H\) such that \(\mathrm P=\mathrm P_{\mathfrak H}\).
It would be interesting to find all right admissible families for a given semigroup.
Saratov State University
named after N. G. Chernyshevsky
Received
27 IX 1961
CITED LITERATURE
\(^{1}\) V. V. Wagner, Matem. sborn., 38(80), No. 2, 203 (1956).
\(^{2}\) E. S. Lyapin, Matem. sborn., 52(94), No. 1, 589 (1960).