It is useful to interpret an equation differently in different contexts. In terms of their application, an equation may be interpreted as a mechanism by which a question may be answered given additional specific information. However, when manipulating an equation, it is better to interpret it as an abstract pattern: a tree. More concretely however, an equation may also be interpreted as an implicit, compressed representation of implicit information.
Given a particular series of variable assignments, an equation may be thought of as a mechanism that converts them into a result. The conversion is built up from simpler, known mappings that are applied recursively to the given values. Each mapping in turn takes any of a pre-arranged series of input values and combines them according to pre-determined rules. By repeatedly replacing those functions whose arguments are of the correct form with their result, the equation may be reduced to a simpler form.
In addition to using equations to determine a result, they may be transformed from one form to another. Each of these forms may be thought of in the abstract as a pattern. All equation patterns share common underlying structure - they can all be represented as a tree structure. Such a tree consists of a collection of nodes all of which have a single parent branch except for the root node of the tree. In the case of an equation, functions are parents of other functions and of values and variables.
More than one equation can represent the same information, i.e. the same results for given inputs. Some equations capture some or all of the information in another equation, while some equations or equation collections contain contradictions. Algebra is concerned with transforming between forms of equations such that at least some of the underlying information is preserved, but is restated in a desired form.
Assigning the correct values to each variable, an equation will evaluate a particular result. The equation may then be thought of as representing the set of all variable assignments and corresponding results. This often a very large or even infinite set. An equation can therefore be thought of as an implicit and compressed representation of this set. Normally there are many representations of this set, i.e. many equivalent equations that correspond to the same set of assignments and results.
Most equations exists within a community of multiple equations. The same set of values can be expressed by one equation or by multiple equations and by joining or splitting equations, a particular part of the implicit information that the equation contains may be made explicit or discarded. What remains may be stated most simply and clearly.