An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H.
An H-design (V,C) of order v and index μ is embedded into a G-design (X,B) of order v+w and index λ if μ≤λ, V⊆X and there is an injective mapping f C→B such that B is subgraph of f(B) for every B∈C.
An H-design (U,C) of order u and index λ is embedded into a G-design (V,B) of order v and index μ if λ≤μ, U⊆V and there is an injective mapping f C→B such that B is a subgraph of f(B) for every B∈C.
A G-design (V,B) is called embedded into a H-design (V∪W,D) if G is a subgraph of H and there is an injective mapping f:B→D such that B is a subgraph of f(B) for every B∈B.
We say that a handcuffed design (V,P) of order v and block size s (P. Hell, A. Rosa, Discrete Math. 2 (1972) 229 252) is embedded in (W,C) if there is an injective mapping f P→C such that B is a subgraph of f(B) for every B∈P.
First, we want to show that is an injective mapping on.
Because f is an injective mapping, f n ( m ) ≠ m for f ( m ) ≠ m.
Also, it is easy to see that T is an injective mapping.
Note that g is an injective mapping, therefore, by Lemma 2.1, x is unique, and hence the result follows.
Then the map ψ is an injective mapping between (A (D,mathbb{C})/operatorname{ker} varphi) and (M (Omega,mathbb{H})), where (u_{1}), (v_{1}) are defined similarly to u, v, respectively, and (M Omega,mathbb{H} )) is the same as that in Section 2.
