By Rédei, L.; Sneddon, I. N.; Stark, M

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10) THEOREM 26. If an element a in a semigroup (with unity element) has an inverse, then there is no other left inverse. If VL has no inverse, then either the set of left inverses or the set of right inverses ofoL is empty. This theorem is obviously equivalent to the next proposition: If a' is a left inverse and a" a right inverse of a, then a' = a". The latter assertion follows from a' = a'(aa") = (a'a)a" = a" , which proves Theorem 26. It is easy to see the formal similarity of Theorems 24 and 26, which also appears in the proofs.

G. 2 + 3 means the number 5. In school mathematics one is accustomed to speak of "subtraction" (denoted by a — P) and "division" denoted by a - r ^ o r — which are also considered to be compositions or more exactly "inverse compositions". We do not regard these as compositions, but we shall discuss them later. 28 COMPOSITIONS 29 If in the set S a composition has been defined, special notations are used for certain elements (connected with this composition). The most im portant of these are "neutral element" and "inverse element".

Chapter IX) semimodules, semirings and semifields play a very im portant role. If S is a structure with two compositions, then we denote by S+ and S x the additive and multiplicative structures, respectively, constituted by the elements of S where, of course, the addition or the multiplication defined in S is understood. Generally we use those concepts and terms relative to S which have a meaning relative to S+ or S x , respectively. , the element 0 of S+ means, as above, the element 0 of S. Similarly, the unity element of Sx (if it exists) is called the unity element of S.