
## Theorem 3.2.

(1) *For any* π, σ ∈ Π(U), if σ ≺ π*, then* G(σ) < G(π). (2) The co-entropy function G reaches the minimum value 0 *for the finest partition* πˆ.

(3) The co-entropy function G *reaches the maximum value* 2 n−2 2 nlog(2n − 2) *for the coarsest partition* πˇ.

Proof. All the clauses follow directly from Theorem 3.1 and Eq. (9).

Similar to Proposition 3.1, we have the following observation.

Proposition 3.2. Let hU, πi and hU, σi be two approximation spaces with |U| = |V|, and let f : U −→ *V be a mapping.* (1) *If f is a monomorphism from* hU, πi to hV, σi, in particular, π  σ*, then* G(π) ≤ G(σ). (2) *If f is a strict monomorphism from* hU, πi to hV, σi, in particular, π ≺ σ*, then* G(π) < G(σ).

(3) *If f is an isomorphism from* hU, πi to hV, σi, then G(π) = G(σ).

Proof. It follows immediately from Proposition 3.1 and Eq. (9).

As a corollary of Theorem 3.2 and Proposition 3.2, we see that G is a partition measure on U in the sense of
[31, Definition 3.4], that is, G is nonnegative and satisfies the following two conditions: G(σ) < G(π) if σ ≺ π; G(π) = G(σ) if there exists an isomorphism from hU, πi to hV, σi.

Note that our information entropy and co-entropy are not directly based on the blocks of a partition. Therefore, in general they do not satisfy the definition of expected granularity proposed in [28].

## 4. Relationship Of Co-Entropies Between Different Universes

In the last section, we have seen that if f is a strict monomorphism from hU, πi to hU, σi, in particular, π ≺ σ, then H(π) > H(σ) and G(π) < G(σ). In this section, we consider the monotonicities of H and G for different universes. In other words, we compare H(π) with H(σ) and G(π) with G(σ) when there exists a strict monomorphism from hU, πi to hV, σi, where |V| > |U|. For convenience, we write hU, πi ֒→ hV, σi if |V| > |U| and there exists a strict monomorphism from hU, πi to hV, σi.

We start with the following observation on the entropy function H and the co-entropy function G reviewed in Section 2.2. Consider hU1, π1i = h{1}, {{1}}i, hU2, π2i = h{1, 2}, {{1}, {2}}i, and hU3, π3i = h{1, 2, 3}, {{1, 3}, {2}}i.

Clearly, hU1, π1i ֒→ hU2, π2i ֒→ hU3, π3i.

It is easy to check by Definition 2.1 that H(π1) = 0, H(π2) = 1, and H(π3) = log 3 −
2 3
< 1. This means that the entropy function H is not monotonic. By the way, we can get by a direct computation that G(π1) = 0, G(π2) = 0, and G(π3) =
1 2
.

Let us continue to discuss the monotonicity of co-entropy function G. Consider hU1, π1i = h{1}, {{1}}i, hU2, π2i =
h{1, 2}, {{1, 2}}i, and hU3, π3i = h{1, 2, 3}, {{1, 2}, {3}}i. Again, we see that hU1, π1i ֒→ hU2, π2i ֒→ hU3, π3i.

It is easy to check by Definition 2.2 that G(π1) = 0, G(π2) = 1, and G(π3) =
2 3
. This shows that the co-entropy function G is not monotonic either. In this case, we can obtain by a direct computation that G(π1) = 0, G(π2) =
1 2
, and G(π3) =
1 2
.

Finally, we address the monotonicity of entropy function H. Consider hU1, π1i = h{1, 2}, {{1, 2}}i, hU2, π2i =
h{1, 2, 3}, {{1, 2}, {3}}i, and hU3, π3i = h{1, 2, 3, 4}, {{1, 2, 4}, {3}}i. Obviously, we have that hU1, π1i ֒→ hU2, π2i ֒→ hU3, π3i.

By a routine computation we can get that H(π1) =
3 2
, H(π2) =
5 2
, and H(π3) =
13 4
−
3 4 log 3 <
5 2
. Consequently, the entropy function H is not monotonic either. On the other hand, it follows from Eq. (9) that G(π1) =
1 2
, G(π2) =
1 2
,
and G(π3) =
3 4
+
3 4 log 3.

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