By Marina Cohen

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Let : M ! a ı b/. a,b2S (a) If has the property (Ri), i D 1, 2, 3 deﬁned for roundings, then fS, property (RGi), i D 1, 2, 3, respectively. ı g has the (b) If the groupoid fM , ıg has a right neutral element e and e 2 S, then (RG1), (RG2), and (RG3) imply (RG). Proof. (a) We omit the proof of this property since it is straightforward. (b) We give the proof in the case of a lower screen. a ı b/. a ı b/ Ä a ı b. , e is also a right neutral element in fS, ı g. a ı b/ Ä a ı b. 2) 38 Chapter 1 First concepts (RG3) yields a (R1) and (R2) ı b Ä a ı b.

X is a lower bound of B. Therefore we have x Ä infM B D x. But by (R3), x Ä x. From both inequalities we get by (O3) x D x. Since : M ! S, then x D infM B 2 S. 15, fS, Äg is a complete inf-subnet of fM , Äg, and for all B Â S, infM B D infS B. (b) Now we show that (S1) and (S2) hold in any complete inf-subnet S Â M . S/ D inf ¿. Therefore (S1) holds. Further, for all A Â S we have by hypothesis infS A D infM A. a/ \ S/ 2 S. a/ \ S/. a/ \ S since it is an element of this set itself. a/\S, which completes the proof of the theorem.

We shall see later that the structures of ordered or weakly ordered or isotonely ordered ringoids or vectoids represent the properties that are invariant with respect to semimorphism. The ringoid is a generalization of the mathematical structure of a ring and the vectoid of that of a vector space. That semimorphisms preserve reasonable mathematical structures is another strong reason for using semimorphisms to deﬁne all arithmetic operations in all subsets of Figure 1. For the power set and the interval spaces in Figure 1 the order relation in (R2) is the subset relation Â.