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Relationship cardinalities can be said to make existence assertions - that such an such entity may or must exist in relation to a given entity. Sometimes it is useful to make further existence assertions beyond what can be asserted by means of relationship cardinality, applying as cardinality does to a single relationship. In this section we introduce a notation for doing so for a certain significant class of existence assertions which involves use of a ‘bowtie’ symbol on each of two or more contributing relationships; visually the bowtie is an extension of the cardinality crowsfoot and the bar used to distinguish identifying relationships. The use of the bowtie was proposed by Bob Appleyard and is a nod in the direction of relational algebra. The combination of uniqueness and existence conditions on entitites of a type conditional upon the proper scoping of relationships is expressed by pullback diagrams in category theory and so we speak of certain diagrams of relationships as pullback diagrams. We now give a number of examples.
In the example in figure 47 the existence assertion is to the effect that in a mathematic matrix there is an element for each combination of a row and a column. According to the uniqueness assertion there is at most one. The bowties assert both the existence and the uniqueness and the square of relationships is said to be a pullback square.
There is a similar shape to the models representing the structure1 of rectangular tables of data. An example is given in figure 48. In the HTML language, and in other computer markup languages, such data tables are communicated row by row rather than column by column.
In some tabular displays the rows or columns of a table, or both, may be grouped together to represent some grouping of the subjects. The structure then has different branches that are hierarchical and joined at the detail level into the recognisable shape of the 2 dimensional matrix structure. One such is illustrated in figure 49.
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The terms network and matrix are commonly used in contrast to the term hierarchical to refer to arrangements of entities not constrained to be hierarchical; for example in organisational structure the term matrix management is used in situations were different dimensions are managed by different management hierarchies and in which individuals therefore have multiple reporting lines. The term hierarchical is etymologically derived from Greek sacred ruler and emerged in its modern sense via its use in medieval times in relation to the church organisation.