As a result, in all the above three cases we always have that G(π1) ≤ G(π2) ≤ G(π3).

We thus conjecture that G(π) ≤ G(σ) whenever hU, πi ֒→ hV, σi. Indeed, it holds true, as we will see later.

To prove the conjecture, it is convenient to introduce the following notion and a key lemma.

Definition 4.1. Let hU, πi be an approximation space and a < *U. The approximation space* hU ∪{a}, π∪{{a}}i is called the one-point extension of hU, πi *by a. We say that* hV, σi *is a one-point extension of* hU, πi if hV, σi = hU∪{a}, π∪{{a}}i for some a.

For example, hU2, π2i = h{1, 2, 3}, {{1, 2}, {3}}i is the one-point extension of hU1, π1i = h{1, 2}, {{1, 2}}i by 3.

The following lemma shows that one-point extension does not change co-entropy.

Lemma 4.1. Let hV, σi *be a one-point extension of* hU, πi. Then G(σ) = G(π).

Proof. Suppose that π = {U1, U2, . . . , Uk} and σ = {U1, U2, . . . , Uk, {a}}, where a < U; assume that A(U, π) =
{(Ai, A
′ i
) | 1 ≤ i ≤ m} and Ai =
X ⊆ U

appπ X, appπX= (Ai, A
′ i
)
with |Ai| = ri. It thus follows that

$\downarrow$ . 
A(V, σ) = A(U, π) ∪Ai ∪ {a}, A
′
i ∪ {a}| 1 ≤ i ≤ m	.

9
For any (Bi, B
′
i
) ∈ A(V, σ), we write Bi for X ⊆ V

appσ X, appσX= (Bi, B
′
i
)
and si for |Bi|. If (Bi, B
′
i
) =
(Ai, A
′ i
) ∈ A(U, π), then we see that Bi = Ai and thus si = riin this case. If (Bi, B
′ i
) =Ai ∪ {a}, A
′
i ∪ {a}∈
nAi ∪ {a}, A
′
i ∪ {a}| 1 ≤ i ≤ mo, then we have that Bi = {X ∪ {a} | X ∈ Ai} and si = ri still holds in this case. Therefore, we get by Definition 3.3 that

$$\begin{array}{r c l}{{{\mathcal G}(\sigma)}}&{{=}}&{{\sum_{i=1}^{m}\frac{r_{i}}{2^{n+1}}\log r_{i}+\sum_{i=1}^{m}\frac{r_{i}}{2^{n+1}}\log r_{i}}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{=}}&{{\sum_{i=1}^{m}\frac{r_{i}}{2^{n}}\log r_{i}}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{=}}&{{{\mathcal G}(\pi),}}\end{array}$$
$$\leq i\leq m\}\,.$$

finishing the proof of the lemma.

For subsequent need, we would like to generalize Definition 4.1 as follows.

Definition 4.2. Let hU, πi and hV, σi *be two approximation spaces. We say that* hV, σi *is a* multi-one-point extension of hU, πi *if there are approximation spaces* hUi, πii, 0 ≤ i ≤ *l, with* hU0, π0i = hU, πi and hUl, πli = hV, σi *such that* each hUi, πii, 1 ≤ i ≤ *l, is a one-point extension of* hUi−1, πi−1i.

For example, hV, σi = h{1, 2, 3, 4}, {{1, 2}, {3}, {4}}i is a multi-one-point extension of hU, πi = h{1, 2}, {{1, 2}}i. In fact, we may take hU0, π0i = hU, πi, hU1, π1i = h{1, 2, 3}, {{1, 2}, {3}}i, and hU2, π2i = hV, σi.

The following fact follows immediately from Lemma 4.1.

Corollary 4.1. If hV, σi *is a multi-one-point extension of* hU, πi*, then* G(σ) = G(π).

In light of the above corollary, let us refer to multi-one-point extensions as one-point extensions for simplicity.

Further, we have the following observation.

Theorem 4.1. *Suppose that there is a monomorphism from* hU, πi to hV, σi*. If there exists* hU
′, π′i *that satisfies the* following two conditions:
(a) *either* hU
′, π′i = hU, πi or hU
′, π′i *is a one-point extension of* hU, πi,
(b) hU
′, π′i *is isomorphic to* hV, σi,