All properties in Deﬁnition 9 are transitive.

Theorem 63. Properties M↑, M↓, R↑, R↓, C↑ and C↓ are transitive.
Proof. We  show  only  the  upwards  cases.  The  downwards  cases  are  analogous.  Let  G1 = (cid:3)S1, E1(cid:4),  G2 = (cid:3)S2, E2(cid:4) and 
G3 = (cid:3)S3, E3(cid:4) be  arbitrary  STGs.  Let  τ1 = (cid:3) f1, R1(cid:4) from  G1 to  G2 and  τ2 = (cid:3) f2, R2(cid:4) from  G2 to  G3 be  two  arbitrary 
transformations such that τ1◦τ2 is a transformation, i.e.  f1◦ f2 is a transformation function. Deﬁne  f3 = f1◦ f2, R3 = R1◦R2
and τ3 = τ1◦τ2 = (cid:3) f3, R3(cid:4).
t∈ f1(s) f2(t). However, 
f1 is M↑ so there is some t ∈ S2 such that  f1(s) = t. Hence,  f3(s) = f2(t) and it follows that | f3(s)| = 1 since  f2 is M↑. 
Hence,  f3 is M↑ since s was chosen arbitrarily.

M↑: Suppose both τ1 and τ2 are M↑. Choose s ∈ S1 arbitrarily. By deﬁnition,  f3(s) = f2( f1(s)) = (cid:2)
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