L. M. Gluskin

Semiheaps with Minimal Left Ideals

(Presented by Academician A. I. Mal’cev, 5 I 1963)

A semiheap \((^1)\) is a set \(S\) on which a ternary algebraic operation is defined, assigning to each triple of elements \(s_1, s_2, s_3 \in S\) an element \([s_1, s_2, s_3] \in S\) and satisfying the following associativity-type condition*:

\[ \forall_{s_i\in S}\ [[s_1s_2s_3]\,s_4s_5]=[s_1\,[s_4s_3s_2]\,s_5]=[s_1s_2\,[s_3s_4s_5]]. \]

The present note is devoted to one of the far-reaching analogies between the theory of semigroups and the theory of semiheaps—the transfer to semiheaps of the theory of completely simple semigroups of Sushkevich—Rees—Clifford \((^2,^3,^11)\).

1. A nonempty subset \(A\) of a semiheap \(S\) is called its left (respectively, right, lateral) ideal if

\[ [SSA]\subseteq A \qquad (\text{respectively } [ASS]\subseteq A,\quad [SAS]\subseteq A). \]

If a subset \(A\subseteq S\) is simultaneously a left, right, and lateral ideal, then it is called simply an ideal of the semiheap \(S\). If a semiheap \(S\) contains an element \(0\) such that \([SS0]=[S0S]=[0SS]=0\), then \(0\) is called the zero of the semiheap \(S\). A left (right) ideal \(A\) of the semiheap \(S\) is called minimal if it is distinct from \(0\) and contains no proper (i.e., distinct from \(A\) and \(0\)) left (right) ideals of the semiheap \(S\).

1. Let \(A\) be a group with an adjoined zero \(o\), i.e. \(A=G\cup\{o\}\), where \(G\) is a group, \(o\) is the zero of the semiheap \(A\), and \(\psi\) is an inverse automorphism of the semiheap \(A\) such that, for some \(s\in G\),

\[ \forall_{x\in A}\ \psi\psi x=sxs^{-1}. \]

If \(s=e\), then \(\psi\) is called an involution \((^1)\) of the semiheap \(A\). In particular, \(A\) admits the canonical involution \(\xi\): \(\xi o=o,\ \xi a=a^{-1}\) for every \(a\in G\).

Further, let \(I,\Lambda\) be two arbitrary sets, and let \(P=(p_{\lambda\chi})\) and \(Q=(q_{ij})\) be a \(\Lambda I\)-matrix \((^3,^5)\) (respectively, an \(I\Lambda\)-matrix) over the semiheap \(A\) such that in each of their rows there exists at least one element distinct from zero and

\[ \forall_{\chi,\lambda\in\Lambda} p_{\lambda\chi}=s^{-1}\cdot \psi p_{\chi\lambda},\qquad \forall_{i,j\in I} q_{ij}=\Psi q_{ji}\cdot s. \]

Denote by \(S\) the set of all triples \((a,i\lambda)\), where \(a\in A,\ i\in I,\ \lambda\in\Lambda\), and regard all triples \((o,i\lambda)\) as equal to one another (denoting them by \(0\)). Introduce in \(S\) the following ternary algebraic operation:

\[ [(a,i\chi)(b,j\lambda)(c,k\mu)]=(ap_{\chi\lambda}\cdot \psi b\cdot q_{jk}c,\ i\mu). \tag{1} \]

With respect to this operation the set \(S\), as is not difficult to verify, is a semiheap with zero \(0\). \(S\) contains no proper ideals, and is the union of its minimal left ideals

\[ L_\lambda=\bigcup_{i\in I}(A,i\lambda) \]

and of its minimal right ideals

\[ R_i=\bigcup_{\lambda\in\Lambda}(A,i\lambda) \]

(here \((A,i\lambda)\) denotes the set of all triples \((a,i\lambda)\) for fixed \(i,\lambda\)). We shall denote such a semiheap by \(S(A,\psi,P_\Lambda,Q_I)\).

1. Suppose that under the conditions of item 2, \(I\) (respectively \(\Lambda\)) is the union of its disjoint nonempty subsets \(I_1,I_2\) (respectively \(\Lambda_1,\)

* For notation, see the article \((^4)\).

\(\Lambda_2\)); \(\psi\) is the canonical involution of the semigroup \(A\). Further, let \(p_{\chi\lambda}=o\) for \((\chi,\lambda)\in \Lambda_k\times \Lambda_k\) and \(q_{ij}=o\) for \((i,j)\in I_k\times I_k\) \((k=1,2)\). Denote

\[ T_k=\bigcup_{i\in I_k,\ \lambda\in \Lambda_k}(A,i\lambda)\quad (k=1,2). \]

The set \(T=T_1\cup T_2\), with respect to operation (1), is also a semigroup without proper ideals. As in Sec. 2, \(T\) coincides with the union of its minimal left ideals and with the union of its minimal right ideals. In contrast to the semigroup of Sec. 2, \(T\) is the union of its proper two-sided (i.e., left and right, but not necessarily lateral) ideals \(T_k\), and moreover \(T_k^{[3]}=[T_kT_kT_k]=0\).

1. A semigroup \(S\) is called \(i\)-simple if it contains no proper ideals and is distinct from the semigroup \(V=\{0,a\}\), where \(0\) is a zero and \(a^{[3]}=0\). If the semigroup \(S\) contains at least one minimal left ideal, then denote by \(K\) the union of all minimal left ideals of the semigroup \(S\).

Theorem 1. If a semigroup \(S\) contains a minimal left ideal and \(K^{[3]}\ne 0\), then \(S\) also contains a minimal right ideal.

Theorem 2. If an \(i\)-simple semigroup with zero \(S\) contains a minimal left or right ideal and \(K^{[3]}\ne 0\), then \(S\) is isomorphic to one of the semigroups of Secs. 2–3.

Theorem 3. If an \(i\)-simple semigroup \(S\) with zero contains a minimal left ideal and contains no nilpotent left ideals or proper two-sided ideals, then it is isomorphic to one of the semigroups \(S(A,\psi,P_\Lambda,Q_I)\).

Theorem 4. If an \(i\)-simple semigroup \(S\) with zero satisfies the minimality condition for left ideals (in particular, if \(S\) is finite), then it is isomorphic to one of the semigroups of Secs. 2–3.

Theorem 5. Every \(i\)-simple semigroup without zero, containing a minimal left or right ideal, is isomorphic to the subsemigroup of all nonzero elements of some semigroup \(S(A,\psi,P_\Lambda,Q_I)\) (in which the matrices \(P\) and \(Q\) contain no zero elements).

1. Let \(S=S(A,\psi,P_\Lambda,Q_I)\), \(S'=S(A',\psi',P'_{\Lambda'},Q'_{I'})\), let \(c\) be an arbitrary element of \(G'=A'\setminus\{0\}\), let \(\varphi\) be an isomorphism of the semigroup \(A\) onto \(A'\), let \(\chi\) and \(\xi\) be one-to-one maps of the set \(I\) onto \(I'\) and of \(\Lambda\) onto \(\Lambda'\), respectively, and let \(r_i\) and \(s_\chi\) be arbitrary elements of \(G\), with \(i'=\chi i\in I'\), \(\lambda'=\xi\lambda\in \Lambda'\), \(z\in A'\). For any such \(i,\lambda\),

\[ p'_{\chi'\lambda'}=s_\chi\cdot \varphi p_{\chi\lambda}\cdot c\cdot \psi' s_\lambda,\qquad q'_{i'j'}=\psi' r_i\cdot c^{-1}\cdot \varphi q_{ij}\cdot r_j,\qquad \psi' z=c^{-1}\cdot \varphi\psi\varphi^{-1}z\cdot c. \]

By a direct check one can verify that the map

\[ f(x,i\chi)=\bigl(r_i^{-1}\cdot \varphi x\cdot s_\chi^{-1},\, i'\chi'\bigr) \]

is an isomorphism of the semigroup \(S\) onto \(S'\). Denote by \(\Phi\) the class of all such isomorphisms.

Theorem 6. Every isomorphism of the semigroup \(S(A,\psi,P_\Lambda,Q_I)\) is contained in the class \(\Phi\).

This theorem is analogous to the known description of isomorphisms of completely simple semigroups \((3,5)\).

A similar mapping of the semigroup \(T\) of Sec. 3 is also its isomorphism if \(\chi I_k=I'_k,\ \xi\Lambda_k=\Lambda'_k\ (k=1,2)\), \(c=e\) (\(e\) is the identity of the semigroup \(A'\)); moreover, all isomorphisms of the semigroup \(T\) are exhausted by such mappings.

1. If a semigroup without proper ideals contains a minimal left (right) ideal and at least one nonnilpotent element, then it cannot be isomorphic to the semigroup \(T\) of Sec. 3 or to the semigroup \(V\) of Sec. 4.

Theorem 7. Let \(S\) be a semiheap without proper ideals, containing a minimal left ideal. If \(S\) contains at least one non-idempotent element (in particular, if \(S\) does not contain a zero), then, under an isomorphism of the semiheap \(S\) (or of the semiheap \(S\cup\{0\}\)) onto the semiheap \(S(A,\psi,P_\Lambda,Q_I)\), as the inverse automorphism \(\psi\) one can choose a certain involution of the semigroup \(A\) (see § 2).

In particular, if the semigroup \(S\) contains no left and right ideals distinct from \(S\), then its elements form a group \(G\) with respect to a certain binary operation, and the operation in \(S\) is determined as follows:

\[ \forall_{a,b,c\in S}\ [abc]=a\cdot p\cdot \psi b\cdot c, \tag{2} \]

where \(p\in G\), and \(\psi\) is an involution of the group \(G\). From this one can easily obtain the structure of an arbitrary heap \((^{6},{}^{1})\).

1. A semiheap \(S\) is called a generalized heap \((^{1})\) if

\[ \forall_{s\in S}\ [sss]=s, \]

\[ \forall_{a,b,c\in S}[abbcc]=[accbb]\wedge[bbcca]=[ccbba] * . \]

Denote by \(U_X\) the \(X X\)-matrix over the semigroup \(A\) in which \(u_{ii}=e\) and \(u_{ij}=0\) for \(i\ne j\). It is easy to verify that the semiheap \(S(A,\xi,U_\Lambda,U_I)\), where \(\xi\) is the canonical involution of the semigroup \(A\), is a generalized heap.

Theorem 8. Let \(S\) be a generalized heap with zero and without proper ideals. If \(S\) contains a minimal left or right ideal, then it is isomorphic to the semiheap \(S(A,\xi,U_\Lambda,U_I)\).

The analogous semiheap without zero is a heap.

1. Theorem 9. Let \(S\) be an \(i\)-simple semiheap containing a minimal left ideal. If \(K^{[3]}=0\), then \(S\) is the union of its subsemiheaps \(K\) and \(B\), \(K\cap B=\{0\}\), and

\[ K^{[3]}=B^{[3]}=[KKB]=[KBB]=[BKK]=[BBK]=\{0\}, \]

\[ \forall_{k\in K}\ [BkB]=B,\qquad \forall_{b\in B}\ [KbK]=K, \]

\[ K\setminus\{0\}=\bigcup_{i\in I,\ \lambda\in\Lambda} A_{i\lambda}, \]

where \(I,\Lambda\) are certain sets, all \(A_{i\lambda}\) are nonempty, and for any \(i,j\in I\), \(\lambda,\mu\in\Lambda\), \(a_{i\lambda}\in A_{i\lambda}\), \(c_{j\mu}\in A_{j\mu}\), \(b\in B\),

\[ [a_{i\lambda}bc_{j\mu}]\ne 0\to \forall_{k\in I}\ [A_{k\lambda}bc_{j\mu}]=A_{k\mu}. \]

1. For the semiheaps of §§ 2–3 there exists an isomorphic representation by means of matrices over groups, analogous to the representation of completely simple semigroups \((^{3},{}^{5})\). To each element \(a=(a,i\lambda)\) from the semiheap \(S\) of § 2 or § 3, put in correspondence a \(\Lambda I\)-matrix \(\bar a=(x_{i\lambda})\) such that \(x_{i\lambda}=a\), \(x_{j\varkappa}=0\) for \(j\ne i\) or \(\varkappa\ne\lambda\). Denote by \(\bar a^{*}\) the \(\Lambda I\)-matrix obtained from \(\bar a\) by transposition and by replacing all elements \(x_{j\varkappa}\) by \(\psi x_{j\varkappa}\). If on the set \(\bar S\) of all such matrices \(\bar a\) we introduce the operation \([\bar a\bar b\bar c]=\bar a P\bar b^{*}Q\bar c\), then the mapping \(a\to\bar a\) is an isomorphism from \(S\) onto \(\bar S\).

2. Let \(S=S(A,\psi,P_\Lambda,Q_I)\); let \(N\) be such a normal divisor of the group \(G=A\setminus\{0\}\) that \(\psi N=N\); \(\bar A=(G/N)\cup\{0\}\); further, let \(\bar\psi(aN)=\psi a\cdot N\) for any \(a\in A\), and \(\bar P_\Lambda=(p_{\varkappa\lambda}N)\), \(\bar Q_I=(q_{ij}N)\). Partition the set \(\Lambda\) into pairwise disjoint subsets \(\bar\lambda\) so that \(p_{\varkappa\lambda}=p_{\varkappa\nu}\) for any \(\lambda,\nu\in\bar\lambda\), and denote \(\bar p_{\varkappa\bar\lambda}=p_{\varkappa\lambda}\), where \(\varkappa\in\bar\varkappa,\lambda\in\bar\lambda\). Let \(\bar\Lambda\) be the class of all

\[ \text{* } \wedge \text{ is conjunction: } \alpha\wedge\beta \text{ means “}\alpha\text{ and }\beta\text{”.} \]

of such subsets \(\bar{\lambda}\), \(P_{\bar{\Lambda}}=(p_{\bar{\chi}\lambda})\). We define the set \(\bar I\) and the matrix \(Q_{\bar I}\) analogously. The mappings \(\varphi_N(a,i\chi)=(aN,i\chi)\), \(\varphi_l(a,i\chi)=(a,i\bar\chi)\) \((i\in \bar i)\), \(\varphi_r(a,i\chi)=(a,\bar i\chi)\) \((\chi\in\bar\chi)\) are homomorphisms of the semigroup \(S\) onto the semigroup \(S(\bar A,\psi,\bar P_{\Lambda},\bar Q_I)\), \(S(A,\psi,P_{\Lambda},Q_{\bar I})\), or, respectively, \(S(A,\psi,P_{\bar\Lambda},Q_I)\). The analogous mappings of the semigroup of item 3 are also its homomorphisms.

Theorem 10. Every homomorphism of the semigroup of item 2 or item 3 can be represented as a superposition of no more than three homomorphisms of the form \(\varphi_N,\varphi_l,\varphi_r\) (cf. with \((^7,^8)\)).

1. It is not difficult to verify that, by virtue of Theorem 10, the following semigroups have no nontrivial homomorphisms: a) the semigroup \(V\) of item 4; b) any semigroup \(S\) of item 6 with action (2), for which the corresponding group \(G\) is simple; c) every semigroup of the type described in items 2, 3, for which the group \(G=A\setminus\{o\}\) is a group with identity, and the matrices \(P\) and \(Q\) do not contain two identical rows. However, in contrast to \((^9,^10)\), items a)—c) give an exhaustive description of semigroups with minimal left ideals and without nontrivial homomorphisms only in the case of semigroups without zero: it is unclear whether, among such semigroups, there can exist a semigroup \(S\) of item 8.

Kommunarsk
Mining and Metallurgical Institute

Received
1 XII 1962

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