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By E. S. Ljapin

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As for the problem (which naturally arises) of describing and classifying all representations of seroigroups, it is necessary here to set up certain preliminary refinements and limitations. Work on a solution of this important problem has only just begun. Let us mention the articles of Stoll [1] and V. V. Vagner [6]. , the class of semigroups of transformations. As we have just shown, every semigroup can be represented in the class E. On the other hand, every multiplicative set representable in E is a semigroup (since every multiplicative set isomorphic to a semigroup is itself a seroigroup).

X, Y E9)() ms E <1>' and elements Zl' Z2' ... , Zi+1(m i ), X = Zl' Y = Zs+1 (i = 1,2, ... , s). It is easy to see that the relation I is an equivalence. , Yen), then X Y(I) by the definition of 1. From this it follows that I is an upper bound for the set of equivalences of '. Assume also for Zl' Z2' ... I Zi""'" Zi+1(mi ) Since m >m i, it follows that (i = 1,2, ... ,s). Zi """,Zi+l(m) Sec. 5] RELATIONS 39 and therefore Zl "" Zs+l(m).

In the study of relations it is extremely important to determine whether they possess certain properties. Let n be a relation in the set m. The relation n is called REFLEXIVE iffor every X E m we have X,,-, X(n). , Zen). I X(n). , Yen) and y,,-, X(n) always follows X = Y. DEFINITION. 4 P. -L. Dubreil-Jacotin, Thiorie algebrique des relations d'equivalence, J. Math. Pures Appl. 18 (1939), 63-95. 36 THE CONCEPT OF A SEMI GROUP [CHAP. 7. Among all types of relations the following two classes are most important to us.

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