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On certain systems which are almost groups. (English) Zbl 0063.03769

From the text: We consider systems \(\mathfrak S\) of elements \(A,B,\ldots\) satisfying the following postulates. I. To every pair of elements \(A,B\) in \(\mathfrak S\) there exists a uniquely determined product \(A\cdot B\) in \(\mathfrak S\). II. \((AB)C=A(BC)\). III. There exists in \(\mathfrak S\) an element \(E\) such that \(EA=A\) for every \(A\). IV. To every \(A\) in \(\mathfrak S\) there exists an \(A'\) in \(\mathfrak S\) such that \(AA'=E\).
The system \(\mathfrak S\) differs from a group only in that it contains a left unit and right inverse instead of a right unit and right inverse. We shall call such a system a left right system, abbreviated \((l, r)\) system. It is proved:
Theorem 1. There exists one and only one idempotent \((l, r)\) system of order \(n\) \((n = 1,2, \ldots)\). It consists of the elements \(A_1, A_2, \ldots, A_n\) with the law of composition \(A_iA_j=A_j\).
Theorem 2. Every \((l, r)\) system is the direct product of an idempotent \((l, r)\) system and a group.

MSC:

20N99 Other generalizations of groups
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