is a homomorphism, since e1 was chosen arbitrarily.

(3) Immediate from (1) and Deﬁnition 12.

For the opposite direction we make the additional assumption of unit-labelled STGs, since the original concepts ignore 

labels.
Theorem 14. Let G1 = (cid:3)S1, E1(cid:4) and G2 = (cid:3)S2, E2(cid:4) be two STGs such that L(E1) = L(E2) = {(cid:2)}. Let τ = (cid:3) f , R(cid:4) be a transformation 
such that R((cid:2), (cid:2)) holds. Then:

1. If f is a homomorphism, then τ is M↑R↑C↑.
2. If f is a strong homomorphism, then τ is M↑R(cid:14)C(cid:14).
3. If f is an embedding, then τ is M(cid:14)R↑C↑.
4. If f is a retraction, then τ is M(cid:14)R↓C↓.

(2) Suppose  f

is a strong homomorphism. Then  f
A-framework-for-analysing-state-abstraction-metho_2022_Artificial-Intelligen