Note: Definition of complete semi-order lattice (Davey and Priestley, 1992) is the following.

A semi-order relation ‘’ is defined on X (X may be infinite). , if A has the supremum and infimum, i.e., there exist , for if , then (supremum) and there exist , for if , then (infimum), then is called a complete semi-order lattice.

there exists a finite sequence and such that , simply .

Now we prove that is the supremum of below.

First, we show that is an equivalence relation. Its reflexivity and symmetry are obvious.

We show its transitivity. Assume . Assume that R satisfies . Then, , i.e., . Thus, is the supremum.

Now we prove that is the infimum of below.

First, we show that is an equivalence relation. From the definition of , its reflexivity and symmetry are obvious.

From and

, we have

, i.e., . has transitivity. Thus, is an equivalence relation.

Now we show that is the infimum. Assume that R is any lower bound of , i.e., for , . Assume , i.e., such that , . From and , we have .

From its transitivity, we have that and are R equivalent, i.e., . Again,, , then is the infimum.

Finally, is a complete semi-order lattice.

Obviously, when , is a quotient set of , where is a quotient set of X corresponding to equivalence relation R. The proposition shows that given a domain X, all its quotient sets compose a complete semi-order lattice based on the inclusion relations of the sets. We can implement various supremum and infimum operations over the lattice.

In Fig. 1.3, if under the equivalence relation R₀, the whole factory is regarded as an equivalence class, i.e., all elements of X are equivalent, then is the coarsest relation among R. If the factory is decomposed into several equivalence classes such as machining, assembly and welding workshops, etc., we have a new equivalence relation . Similarly, each workshop can be further divided into fine-grained cells, and so on. Finally, we have such that . This means that is the finest relation among R. And we have

Generally, assume that G is a tree with n levels shown in Fig. 1.4. Let X be all leaves of G and be a node of G. denotes all leaves of the subtree rooted at p. Let be a set of nodes at depth i. Then each A_i corresponds to a partition of X, i.e., for , is an equivalence class and corresponds to an equivalence relation .

Obviously, .

Conversely, a sequence of monotonic relations can be represented by a tree with n levels.

If a function f (x) is assigned to each leaf , then a heuristic search of G can be regarded as a hierarchical inference over a sequence of monotonic equivalence relations, that is, a heuristic tree search can be transformed into a hierarchical problem solving. We will further discuss it in Chapter 6.

Compared to other methods such as rough set theory, quotient space theory pays more extension to the concept of ‘structures’. So it is needed to investigate families of all quotient spaces composed from topologic space and to establish the corresponding structural theorem. We will establish three different complete lattice structures of families of quotient spaces and their properties, relations and applications below.

is a quotient space, where is a quotient topology on induced from topologic space . In the following discussion, in order to differentiate from other topologies, symbol with square brackets [ ] is used to denote a quotient topology induced from a corresponding quotient set.

, where is a set of all quotient topologies induced from quotient sets.

From the basic theorem, we have the theorem above.

If the topology on quotient sets is unrestricted, i.e., it is not limited to the quotient topology induced from quotient set, the theorem does not necessarily hold. This means that the supremum (or infimum) of a family of topologies is not a quotient topology induced from quotient set. Namely, there exists a subset of U such that it does not have the supremum (infimum) within U. The following example shows the fact.

In the other hand, the supremum of and is . The induced topology from X₃ is . Thus, .

First, two lemmas are introduced below (Eisenberg, 1974).

It’s noted that for any subset on V, based on Definitions 1.13 and 1.14 we can also define its corresponding supremum and infimum quotient spaces, respectively.

is any subset from W, X′ is the supremum of , and is a nature projection. From Lemma 1.2, a minimal topology T′ exists on X′ such that it makes all mappings continuous. Since is continuous, is an open mapping, i.e., open sets are mapped as open sets. We have that all open sets on are mapped as open sets on . From Definition 1.12, we have . T′ is the upper bound of . Since T′ is the coarsest topology, we have that is the supremum of A.

Similarly, for any subset B, there exists its infimum. Therefore, W is a complete lattice.

Similarly, we can prove that V is a complete semi-order lattice.

Obviously, V is a sub-lattice of W and a complete lattice obtained by the supremum and infimum operations over elements of U. And W is a complete lattice composed by elements of U and all lower topologies of the induced quotient topologies on U.

The combination method of quotient spaces is the main one in quotient space theory. V is the maximal complete lattice that can be produced by using the supremum and infimum operations over the induced quotient topologies of U. This is just a combination method of quotient spaces and similar to the method that quotient topologies are produced by topological bases. The supremum and infimum operations over induced quotient topologies of U are closed on V, which ensures the completeness of the family of quotient spaces that is produced by the combination method.

If the quotient spaces are not only constructed by domain granulation (quotient sets) but also by topology granulation in U, then we have W, the complete lattice composed by all possible quotient spaces from the above method and from the supremum and infimum operations over the quotient spaces.

In order to discuss the relation among U, V and W, we introduce the concept of ‘minimal open set’ below.

Obviously, . We’ll prove below that in general the relation among U, V and W is a proper subset one. For example, in Section 1.4.2.1, quotient space is in V but not in U, so U is a proper subset of V.

⇐ Sufficient condition: Since is a trivial (or discrete) topology, i.e., all quotient topologies on any quotient set are trivial or discrete, .

⇒Reduction to absurdity: Assume that T is neither a trivial nor a discrete topology. We show below that . Let . From Lemma 1.3, for each , a non-empty bipartition of exists and one of its bipartition is an open set at most, i.e., a minimal open set. The open set is left, otherwise is divided into a set of singletons. After the treatment, T is denoted as , where is either a singleton or a minimal open set.

If T has only one component , then T is a trivial topology. This is a contradiction.

If each is a singleton, then T is a discrete topology. This is also a contradiction.

Thus, there exist some ’s such that . Let be the one having the minimal cardinality among ’s. Now we discuss the following two cases.

Then, we have induced quotient topologies and the supremum topology .

On the other hand, we have the supremum of and . Since is the induced quotient topology from , , then . This is a contradiction.

If C is a singleton, there exists a non-empty bipartition on . Construct . Then, are not open. Otherwise, assume that is an open set. From Lemma 1.3, is either a discrete topology or a minimal open set. But the former is impossible since C is a singleton. The latter is impossible as well since is a proper sunset of X. This is a contradiction. Therefore, are not open sets.

Similarly, are not open sets either.

We have . Then .

On the other hand, , we have ,

Then, . This is a contradiction.

The three semi-order lattices given in the section correspond to three different multi-granular worlds. The completeness of the lattices provides a theoretical foundation for the translation, decomposition, combination operations over the multi-granular worlds.