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A hierarchy (from the Greek hierarchia, “rule of a high priest”, from hierarkhes, “leader of sacred rites”) is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being “above”, “below”, or “at the same level as” one another.

A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one’s immediate superior or to one of one’s subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Indirect hierarchical links can extend “vertically” upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy which are not linked vertically to one another nevertheless can be “horizontally” linked through a path by traveling up the hierarchy to find a common direct or indirect superior, and then down again. This is akin to two co-workers or colleagues; each reports to a common superior, but they have the same relative amount of authority. Organizational forms exist that are both alternative and complementary to hierarchy. Heterarchy is one such form.

A hierarchy is typically depicted as a pyramid, where the height of a level represents that level’s status and width of a level represents the quantity of items at that level relative to the whole. For example, the few Directors of a company could be at the apex, and the base could be thousands of people who have no subordinates.

These pyramids are typically diagrammed with a tree or triangle diagram (but note that not all triangle/pyramid diagrams are hierarchical, for example, the 1992 USDA food guide pyramid), both of which serve to emphasize the size differences between the levels. An example of a triangle diagram appears to the right. An organizational chart is the diagram of a hierarchy within an organization, and is depicted in tree form below.

More recently, as computers have allowed the storage and navigation of ever larger data sets, various methods have been developed to represent hierarchies in a manner that makes more efficient use of the available space on a computer’s screen. Examples include fractal maps, TreeMaps and Radial Trees.

Visual hierarchy
In the design field, mainly graphic design, successful layouts and formatting of the content on documents are heavily dependent on the rules of visual hierarchy. Visual hierarchy is also important for proper organization of files on computers.

An example of visually representing hierarchy is through the Nested clusters. The Nested clusters represents hierarchical relationships by using layers of information. The child element is within the parent element, such as in a Venn diagram. This structure of representing hierarchy is most effective in representing simple relationships. For example, when directing someone to open a file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder. They will keep opening files within the folders until the designated file is located.

For more complicated hierarchies, the stair structure represents hierarchical relationships through the use of visual stacking. Visually imagine the top of a downward staircase beginning at the left and descending on the right. The child elements are towards the bottom of the stairs and the parent elements are at the top. This structure is effective when representing more complicated hierarchies where steps are not placed in obvious sequences. Further steps are concealed unless all of the steps are revealed in sequence. In the computer desktop example, a file that is being sought after can only be found once another file is opened. The link for the desired file is within another document. All the steps must be completed until the final destination is reached.

Informal representation
In plain English, a hierarchy can be thought of as a set in which:[1]

No element is superior to itself, and
One element, the hierarch, is superior to all of the other elements in the set.
The first requirement is also interpreted to mean that a hierarchy can have no circular relationships; the association between two objects is always transitive. The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.

Mathematical representation
Main article: Hierarchy (mathematics)
Mathematically, in its most general form, a hierarchy is a partially ordered set or poset.[7] The system in this case is the entire poset, which is constituted of elements. Within this system, each element shares a particular unambiguous property. Objects with the same property value are grouped together, and each of those resulting levels is referred to as a class.

“Hierarchy” is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity. Operations such as addition, subtractio