On certain systems which are almost groups

We consider systems ©of elements A,By • • • satisfying the following postulates. I. To every pair of elements A, B in © there exists a uniquely determined product A B in ©. II. (AB)C=A(BC). III. There exists in © an element E such that E A —A for every A. IV. To every A in © there exists an A' in © such that AA'~E. The system © 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 (/, r) system. DEFINITION 1. An element F is called an idempotent if F~F. PROPOSITION 1. If Fis idempotent then FE — E. PROOF. From FF=*F we have FFF^^FF-, hence FE**E. PROPOSITION 2. If FE = E then F is idempotent For FF=FEF = EF=F. PROPOSITION 3. The idempotents of a (/, r) system form a (/, r) system. PROOF. If F and F' are idempotent then by Proposition 1 FFE « FE JS,