    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. ALGEBRA

1.1. Relations

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

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

• reflexive if, for all s, s R s;
• symmetric if s R t implies t R s;
• transitive if s R t and t R u implies s R u.
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 .

Then for any i such that . and 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 Last update: 3 August, 2004