1. Algebra
      1.1. Relations
      1.2. Partitions
            1.2.1. Definition
            1.2.2. Sample partition
            1.2.3. Operations with partitions
            1.2.4. Examples of operations
            1.2.5. Applet on partitions
2. Abstract Automata
3. Abstract Network
4. Partition Pairs and Pair Algebra
5. Construction of an Abstract Network
6. Structured Network
7. Additive Decomposition



1.1. Relations

Def A relation between a set S and a set T is a subset R of ST; and for (s,t) in R we write s R t. Thus R={(s,t)|s R t}.

A relation R on SS (sometimes called simply a relation on S) is:

A relation R on S is an equivalence relation on S if R is reflexive, symmetric and transitive.
If R is an equivalence relation on S, then for every s in S, the set Bk(s)={q|s R q} is an equivalence class (i.e., the equivalence class defined by s).

1.2. Partitions

1.2.1. Definition of partitions

Def A partition on S is a collection of disjoint subsets of S whose set union is S, i.e.  such that and 

The partition is the measure of information.

We refer to the sets of  as blocks of  and designate the block which contains s by .

1.2.2. Sample partition

1.2.3. Operations with partitions

If s and t are in the same block of , we write:


The computation of 
The computation of  : to compute  we proceed inductively. Let

and for i>1 let



for any i such that .

For  and on S we say that  is larger than or equal to  and write  if and only if every block of  is contained in a block of . Thus  if and only if and if and only if .

Operations "·" , "+" and the ordering of partitions form the basic link between machine concepts and algebra.

Examples of operations
  Applet on Partitions
Construction of an Abstract Network
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Last update: 3 August, 2004