层次

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A hierarchy (from the Greek ἱεραρχία hierarkhia, "rule of a high priest", from hierarkhes, "president 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. Hierarchy is an important concept in a wide variety of fields, such as philosophy, mathematics, computer science, organizational theory, systems theory, and the social sciences (especially political philosophy).

A hierarchy (from the Greek ἱεραρχία hierarkhia, "rule of a high priest", from hierarkhes, "president 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. Hierarchy is an important concept in a wide variety of fields, such as philosophy, mathematics, computer science, organizational theory, systems theory, and the social sciences (especially political philosophy).

“层次 hierarchy”（从古希腊语 ἱεραρχία hierarkhia“大祭司的统治”和hierarkhes“神圣仪式的主持”而来）是一组元素（对象、名称、值、类别等）的结构，表示相互之间“之上”“之下”或“同级”关系。在哲学、数学、计算机科学、系统论和社会科学（尤其是政治哲学）中，层次都是很重要的概念。

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. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy that 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 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. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy that 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.

层次结构可直接或间接地、竖向或对角地连接实体。大体上呈层次结构的系统尽管也可对替代层次作合并，层次结构中唯一的直接关系就是其直接上级或直接下级关系。层次联系可通过同向的多个联系汇成的路径向上或向下发展。层次结构中所有没有垂直相连的部分都可以通过水平路径（向上找到共同直接或间接上级再向下）相连。就像两个同事、同僚向共同上级汇报，但他们各自都有同等的相对权限。组织形式的存在对层次既是替代也是补充。异质结构也是这样一种形式。

名称 Nomenclature

See also: Glossary of graph theory, Taxonomy (general), and Structure

Hierarchies have their own special vocabulary. These terms are easiest to understand when a hierarchy is diagrammed (see below).

Hierarchies have their own special vocabulary. These terms are easiest to understand when a hierarchy is diagrammed (see below).

层次有其特殊词汇，这些术语在层次图解（见下）中较易理解。

In an organizational context, the following terms are often used related to hierarchies:

In an organizational context, the following terms are often used related to hierarchies:

在组织性的语境下，以下术语经常用于层次性:

• Object: one entity (e.g., a person, department or concept or element of arrangement or member of a set)
• 对象 Object：一个实体（例如一种成员或配置集的个人、部门、概念、元素）

• System: the entire set of objects that are being arranged hierarchically (e.g., an administration)
• 系统 System：层次性配置的整组对象集（例如管理体系）

• Dimension: another word for "system" from on-line analytical processing (e.g. cubes)
• 维度 Dimension：联机分析处理（例如立方体）中“系统”的同义词

• Member: an (element or object) at any (level or rank) in a (class-system, taxonomy or dimension)
• 成员 Member：分类系统、分类学或维度中任一层级或位阶一个元素或对象

• 位置相关的术语 Terms about Positioning

• • Rank: the relative value, worth, complexity, power, importance, authority, level etc. of an object
  • 位阶 Rank：一个对象的相对值、价值、复杂度、力量、重要度、权威性、等级等

• • Level or Tier: a set of objects with the same rank OR importance
  • 等级 Level：同等位阶或重要度的一组对象

• • Ordering: the arrangement of the (ranks or levels)
  • 排序 Ordering：等级或位阶的配置

• • Hierarchy: the arrangement of a particular set of members into (ranks or levels). Multiple hierarchies are possible per (dimension taxonomy or Classification-system), in which selected levels of the dimension are omitted to flatten the structure
  • 层次 Hierarchy：特定成员组位阶或等级的配置。通过忽视所选等级以扁平化结构，一个维度、分类学或体系分类的多层次是可能的

• 位置相关的术语Terms about Placement

• • Hierarch, the apex of the hierarchy, consisting of one single orphan (object or member) in the top level of a dimension. The root of an inverted-tree structure
  • 顶 Hierarch：层次的顶，位于维度中顶层的一个独立个体（对象或成员）。倒树形结构的根源

• • Member, a (member or node) in any level of a hierarchy in a dimension to which (superior and subordinate) members are attached
  • 构件 Member：维系（上级或下级）构件的维度中层次结构里任一层的一个构件或节点

• • Orphan, a member in any level of a dimension without a parent member. Often the apex of a disconnected branch. Orphans can be grafted back into the hierarchy by creating a relationship (interaction) with a parent in the immediately superior level
  • 孤体 Orphan：一个没有父级的构件。通常是孤立分支的顶。孤体可通过创建与父级的关系而重新嫁接到层次中。

• • Leaf, a member in any level of a dimension without subordinates in the hierarchy
  • 叶 Leaf：维度中任一层里没有下级的构件

• • Neighbour: a member adjacent to another member in the same (level or rank). Always a peer.
  • 近邻 Neighbour：同一等级或位阶中相邻与另一构件的构件，通常是同级。

• • Superior: a higher level or an object ranked at a higher level (A parent or an ancestor)
  • 上级 Superior：更高等级或更高等级中的对象（父级或祖先级）

• • Subordinate: a lower level or an object ranked at a lower level (A child or a descendant)
  • 下级 Subordinate：更低等级或更低等级中的对象（子级或后裔级）

• • Collection: all of the objects at one level (i.e. Peers)
  • 群 Collection：一等级中的所有对象（例如同级成员）

• • Peer: an object with the same rank (and therefore at the same level)
  • 同级 Peer：同一位阶（即同一等级）的对象

• • Interaction: the relationship between an object and its direct superior or subordinate (i.e. a superior/inferior pair)
  • 交互 Interaction：对象与其直接上级或下级的关系（例如一对上下级）

• • • a direct interaction occurs when one object is on a level exactly one higher or one lower than the other (i.e., on a tree, the two objects have a line between them)
    • ，一个对象比另一个刚好高一等级或第一等级时（例如在树形图中两个对象间有连线）发生直接 Direct交互

• • Distance: the minimum number of connections between two objects, i.e., one less than the number of objects that need to be "crossed" to trace a path from one object to another
  • 距离 Distance：两个对象间的最小连接数，例如若连接数比对象数少1则需要一个对象到另一个路径交叉

• • Span: a qualitative description of the width of a level when diagrammed, i.e., the number of subordinates an object has
  • 跨度 Span：层级宽度图表化的量化表述，例如一个对象所拥有的下级的数量。

• 自然相关的术语 Terms about Nature

• • Attribute: a heritable characteristic of (members and their subordinates) in a level (e.g. hair-colour)
  • 属性 Attribute：等级中（成员或其下级）的遗传特征（例如发色）

• • Attribute-value: the specific value of a heritable characteristic (e.g. Auburn)
  • 属性值 Attribute-value：遗传特征的具体值（例如棕发）

In a mathematical context (in graph theory), the general terminology used is different.

In a mathematical context (in graph theory), the general terminology used is different.

在数学语境下（图论）所用的一般术语则不同。

Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with data structures, objects are known as nodes, superiors are called parents and subordinates are called children. In a business setting, a superior is a supervisor/boss and a peer is a colleague.

Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with data structures, objects are known as nodes, superiors are called parents and subordinates are called children. In a business setting, a superior is a supervisor/boss and a peer is a colleague.

大多数层次结构按其主体不同而适用更具体的词汇，但其背后的理念是相同的。例如，在数据结构中，对象成为节点，上级称为父级，下级成为子级。在商业环境中，上级是主管/老板，同级即同事。

支化度 Degree of branching

Degree of branching refers to the number of direct subordinates or children an object has (in graph theory, equivalent to the number of other vertices connected to via outgoing arcs, in a directed graph) a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes: linear and branching.

Degree of branching refers to the number of direct subordinates or children an object has (in graph theory, equivalent to the number of other vertices connected to via outgoing arcs, in a directed graph) a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes: linear and branching.

支化度是指对象所有直接下级或子项的数量（在图论中，即有向图通过外向弧所连接的顶点数）。层次可按其“最大度”分类，系统中支化度的最高值表示系统整体，这种分类方式分化出两个分野：线性和支化。

In a linear hierarchy, the maximum degree is 1. In other words, all of the objects can be visualized in a line-up, and each object (excluding the top and bottom ones) has exactly one direct subordinate and one direct superior. Note that this is referring to the objects and not the levels; every hierarchy has this property with respect to levels, but normally each level can have an infinite number of objects. An example of a linear hierarchy is the hierarchy of life.

In a linear hierarchy, the maximum degree is 1. In other words, all of the objects can be visualized in a line-up, and each object (excluding the top and bottom ones) has exactly one direct subordinate and one direct superior. Note that this is referring to the objects and not the levels; every hierarchy has this property with respect to levels, but normally each level can have an infinite number of objects. An example of a linear hierarchy is the hierarchy of life.

线性层次结构的最大度是1。即所有对象都可呈现在一个列表中，每个对象(顶部和底部的除外)只有一个直接下级和一个直接上级。注意，这里说的是对象而不是级别；就级别而言每个层次结构都有这种属性，但每个级别通常可以有无限个对象。生命层次结构就是线性层次的一个例子。

In a branching hierarchy, one or more objects has a degree of 2 or more (and therefore the minimum degree is 2 or higher). For many people, the word "hierarchy" automatically evokes an image of a branching hierarchy. Branching hierarchies are present within numerous systems, including organizations and classification schemes. The broad category of branching hierarchies can be further subdivided based on the degree.

In a branching hierarchy, one or more objects has a degree of 2 or more (and therefore the minimum degree is 2 or higher). For many people, the word "hierarchy" automatically evokes an image of a branching hierarchy. Branching hierarchies are present within numerous systems, including organizations and classification schemes. The broad category of branching hierarchies can be further subdivided based on the degree.

在分支层次结构中，至少有一个对象的支化度在2或以上（即支化度下限是2或更高）。对许多人来说，“层次”这个词自动唤起了支化层次结构的印象。支化层次存在于很多系统中，包括组织、分类法。广义上的分支层次可据其分支度进一步细分。

A flat hierarchy is a branching hierarchy in which the maximum degree approaches infinity, i.e., that has a wide span. Most often, systems intuitively regarded as hierarchical have at most a moderate span. Therefore, a flat hierarchy is often not viewed as a hierarchy at all. For example, diamonds and graphite are flat hierarchies of numerous carbon atoms that can be further decomposed into subatomic particles.

A flat hierarchy is a branching hierarchy in which the maximum degree approaches infinity, i.e., that has a wide span. Most often, systems intuitively regarded as hierarchical have at most a moderate span. Therefore, a flat hierarchy is often not viewed as a hierarchy at all. For example, diamonds and graphite are flat hierarchies of numerous carbon atoms that can be further decomposed into subatomic particles.

扁平层次是指最大度接近无穷的支化层次结构，例如跨度很大的层次结构。分层系统一般最多只有中等的跨度，故扁平层次结构通常根本不被视为层次结构。例如钻石和石墨是由许多碳原子组成的扁平层次结构，而这些碳原子可进一步分解为亚原子粒子。

An overlapping hierarchy is a branching hierarchy in which at least one object has two parent objects. For example, a graduate student can have two co-supervisors to whom the student reports directly and equally, and who have the same level of authority within the university hierarchy (i.e., they have the same position or tenure status).

An overlapping hierarchy is a branching hierarchy in which at least one object has two parent objects. For example, a graduate student can have two co-supervisors to whom the student reports directly and equally, and who have the same level of authority within the university hierarchy (i.e., they have the same position or tenure status).

重叠的层次结构是指其中至少有一个对象具有两个父级对象的分支层次结构。例如，一个研究生可以有两个联合主管，学生可同等地直接向他们汇报，且二人在大学层级制度中具有同等的权力水平(例如他们都拥有相同的职位或任期)。

术语历史 History of the term

Possibly the first use of the English word "hierarchy" cited by the Oxford English Dictionary was in 1881, when it was used in reference to the three orders of three angels as depicted by Pseudo-Dionysius the Areopagite (5th–6th centuries). Pseudo-Dionysius used the related Greek word (ἱεραρχία hierarchia) both in reference to the celestial hierarchy and the ecclesiastical hierarchy. The Greek term ἱεραρχία means "rule of a high priest" (from ἱεράρχης hierarches, meaning "president of sacred rites, high-priest" and that from ἱερεύς hiereus, "priest" and ἀρχή arche, amongst others "first place or power, rule"), and Dionysius is credited with first use of it as an abstract noun. Since hierarchical churches, such as the Roman Catholic (see Catholic Church hierarchy) and Eastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with God as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods in secular settings.

Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus Digital Library</ref> and that from ἱερεύς hiereus, "priest" and ἀρχή arche, amongst others "first place or power, rule"), and Dionysius is credited with first use of it as an abstract noun. Since hierarchical churches, such as the Roman Catholic (see Catholic Church hierarchy) and Eastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with God as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods in secular settings.

“层次”英语单词的首次使用可能是在1881年的牛津英语词典里，被用来指伪狄奥尼修斯 Pseudo-Dionysius the Areopagite（5-6世纪）所描述的3个天使的3个位阶。伪狄奥尼修斯使用有关的希腊词汇(ἱεραρχία hierarchia) 来指代天界位阶和教会位阶。希腊语 ἱεραρχία表示大祭司的统治（来自ἱερεύς hiereus, “祭司”及 ἀρχή arche等“最初的地方或力量，统治”所衍生出的 ἱεράρχης hierarches，“神圣仪式的主持，“高阶祭司”）。伪狄奥尼修斯首次将其用作抽象名词概念，自阶层式教堂以来，例如罗马天主教和东正教教堂都有着现代语义上的层次性结构（传统意义上层次结构的顶部或头部表示神），该术语在长期语境下用于指称类似的结构方式。

层次的视觉呈现 Visually representing hierarchies

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Maslow's hierarchy of human needs. This is an example of a hierarchy visualized with a triangle diagram.

Maslow's hierarchy of human needs. This is an example of a hierarchy visualized with a triangle diagram.]]

马斯洛的人类需求层次理论，是一个用三角图表呈现层次结构的离子。

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.

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.

层次结构通常被描述为金字塔形，其中高度代表层次状态而宽度代表该层的元素多寡。例如，一个公司的几个董事处于顶峰，而基层可能是成千上万没有下级的人。

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 in 模板:Section link, below.

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 in , below.

这样的金字塔通常用树形或三角形图表绘制（注意，并非所有的三角形/金字塔形图表都是层次分明的；例如1992年美国农业部膳食指南金字塔），两种图表都表现了各层次的大小差异。右侧是一个三角形图表的例子。组织结构图是组织内部层次结构的图表，在下面的树形图中有所描述。

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.

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.

最近，随着计算机的发展使存储和索引更大的数据集成为可能，人们开发出各种方法来表示层次结构，以便更有效地利用计算机屏幕空间。例如分形图、树形图和辐射状树图。

视觉层次 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.

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.

在设计领域，主要是平面设计，文件内容排版和格式的成功很大程度上取决于视觉层次规则。视觉层次对于像样的电脑文件组织也至关重要。

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.

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.

嵌套集群是层次结构的一个可视化示例。嵌套集群用信息层来表示层次关系。子元素位于父元素之内，例如维恩图 Venn diagram。这种结构可有效表示简单层次关系。例如，当需要在打开一个电脑桌面的文件时，可能要先让操作者打开主文件夹，然后是主文件夹中的子文件夹，不断打开直到找到指定文件。

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.

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.

对于更复杂的层次结构，楼梯结构用堆叠效果表示层次关系。请从视觉上想象一下从左到右下楼楼梯的顶部，子元素位于楼梯底部而父元素位于顶部。当表示更复杂的层次结构时，这种结构非常有效，尤其当台阶没有位于明显的序列中时。下一台阶必须基于所有台阶都顺序显示后才会出现。在计算机桌面的例子中，只有先打开别的文件才能找到目标文件，因为通向目标文件的链接位于另一个文件内。所有步骤都必须完成才能到达最终目标。

通俗表达 Informal representation

In plain English, a hierarchy can be thought of as a set in which:

In plain English, a hierarchy can be thought of as a set in which:

在日常英语中，层次可以被看作是一个集合，其中:

1. No element is superior to itself, and

No element is superior to itself, and

任何元素都不高于自身，且

1. One element, the hierarch, is superior to all of the other elements in the set.

One element, the hierarch, is superior to all of the other elements in the set.

其中有一个层次元素，高于集合中的所有其他元素。

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 first requirement is also interpreted to mean that a hierarchy can have no circular relationships; the association between two objects is always transitive.

第一个前提也可理解为层次结构不能有循环关系; 两个对象之间的关联总是可传递的。

The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.

The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.

第二个要求则确定了层次结构必须有一个所有元素都共有的代表或根源。

数学表达 Mathematical representation

Main article: 层次（数学） Hierarchy (mathematics)

Mathematically, in its most general form, a hierarchy is a partially ordered set or poset. 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.

|isbn=3-540-61534-2|year=1996|publisher=Springer|editor-last=Eklund|editor-first=Peter G.|editor2-last=Ellis|editor2-first=Gerard|editor3-last=Mann|editor3-first=Graham|series=Lecture Notes in Artificial Intelligence 115|location=Germany}}</ref> 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.

"Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity.

Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to a problem. The use of parenthesis is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example:

Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to a problem. The use of parenthesis is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example:

(2 + 5) × (7 - 4).

(2 + 5) × (7 - 4).

(2 + 5) × (7 - 4).

In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with the problem. These rules are largely dominant in algebraic problems, ones that include several steps to solve. The use of hierarchy in mathematics is beneficial to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations.

In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with the problem. These rules are largely dominant in algebraic problems, ones that include several steps to solve. The use of hierarchy in mathematics is beneficial to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations.

数学上，层次结构最一般的形式是偏序集 Partially Ordered Set或Poset。元素所组成的系统即整个偏序集。系统内每个元素都共享一些具体无歧义的属性。相同属性的元素可集结成群，最终形成类别层次。

“层次”专用作指称随复杂度增加而组织的偏序集。例如加法、减法、乘法和除法通常按特定序列实施。一般乘除先于加减，括号也是层次的一种表示，表达了哪些运算优先。例如在(2 + 5) × (7 - 4)中，按数学的层次规则本应先作5乘以7，但加入了括号，则表示了应当先对括号内作运算。在需要多步求解的代数问题上这些规则占主导地位。在数学中使用层次结构有利于快速高效地解决问题，而不必经理漫长的剖析过程。现在认为大多数这类规则是求解特定方程的适当方法。

亚型 Subtypes

嵌套层次 Nested hierarchy

= = = 嵌套层次结构 = = = = < ! -- 如果您更改此标题，请更改链接到该标题的文章中的 wiki 链接! -- >

Matryoshka dolls, also known as nesting dolls or Russian dolls. Each doll is encompassed inside another until the smallest one is reached. This is the concept of nesting. When the concept is applied to sets, the resulting ordering is a nested hierarchy.

Matryoshka dolls, also known as nesting dolls or Russian dolls. Each doll is encompassed inside another until the smallest one is reached. This is the concept of nesting. When the concept is applied to sets, the resulting ordering is a nested hierarchy.

[玛特罗什卡套娃，又称嵌套娃娃或俄罗斯套娃。每个洋娃娃都被包裹在另一个洋娃娃里面，直到最小的洋娃娃。这就是嵌套的概念。当这个概念应用于集合时，其结果排序是一个嵌套的层次结构。]]

A nested hierarchy or inclusion hierarchy is a hierarchical ordering of nested sets. The concept of nesting is exemplified in Russian matryoshka dolls. Each doll is encompassed by another doll, all the way to the outer doll. The outer doll holds all of the inner dolls, the next outer doll holds all the remaining inner dolls, and so on. Matryoshkas represent a nested hierarchy where each level contains only one object, i.e., there is only one of each size of doll; a generalized nested hierarchy allows for multiple objects within levels but with each object having only one parent at each level. The general concept is both demonstrated and mathematically formulated in the following example:

A nested hierarchy or inclusion hierarchy is a hierarchical ordering of nested sets. The concept of nesting is exemplified in Russian matryoshka dolls. Each doll is encompassed by another doll, all the way to the outer doll. The outer doll holds all of the inner dolls, the next outer doll holds all the remaining inner dolls, and so on. Matryoshkas represent a nested hierarchy where each level contains only one object, i.e., there is only one of each size of doll; a generalized nested hierarchy allows for multiple objects within levels but with each object having only one parent at each level. The general concept is both demonstrated and mathematically formulated in the following example:

嵌套层次结构或包含层次结构是嵌套集的层次结构排序。嵌套的概念在俄罗斯套娃中得到了体现。每个洋娃娃都被另一个洋娃娃包裹着，一直到最外层的。外部的洋娃娃包含所有其内部的洋娃娃，往外一层的洋娃娃也包含所有其内部剩余的洋娃娃，如此反复。套娃结构是一个每层都只有一个对象的嵌套层次结构，例如在套娃中同样大小的洋娃娃只有一个；广义的嵌套层次可以每层都有多个对象，但每层里的对象都只有一个父级对象。嵌套层次的一般概念的论证及数学表达如下：

[math]\displaystyle{ \text{square} \subset \text{quadrilateral} \subset \text{polygon} \subset \text{shape} \, }[/math]

[math]\displaystyle{  \text{square} \subset \text{quadrilateral} \subset \text{polygon} \subset \text{shape} \,  }[/math]

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A square can always also be referred to as a quadrilateral, polygon or shape. In this way, it is a hierarchy. However, consider the set of polygons using this classification. A square can only be a quadrilateral; it can never be a triangle, hexagon, etc.

A square can always also be referred to as a quadrilateral, polygon or shape. In this way, it is a hierarchy. However, consider the set of polygons using this classification. A square can only be a quadrilateral; it can never be a triangle, hexagon, etc.

正方形可称为四边形、多边形或形状。这样看来是层次性的。然而若多边形集采用这种分类法，则正方形只能是四边形，而不会是三角形、六边形等。

Nested hierarchies are the organizational schemes behind taxonomies and systematic classifications. For example, using the original Linnaean taxonomy (the version he laid out in the 10th edition of Systema Naturae), a human can be formulated as:

Nested hierarchies are the organizational schemes behind taxonomies and systematic classifications. For example, using the original Linnaean taxonomy (the version he laid out in the 10th edition of Systema Naturae), a human can be formulated as:

嵌套层次结构是分类学和系统分类背后的组织性方案。例如，在最初的林奈分类法（他在《自然系统》第10版中所列）中，人类可被归为

[math]\displaystyle{ \text{H. sapiens} \subset \text{Homo} \subset \text{Primates} \subset \text{Mammalia} \subset \text{Animalia} }[/math]

[math]\displaystyle{ \text{H. sapiens} \subset \text{Homo} \subset \text{Primates} \subset \text{Mammalia} \subset \text{Animalia} }[/math]

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Taxonomies may change frequently (as seen in biological taxonomy), but the underlying concept of nested hierarchies is always the same.

Taxonomies may change frequently (as seen in biological taxonomy), but the underlying concept of nested hierarchies is always the same.

分类法可能会频繁改变（如在生物分类法中所见），但嵌套层次结构这一基本概念始终不变。

In many programming taxonomies and syntax models (as well as fractals in mathematics), nested hierarchies, including Russian dolls, are also used to illustrate the properties of self-similarity and recursion. Recursion itself is included as a subset of hierarchical programming, and recursive thinking can be synonymous with a form of hierarchical thinking and logic.

In many programming taxonomies and syntax models (as well as fractals in mathematics), nested hierarchies, including Russian dolls, are also used to illustrate the properties of self-similarity and recursion. Recursion itself is included as a subset of hierarchical programming, and recursive thinking can be synonymous with a form of hierarchical thinking and logic.

在许多编程分类法和句法模型(以及数学中的分形)中，包括俄罗斯娃娃，嵌套的层次结构也用作自相似和递归性质的呈现。递归本身是层次性编程的一个子集，递归思维可用作层次思维和逻辑的同义词。

包容层次 Containment hierarchy

A containment hierarchy is a direct extrapolation of the nested hierarchy concept. All of the ordered sets are still nested, but every set must be "strict"—no two sets can be identical. The shapes example above can be modified to demonstrate this:

A containment hierarchy is a direct extrapolation of the nested hierarchy concept. All of the ordered sets are still nested, but every set must be "strict"—no two sets can be identical. The shapes example above can be modified to demonstrate this:

包容层次是嵌套层次概念的直接外推。所有的有序集仍然是嵌套的，但是每个集合必须是“严格”的——不存在两个相同的集合。上面的图形示例可稍作改变来论证:

[math]\displaystyle{ \text{square} \subsetneq \text{quadrilateral} \subsetneq \text{polygon} \subsetneq \text{shape} \, }[/math]

[math]\displaystyle{  \text{square} \subsetneq \text{quadrilateral} \subsetneq \text{polygon} \subsetneq \text{shape} \,  }[/math]

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The notation [math]\displaystyle{ x \subsetneq y \, }[/math] means x is a subset of y but is not equal to y.

The notation [math]\displaystyle{ x \subsetneq y \, }[/math] means x is a subset of y but is not equal to y.

符号 < math > x subsetneq y，</math > 表示x是y的子集，但不等于y。

A general example of a containment hierarchy is demonstrated in class inheritance in object-oriented programming.

A general example of a containment hierarchy is demonstrated in class inheritance in object-oriented programming.

面向对象程序设计的类继承是包容层次的一个例证。

Two types of containment hierarchies are the subsumptive containment hierarchy and the compositional containment hierarchy. A subsumptive hierarchy "subsumes" its children, and a compositional hierarchy is "composed" of its children. A hierarchy can also be both subsumptive and compositional模板:Example needed.

Two types of containment hierarchies are the subsumptive containment hierarchy and the compositional containment hierarchy. A subsumptive hierarchy "subsumes" its children, and a compositional hierarchy is "composed" of its children. A hierarchy can also be both subsumptive and compositional.

包容层次有包含包容层次和组合包容层次两种。包含包容层次“包含”其子级，而组合包容层次“组合”其子级。层次结构也可能同时是包含的和组合的。

包容层次结构 Subsumptive containment hierarchy

A subsumptive containment hierarchy is a classification of object classes from the general to the specific. Other names for this type of hierarchy are "taxonomic hierarchy" and "IS-A hierarchy". The last term describes the relationship between each level—a lower-level object "is a" member of the higher class. The taxonomical structure outlined above is a subsumptive containment hierarchy. Using again the example of Linnaean taxonomy, it can be seen that an object that is part of the level Mammalia "is a" member of the level Animalia; more specifically, a human "is a" primate, a primate "is a" mammal, and so on. A subsumptive hierarchy can also be defined abstractly as a hierarchy of "concepts". For example, with the Linnaean hierarchy outlined above, an entity name like Animalia is a way to group all the species that fit the conceptualization of an animal.

A subsumptive containment hierarchy is a classification of object classes from the general to the specific. Other names for this type of hierarchy are "taxonomic hierarchy" and "IS-A hierarchy". The last term describes the relationship between each level—a lower-level object "is a" member of the higher class. The taxonomical structure outlined above is a subsumptive containment hierarchy. Using again the example of Linnaean taxonomy, it can be seen that an object that is part of the level Mammalia "is a" member of the level Animalia; more specifically, a human "is a" primate, a primate "is a" mammal, and so on. A subsumptive hierarchy can also be defined abstractly as a hierarchy of "concepts". For example, with the Linnaean hierarchy outlined above, an entity name like Animalia is a way to group all the species that fit the conceptualization of an animal.

包容层次结构 subsumptive containment hierarchy是对象类从一般到特定的分类。这类层次结构的其他名称是“分类层次结构”和“ IS-A 层次结构”。最后一个术语描述了每个级别之间的关系——较低级别的对象“是”较高级别类的成员。上面概述的分类结构是一个包容性的层次结构。再次使用林奈分类系统的例子，可以看到，属于哺乳动物 Mammalia 等级的物体“是”动物等级的成员；更具体地说，人类“是”灵长类动物，灵长类动物“是”哺乳动物等等。也可以抽象地将包含的层次结构定义为“概念”的层次结构。例如，根据上述的林奈分类系统，像动物这样的实体名称是对所有符合动物概念的物种进行分类的一种方法。

组合包容层次结构 Compositional containment hierarchy

A compositional containment hierarchy is an ordering of the parts that make up a system—the system is "composed" of these parts. Most engineered structures, whether natural or artificial, can be broken down in this manner.

A compositional containment hierarchy is an ordering of the parts that make up a system—the system is "composed" of these parts. Most engineered structures, whether natural or artificial, can be broken down in this manner.

组合包容层次结构 Compositional containment hierarchy是组成系统的各个部分的顺序——系统是由这些部分“组成”的。大多数工程结构，无论是自然的还是人工的，都可以用这种方式分解。

The compositional hierarchy that every person encounters at every moment is the hierarchy of life. Every person can be reduced to organ systems, which are composed of organs, which are composed of tissues, which are composed of cells, which are composed of molecules, which are composed of atoms. In fact, the last two levels apply to all matter, at least at the macroscopic scale. Moreover, each of these levels inherit all the properties of their children.

The compositional hierarchy that every person encounters at every moment is the hierarchy of life. Every person can be reduced to organ systems, which are composed of organs, which are composed of tissues, which are composed of cells, which are composed of molecules, which are composed of atoms. In fact, the last two levels apply to all matter, at least at the macroscopic scale. Moreover, each of these levels inherit all the properties of their children.

每个人每时每刻都会遇到的组合层次就是生命的层次 hierarchy of life。每个人都可以归结为器官系统 organ system，器官系统是由组织构成的，组织是由细胞构成的，分子是由原子构成的。事实上至少在宏观，最后两个层次适用于所有物质。此外，每个级别都继承了它们子级别的所有属性。

In this particular example, there are also emergent properties—functions that are not seen at the lower level (e.g., cognition is not a property of neurons but is of the brain)—and a scalar quality (molecules are bigger than atoms, cells are bigger than molecules, etc.). Both of these concepts commonly exist in compositional hierarchies, but they are not a required general property. These level hierarchies are characterized by bi-directional causation. Upward causation involves lower-level entities causing some property of a higher level entity; children entities may interact to yield parent entities, and parents are composed at least partly by their children. Downward causation refers to the effect that the incorporation of entity x into a higher-level entity can have on x's properties and interactions. Furthermore, the entities found at each level are autonomous.

In this particular example, there are also emergent properties—functions that are not seen at the lower level (e.g., cognition is not a property of neurons but is of the brain)—and a scalar quality (molecules are bigger than atoms, cells are bigger than molecules, etc.). Both of these concepts commonly exist in compositional hierarchies, but they are not a required general property. These level hierarchies are characterized by bi-directional causation. Upward causation involves lower-level entities causing some property of a higher level entity; children entities may interact to yield parent entities, and parents are composed at least partly by their children. Downward causation refers to the effect that the incorporation of entity x into a higher-level entity can have on xs properties and interactions. Furthermore, the entities found at each level are autonomous.

在这个特定的例子中，还有一些涌现特性 emergent properties——在较低层次上看不到的功能（例如认知 cognition不是神经元的特性，而是大脑的特性）——以及标度特性（分子比原子大，细胞比分子大等等）。这两个概念通常都存在于组合层次结构中，但它们不是必要的一般属性。这些层次结构是拥有属性的双向因果关系。涉及较低层次实体的上向因果关系 Upward causation导致较高层次实体的某些属性；子实体可以相互作用产生父实体，并且父实体至少部分由他们的子实体组成。下向因果关系 Downward causation是指将实体 x 并入更高级别的实体可能对 x 的属性和相互作用产生的影响。此外，每个级别上的实体都是自治的 autonomous。

Contexts and applications 语境和应用

= = 上下文和应用 = = < ! ——如果你更改了这个部分的标题，请也更改文章中链接到它的 wikileink！-->

According to Kulish, V. V. (2002), almost every system of organization applied to the world is arranged hierarchically. By their common definitions, every nation has a government and every government is hierarchical. Socioeconomic systems are stratified into a social hierarchy (the social stratification of societies), and all systematic classification schemes (taxonomies) are hierarchical. Most organized religions, regardless of their internal governance structures, operate as a hierarchy under God. Many Christian denominations have an autocephalous ecclesiastical hierarchy of leadership. Families are viewed as a hierarchical structure in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in a family tree) and inheritance (succession and heirship). All the requisites of a well-rounded life and lifestyle can be organized using Maslow's hierarchy of human needs. Learning must often follow a hierarchical scheme—to learn differential equations one must first learn calculus; to learn calculus one must first learn elementary algebra; and so on. Even nature itself has its own hierarchies, as numerous schemes such as Linnaean taxonomy, the organization of life, and biomass pyramids attempt to document. Hierarchies are so infused into daily life that they are viewed as trivial.

According to Kulish, V. V. (2002), almost every system of organization applied to the world is arranged hierarchically. By their common definitions, every nation has a government and every government is hierarchical. Socioeconomic systems are stratified into a social hierarchy (the social stratification of societies), and all systematic classification schemes (taxonomies) are hierarchical. Most organized religions, regardless of their internal governance structures, operate as a hierarchy under God. Many Christian denominations have an autocephalous ecclesiastical hierarchy of leadership. Families are viewed as a hierarchical structure in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in a family tree) and inheritance (succession and heirship). All the requisites of a well-rounded life and lifestyle can be organized using Maslow's hierarchy of human needs. Learning must often follow a hierarchical scheme—to learn differential equations one must first learn calculus; to learn calculus one must first learn elementary algebra; and so on. Even nature itself has its own hierarchies, as numerous schemes such as Linnaean taxonomy, the organization of life, and biomass pyramids attempt to document. Hierarchies are so infused into daily life that they are viewed as trivial.

根据 Kulish v. v. (2002) 的研究，几乎世界上所有的组织系统都是等级分明的。按照他们的共同定义，每个国家 nation都有一个政府，每个政府都是层级制度。社会经济系统被划分为一个社会层级（社会阶层 social stratification） ，所有的系统分类方案(分类法)都是分等级的。大多数有组织的宗教，不管它们的内部管理结构如何，都在上帝之下按照层级运作。许多基督教教派 Christian denominations都有一个独立的教会领导层次。从表亲关系（例如，第一代表兄弟，第二代表兄弟等等）、祖先(由家谱 family tree描述)和继承(继承和继承权)的角度来看，家庭可以看作是一种等级结构。一个全面的生活和生活方式的所有必需品都可以用马斯洛人类需求层次 Maslow's hierarchy of human needs来组织。学习通常必须遵循一个层次框架——学习微分方程必须先学习微积分；学习微积分必须先学习初等代数等等。甚至自然本身也有它自己的层级，许多方案试图记录下来，例如林奈分类学、生命的组织和生物量金字塔。层级制度如此深入到日常生活中，以至于人们认为它们微不足道。

While the above examples are often clearly depicted in a hierarchical form and are classic examples, hierarchies exist in numerous systems where this branching structure is not immediately apparent. For example, most postal code systems are hierarchical. Using the Canadian postal code system as an example, the top level's binding concept is the "postal district", and consists of 18 objects (letters). The next level down is the "zone", where the objects are the digits 0–9. This is an example of an overlapping hierarchy, because each of these 10 objects has 18 parents. The hierarchy continues downward to generate, in theory, 7,200,000 unique codes of the format A0A 0A0 (the second and third letter position allow 20 objects each). Most library classification systems are also hierarchical. The Dewey Decimal System is regarded as infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.

While the above examples are often clearly depicted in a hierarchical form and are classic examples, hierarchies exist in numerous systems where this branching structure is not immediately apparent. For example, most postal code systems are hierarchical. Using the Canadian postal code system as an example, the top level's binding concept is the "postal district", and consists of 18 objects (letters). The next level down is the "zone", where the objects are the digits 0–9. This is an example of an overlapping hierarchy, because each of these 10 objects has 18 parents. The hierarchy continues downward to generate, in theory, 7,200,000 unique codes of the format A0A 0A0 (the second and third letter position allow 20 objects each). Most library classification systems are also hierarchical. The Dewey Decimal System is regarded as infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.

虽然上面的例子通常以层次形式清楚地描述并且很经典，但是层次结构也存在于许多分支结构并不是很明显的系统中。例如，大多数邮政编码 postal code系统是分层的。以加拿大邮政编码系统为例，顶层的概念是“邮区 postal district”，由18个对象(字母)组成。下一层是“地区 zone”，其中的对象是数字0-9。这是重叠层次结构的一个例子，因为这10个对象中的每一个都有18个父对象。层次继续向下生成，理论上存在720万个 A0A 0A0 (第二个和第三个字母的位置每个允许20个对象)格式的唯一代码。大多数图书分类法系统也是分层级的。杜威十进制图书分类法 Dewey Decimal System被认为是无限层次的，因为在小数点之后可以使用的数字数量是没有限制的。

文件:Organizational chart.svg

A simple organizational hierarchy depicted in the form of a tree. Diagrams like this are called organizational charts.

organizational hierarchy depicted in the form of a tree. Diagrams like this are called organizational charts.]]

以树的形式描述的组织层次结构。这样的图被称为组织结构图 organizational charts。

Organizations 组织

Main articles: Organizational structure and Hierarchical organization

Organizations can be structured as a dominance hierarchy. In an organizational hierarchy, there is a single person or group with the most power and authority, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including governments, companies, militia and organized religions. The units or persons within an organization are depicted hierarchically in an organizational chart.

Organizations can be structured as a dominance hierarchy. In an organizational hierarchy, there is a single person or group with the most power and authority, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including governments, companies, militia and organized religions. The units or persons within an organization are depicted hierarchically in an organizational chart.

组织 Organizations可以被构建成一个支配等级 dominance hierarchy。在一个组织的层次结构中，有一个人或一个群体拥有最大的权力和权威，每个后续的层次代表一个较小的权威。大多数组织都是这样构建的，包括政府、公司、民兵和有组织的宗教。组织中的单位或人员在组织结构图中按等级进行描述。

In a reverse hierarchy, the conceptual pyramid of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This mode represents the idea that members of the higher rankings are responsible for the members of the lower rankings.

In a reverse hierarchy, the conceptual pyramid of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This mode represents the idea that members of the higher rankings are responsible for the members of the lower rankings.

在反向层次结构 reverse hierarchy中，权力的概念金字塔是颠倒的，因此顶点在底部，底部在顶部。这种模式代表了等级较高的成员对等级较低的成员负责的思想。

Life 生命

Main article: Biological organisation § Fundamentals

Empirically, we observe in nature a large proportion of the (complex) biological systems, they exhibit hierarchic structure. On theoretical grounds we could expect complex systems to be hierarchies in a world in which complexity had to evolve from simplicity. System hierarchies analysis performed in the 1950s, laid the empirical foundations for a field that would be, from the 1980s, hierarchical ecology.

Empirically, we observe in nature a large proportion of the (complex) biological systems, they exhibit hierarchic structure. On theoretical grounds we could expect complex systems to be hierarchies in a world in which complexity had to evolve from simplicity. System hierarchies analysis performed in the 1950s, laid the empirical foundations for a field that would be, from the 1980s, hierarchical ecology.

在实践中，我们观察到自然界中很大一部分(复杂的)生物系统呈现出层次结构。从理论上讲，在一个复杂性必须从简单性演化而来的世界中，我们可以预期复杂系统具有层次结构。20世纪50年代进行的系统层次分析，为从20世纪80年代开始的层次生态学领域奠定了经验基础。

The theoretical foundations are summarized by Thermodynamics. When biological systems are modeled as physical systems, in its most general abstraction, they are thermodynamic open systems that exhibit self-organised behavior, and the set/subset relations between dissipative structures can be characterized in a hierarchy.

The theoretical foundations are summarized by Thermodynamics. When biological systems are modeled as physical systems, in its most general abstraction, they are thermodynamic open systems that exhibit self-organised behavior, and the set/subset relations between dissipative structures can be characterized in a hierarchy.

这些理论基础由热力学的方法来总结。当生物系统被模拟为物理系统时，在其最一般的抽象中，它们是表现出自组织行为的热力学开放系统，耗散结构 dissipative structures之间的集合/子集关系可以用层次来刻画。

Computer graphic imaging 计算机图形成像

Main article: 3D modeling

CGI and computer animation programs mostly use hierarchies for models. On a 3D model of a human for example, the chest is a parent of the upper left arm, which is a parent of the lower left arm, which is a parent of the hand. This is used in modeling and animation for almost everything built as a 3D digital model.

CGI and computer animation programs mostly use hierarchies for models. On a 3D model of a human for example, the chest is a parent of the upper left arm, which is a parent of the lower left arm, which is a parent of the hand. This is used in modeling and animation for almost everything built as a 3D digital model.

CGI 和计算机动画程序 computer animation programs大多使用层次结构建模。例如在一个人体的三维模型中，胸部是左上臂的母体，而左上臂是左下臂的母体。层次结构用于几乎所有三维数字模型的建模和动画。

Linguistics 语言学

Many grammatical theories, such as phrase-structure grammar, involve hierarchy.

Many grammatical theories, such as phrase-structure grammar, involve hierarchy.

许多语法理论都涉及层次，例如短语结构语法。

Direct–inverse languages such as Cree and Mapudungun distinguish subject and object on verbs not by different subject and object markers, but via a hierarchy of persons.

Direct–inverse languages such as Cree and Mapudungun distinguish subject and object on verbs not by different subject and object markers, but via a hierarchy of persons.

克里语(Cree)和马普敦根语(Mapudungun)等正动-反动语言 Direct–inverse languages不是通过不同的主语和宾语标记来区分动词的主语和宾语，而是通过人的层级来区分。

In this system, the three (or four with Algonquian languages) persons are placed in a hierarchy of salience. To distinguish which is subject and which object, inverse markers are used if the object outranks the subject.

In this system, the three (or four with Algonquian languages) persons are placed in a hierarchy of salience. To distinguish which is subject and which object, inverse markers are used if the object outranks the subject.

在这个系统中，三个人(或者阿尔冈昆诸语言中四个人)被放在显著的等级中。为了区分哪个是主语，哪个是宾语，如果宾语的级别高于主语，则使用反向标记。

On the other hand, languages include a variety of phenomena that are not hierarchical. For example, the relationship between a pronoun and a prior noun phrase to which it refers, commonly crosses grammatical boundaries in non-hierarchical ways.

On the other hand, languages include a variety of phenomena that are not hierarchical. For example, the relationship between a pronoun and a prior noun phrase to which it refers, commonly crosses grammatical boundaries in non-hierarchical ways.

另一方面，语言包括多种不分等级的现象。例如，代词和它所指的名词短语之间的关系，通常以非等级的方式跨越语法界限。

Music 音乐

The structure of a musical composition is often understood hierarchically (for example by Heinrich Schenker (1768–1835, see Schenkerian analysis), and in the (1985) Generative Theory of Tonal Music, by composer Fred Lerdahl and linguist Ray Jackendoff). The sum of all notes in a piece is understood to be an all-inclusive surface, which can be reduced to successively more sparse and more fundamental types of motion. The levels of structure that operate in Schenker's theory are the foreground, which is seen in all the details of the musical score; the middle ground, which is roughly a summary of an essential contrapuntal progression and voice-leading; and the background or Ursatz, which is one of only a few basic "long-range counterpoint" structures that are shared in the gamut of tonal music literature.

The structure of a musical composition is often understood hierarchically (for example by Heinrich Schenker (1768–1835, see Schenkerian analysis), and in the (1985) Generative Theory of Tonal Music, by composer Fred Lerdahl and linguist Ray Jackendoff). The sum of all notes in a piece is understood to be an all-inclusive surface, which can be reduced to successively more sparse and more fundamental types of motion. The levels of structure that operate in Schenker's theory are the foreground, which is seen in all the details of the musical score; the middle ground, which is roughly a summary of an essential contrapuntal progression and voice-leading; and the background or Ursatz, which is one of only a few basic "long-range counterpoint" structures that are shared in the gamut of tonal music literature.

音乐作品的结构通常是按层次来理解的（例如 Heinrich Schenker （1768-1835，见 Schenkerian 分析） ，以及 Fred Lerdahl 和语言学家 Ray Jackendoff 1985年的《调性音乐的生成理论 Generative Theory of Tonal Music》）。一段乐曲中所有音符的总和被理解为一个包含所有音符的曲面，它可以简化为更加稀疏和更加基本的运动类型。在申克的理论中运作的结构层次包括前景层次，这在音乐乐谱的所有细节中都可以看到；中间层次，大致上是对位进行和主音的总结; 背景层次，即 Ursatz，这是调性音乐著作中仅有的几个共享的基本“远距离对位”结构之一。

The pitches and form of tonal music are organized hierarchically, all pitches deriving their importance from their relationship to a tonic key, and secondary themes in other keys are brought back to the tonic in a recapitulation of the primary theme. Susan McClary connects this specifically in the sonata-allegro form to the feminist hierarchy of gender (see above) in her book Feminine Endings, even pointing out that primary themes were often previously called "masculine" and secondary themes "feminine."

The pitches and form of tonal music are organized hierarchically, all pitches deriving their importance from their relationship to a tonic key, and secondary themes in other keys are brought back to the tonic in a recapitulation of the primary theme. Susan McClary connects this specifically in the sonata-allegro form to the feminist hierarchy of gender (see above) in her book Feminine Endings, even pointing out that primary themes were often previously called "masculine" and secondary themes "feminine."

调性音乐 tonal music的音高 pitches和形式是按等级组织的，所有音高的重要性来自于它们与主调 tonic key的关系，其他调中的次要主题在主要主题的再现中被带回主调。苏珊·麦克拉里（Susan McClary）在她的《女性结局（(Feminine Endings）一书中将这一点在奏鸣曲式中具体地与女性主义的性别层次结构(见上文)联系起来，甚至指出，主要的主题通常以前被称为“阳性” ，而次要主题被称为“阴性”。

Criticisms of views, concerning distinctions of type and categories well as distinguishability 关于类型和范畴的区别以及区别性的观点的批判

In the work of diverse theorists such as William James (1842–1910), Michel Foucault (1926–1984) and Hayden White, important critiques of hierarchical epistemology are advanced. James famously asserts in his work "Radical Empiricism" that clear distinctions of type and category are a constant but unwritten goal of scientific reasoning, so that when they are discovered, success is declared. But if aspects of the world are organized differently, involving inherent and intractable ambiguities, then scientific questions are often considered unresolved.

In the work of diverse theorists such as William James (1842–1910), Michel Foucault (1926–1984) and Hayden White, important critiques of hierarchical epistemology are advanced. James famously asserts in his work "Radical Empiricism" that clear distinctions of type and category are a constant but unwritten goal of scientific reasoning, so that when they are discovered, success is declared. But if aspects of the world are organized differently, involving inherent and intractable ambiguities, then scientific questions are often considered unresolved.

在威廉·詹姆斯(1842-1910)、米歇尔·福柯(1926-1984)和海登·怀特等不同理论家的著作中，提出了对等级认识论的重要批判。詹姆斯在他的著作“彻底经验主义 Radical Empiricism”中断言，明确类型和范畴的区分是科学推理的一个恒定但不成文的目标，因此当他们被发现的时候，就是宣告成功的时候。但是，如果世界的各个方面以不同的方式组织起来，涉及到固有的和难以解决的模糊性，那么科学问题通常被认为是没有解决的。

Hierarchy in ethics emerged in Western Europe, West Asia and North Africa around the 1600s. In this aspect, the term hierarchy refers to how distinguishable they are from real to unreal. Feminists, Marxists, anarchists, communists, critical theorists and others, all of whom have multiple interpretations, criticize the hierarchies commonly found within human society, especially in social relationships. Hierarchies are present in all parts of society: in businesses, schools, families, etc. These relationships are often viewed as necessary. Entities that stand in hierarchical arrangements are animals, humans, plants, etc.

Hierarchy in ethics emerged in Western Europe, West Asia and North Africa around the 1600s. In this aspect, the term hierarchy refers to how distinguishable they are from real to unreal. Feminists, Marxists, anarchists, communists, critical theorists and others, all of whom have multiple interpretations, criticize the hierarchies commonly found within human society, especially in social relationships. Hierarchies are present in all parts of society: in businesses, schools, families, etc. These relationships are often viewed as necessary. Entities that stand in hierarchical arrangements are animals, humans, plants, etc.

17世纪左右，伦理学的等级制度在西欧、西亚和北非出现。在这方面，术语层次是指它们从真实到非真实是如何区分的。女权主义者 Feminists、马克思主义者 Marxists、无政府主义者 anarchists、共产主义者 communists、批判理论家 critical theorists以及其他人，他们对人类社会中普遍存在的等级制度，尤其是社会关系中的等级制度，都有不同的解释。等级制度存在于社会的各个部分：企业、学校、家庭等等。这些关系通常被认为是必要的。站在等级结构中的实体是动物、人类、植物等等。

Ethics, behavioral psychology, philosophies of identity 伦理，行为心理学，同一哲学 philosophies of identity

Career-oriented purposes can be diagrammed using a hierarchy describing how less important actions support a larger goal.

Career-oriented purposes can be diagrammed using a hierarchy describing how less important actions support a larger goal.

以事业为导向的目标可以用层次来描述，描述次要的行动如何支持一个更大的目标。

In ethics, various virtues are enumerated and sometimes organized hierarchically according to certain brands of virtue theory.

In ethics, various virtues are enumerated and sometimes organized hierarchically according to certain brands of virtue theory.

伦理学 ethics中列举的各种美德 virtues，有时根据某些美德理论有层次地组织起来。

In some of these random examples, there is an asymmetry of 'compositional' significance between levels of structure, so that small parts of the whole hierarchical array depend, for their meaning, on their membership in larger parts. There is a hierarchy of activities in human life: productive activity serves or is guided by the moral life; the moral life is guided by practical reason; practical reason (used in moral and political life) serves contemplative reason (whereby we contemplate God). Practical reason sets aside time and resources for contemplative reason.

In some of these random examples, there is an asymmetry of 'compositional' significance between levels of structure, so that small parts of the whole hierarchical array depend, for their meaning, on their membership in larger parts. There is a hierarchy of activities in human life: productive activity serves or is guided by the moral life; the moral life is guided by practical reason; practical reason (used in moral and political life) serves contemplative reason (whereby we contemplate God). Practical reason sets aside time and resources for contemplative reason.

在一些随机的例子中，结构层次之间的“组合”意义是不对称的，因此整个层次数组的小部分依赖于它们在更大部分中的隶属关系。人类生活中的活动有一种层次：生产活动服务于或被道德生活所指导；道德生活被实践理性所指导；实践理性(用于道德和政治生活)服务于冥想理性(我们借以思考上帝)。实用理性为深思的理由留出时间和资源。

Examples of other applications 其他应用的例子

<！-- 本节仅供直接举例说明等级制度，仅供上文未概述的例子参考。方法学属于下面的适当部分。相关概念列在“请参阅”部分。如果您不确定链接属于哪个位置，请将其放在“请参阅”部分。提示: 如果你不能很容易地说出它的层次结构是什么，它就不属于这里(但这并不意味着如果你能说出它的层次结构是什么，它就属于这里)。-->

Information-based 基于信息

= = = = = 信息为基础 = = = = < ! -- 是的，连字符是正确的，即使按照 wp: hyphen ー这个类别的逻辑标题是“信息为基础的层次结构” -- >

• Library classification 图书馆分类法

• • Dewey Decimal Classification 杜威十进制分类法

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

• Roads 道路

• • Streets 街道

• Settlement hierarchy 聚落等级

• • As of 2010 截至2010年

• • As of 2100 (estimate according to Doxiadis, 1968) 截至2010年（根据Doxiadis 1968年的估计）

Linguistics-based 基于语言学

= = = 基于语言学的 = = = = = < ! -- 是的，连字符是正确的，即使按 wp: hyphen ー这个类别的逻辑标题是“基于语言学的层次” -- >

• Language family tree 语言家谱

• Levels of adequacy for evaluating grammars 评价语法的充分性等级

• Direct–inverse languages 正对-反对语言

• Structural linguistics 结构语言学

• • Parse tree 分析树

• • Formal grammars 形式语法

• • Abstract syntax tree 抽象语法树

• Evolution of basic color terminology in languages 语言中基本颜色术语的演变

Power- or authority-based 基于权利或权威

= = = 权力-或基于权威 = = = = = < ! -- 是的，连字符是正确的，即使按照 wp: hyphen ー这个类别的逻辑标题是“权力-或基于权威的层次结构” -- >

• Aristocratic hierarchies 贵族等级

• • In Europe 欧洲

• • In China 中国

• Ecclesiastical hierarchies 教会等级

• • Catholic Church hierarchy 天主教会圣统制

• • LDS Church hierarchy 摩门教教堂层次结构

• • Kimbanguist Church hierarchy Kimbanguist教堂层次结构

• • Raëlism Church hierarchy Raëlism教堂层次结构

• • see also autocephaly 另见自治 autocephaly

• Administrative branch of government geographical hierarchies

• • Administrative divisions of China

• • Administrative divisions of India

• • Administrative divisions of the United States

• • Administrative divisions of Russia

• Political party hierarchies 政党的层次结构

• • Nazi Party 纳粹党

• • • SS

• • • Hierarchy of subdivisions within the Gau Gau内部的细分层次结构

• • Communist Party of the Soviet Union 苏联共产党

• • Communist Party of China 中国共产党

• Chain of command 指挥系统

• • Military ranks 军衔

• • Military units 部队

• • U.S. Military Combatant Commands 美国军事战斗司令部

• Intraspecial dominance 种内优势

• • Pecking order 啄食顺序

• Social classes 社会阶级

• • Caste system in India 印度种姓制度

• • Hierarchical structure of Feudal Japan 封建日本的等级结构

• • White racist hierarchy 白人种族主义的层次结构

• • Hierarchy of Exclusion 层次结构的排斥 (Ender's Game 安德的游戏)

= = = = 基于价值的价值 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

• Hierarchy of genres in art 艺术流派的等级

• Evidence 证据

• Human needs 人类需求

• Precious substances 贵重物品

• Judicial hierarchy of social values 社会价值的司法等级

Perception-based 基于感知

= = = 基于感知的感知 = = = = = < ! -- 是的，连字符是正确的，即使按照 wp: hyphen ー这个类别的逻辑标题是“基于感知的层级” -- >

• Color wheel 色环

• • Primary colors 三原色

• • • Secondary colors 二次色

• • • • Tertiary colors 三次色

History-based 基于历史

= = = 历史为基础 = = = = < ! -- 是的，连字符是正确的，即使按 wp: hyphen ー这个类别的逻辑标题是“基于历史的层级” -- >

• Three-age system 史前三时代分类系统

• Cyclic theory of civilization 文明的循环说

• • Oswald Spengler 奥斯瓦尔德·斯宾格勒

• • Arnold J. Toynbee

• Spiral dynamics 螺旋动力学

Science-based 基于科学

< ! -- wp: hyphen ー这个类别的逻辑标题是"科学层次">

• Hierarchy of organization within the Universe 宇宙中的组织层次

• Star systems 恒星系统

• Biological classification 生物分类

• Biological organization 生物组织

• Phylogenetic tree 进化树

• Evolutionary development 进化发展

• Hierarchy of ecological georegions 生态地理等级

Technology-based 基于技术

• Memory hierarchy 存储层次结构

• • Cache hierarchy 缓存层次

• Clusters 聚类

• Class constructs 类构造

• Data organization 数据组织

• • Hierarchical query 层次查询

• Data storage 数据存储

• • Computer files (Macintosh) 计算机文件(Macintosh)

• Devices 设备

• IP addresses IP地址

• Memory 内存

• • Virtual memory allocation 虚拟内存分配

• Networks 网络

• Radio cells 无线电单元

• States (configurations) 状态（配置）

• Web addresses 网页地址

• Structure 结构

• • Data Structure 数据结构

• Inheritance (object-oriented programming) 继承（面向对象编程）

Religion-based 基于宗教

= = = = 基于宗教的 = = = = = = = = = = 是的，连字符是正确的，即使按照 wp: hyphen ー这个类别的逻辑标题是“基于宗教和神话的等级制度” -- >

• Levels of consciousness 意识层次

• • Chakras 轮穴

• • Great chain of being 存在之链

• • G.I. Gurdjieff

• • Timothy Leary 蒂莫西·利里

• Levels of spiritual development 精神发展层次

• • In Theravada Buddhism 南传上座部佛教

• • In Mahayana Buddhism 大乘佛教

• • In Theosophy 神智学

• Ages in the evolution of society 社会发展的时代

• • In Astrology 占星术

• • In Hellenism (the Ancient Greek Religion) 希腊文化(古希腊的宗教)

• • Dispensations in Protestantism 新教的豁免

• • Dispensations in Mormonism 摩门教特许

• Degrees of communion between various Christian churches 不同基督教会之间的交流程度

• UFO religions UFO宗教

• • Command hierarchy of the Ashtar Galactic Command flying saucer fleet 阿斯塔银河司令部”飞碟舰队的指挥等级

• Deities 神灵

• • In Japanese Buddhism 日本佛教

• • In Theosophy 神智学

• Angels 天使

• • In Christianity 基督教

• • In Islam 伊斯兰教

• • In Judaism 犹太教

• • • Kabbalistic 卡巴拉教派

• • In Zoroastrianism 琐罗亚斯德教

• Devils and Demons 撒旦与恶魔

• • Devils

• • Demons

• Hells 地狱

• • In Catholicism (Nine Levels of Hell) 天主教(九层地狱)

• • In Buddhism (Sixteen Levels of Hell) 佛教(十六层地狱)

• Religions in society 社会上的宗教

• (organizational hierarchies are listed under 组织和层次列举于"Power- or authority-based")

Methods using hierarchy 使用层次结构的方法

= = = = 使用层次结构的方法 = = = < ! -- 参见“ further applications” -- >

{ columns-list

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• Analytic Hierarchy Process 层次分析法

• • Hierarchical Decision Process 层次决策过程

• Hierarchic Object-Oriented Design 分层的面向对象设计

• Hierarchical Bayes model 贝叶斯层次模型

• Hierarchical clustering 层次聚类

• • Hierarchical clustering of networks 网络层次聚类

• Hierarchical constraint satisfaction 层次约束满足

• Hierarchical linear modeling 多层阶线性模型

• Hierarchical modulation 分层调制

• Hierarchical proportion 分层比例

• Hierarchical radial basis function 层次径向基函数

• Hierarchical storage management 分级存储管理

• Hierarchical task network 分层任务网络

• Hierarchical temporal memory 皮质学习算法

--Jxzhou（讨论）根据上下文感觉应该是计算机方面的术语，memory是内存，但是维基百科这个词条对应的连接也是神经生物学相关的HTM算法

• Hierarchical token bucket 分层令牌桶

• Hierarchical visitor pattern 分层访问者模式

• Presentation-abstraction-control 表示-抽象-控制

• • Hierarchical-Model-View-Controller 层次-模型-视图-控制器

}}

See also 参见

= = 也见 = = < ! -- 参见“ further applications” -- >

{ columns-list

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• Anarchy 无政府状态

• Class browser 类浏览器

• Forms of government 政体

• Graph theory 图论

• Heterarchy 差异化结构

• Hierarchical classifier 层次化分类器

• Hierarchical epistemology 分层认识论

• Hierarchical hidden Markov model 分层隐马尔科夫模型

• Hierarchical INTegration 阶层统整

• Hierarchical Music Specification Language 分级音乐规范语言

• Hierarchy Open Service Interface Definition 层次结构开放服务接口定义

• Hierarchy problem 等级问题

• 模板:Section link

• Instrumental value 工具价值

• Layer (disambiguation) 层（消歧）

• Multilevel model 多层模型

• Multitree 多树

• Ordinary (officer) 牧师

• • 模板:Section link

• • List of Coptic Orthodox Popes of Alexandria 亚历山大科普特东正教教皇名单

• Peter Principle 彼得原理

• Ring (computer security) 环（计算机安全）

• Social dominance theory 社会支配论

}}

Strucure-related concepts 结构相关概念

(For example, in 模板:Section link)

(For example, in )

(例如，在

• Is-a

• • Hypernymy 上位关系 (and supertype 超类型)

• • Hyponymy 下义关系 (and subtype 亚类型)

• Has-a

• • Holonymy 整体-部分

• • Meronymy 部分-整体

Further reading

• Ahl, Valerie; Allen, Timothy F. H. (1996). Hierarchy Theory. New York: Columbia University Press. ISBN 0-231-08481-1. 

• Akl, Selim G.; Taylor, Peter D. (1983). Cryptographic solution to a multilevel security problem (PDF). Advances in Cryptology: Proceedings of CRYPTO '82. New York: Plenum Publishing Corporation. pp. 237–249. ISBN 0-306-41366-3. Archived from the original (PDF) on 2002-12-29.

• Carbone, June; Cahn, Naomi (2013). "Is marriage for rich men?". Nevada Law Journal. William S. Boyd School of Law. 13 (2): 6.

• Ckurshumova, Wenzislava (2007). "Regulatory hierarchies in auxin signal transduction and vascular tissue development". Dissertation Abstracts International. University of Toronto. 68 (5): section B. ISBN 978-0-494-27682-2. Ph.D. dissertation.{{cite journal}}: CS1 maint: postscript (link)

• Galindo, Cipriano; Fernández-Madrigal, Juan-Antonio (2007). Kacprzyk, Janusz. ed. Multiple Abstraction Hierarchies for Mobile Robot Operation in Large Environments. Studies in Computational Intelligence. Berlin: Springer Berlin Heidelberg. ISBN 978-3-540-72688-3. 

• Nelson, Julie (1992). "Gender, Metaphor and the Definition of Economics". Economics and Philosophy. 8 (1): 103–25. doi:10.1017/S026626710000050X.

• Pumain, Denise (2006). Hierarchy in Natural and Social Sciences. New York, New York: Springer-Verlag. ISBN 978-1-4020-4126-6. 

• Rosenbaum, A. (2000) (in French). Les représentations hiérarchiques en philosophie. Paris: Desclee de Brouwer. 

• Shahbaba, Babak (2007). "Improving classification models when a class hierarchy is available". Dissertation Abstracts International. University of Toronto. 68 (6): section B. ISBN 978-0-494-28076-8. Ph.D. dissertation.{{cite journal}}: CS1 maint: postscript (link)

• • Also includes full copies of:

• • Shahbaba, Babak; Neal, Radford M. (2007). "Improving Classification When a Class Hierarchy is Available Using a Hierarchy-Based Prior" (PDF). Bayesian Analysis. Carnegie Mellon University, Pittsburgh, Pennsylvania: International Society for Bayesian Analysis. 2 (1): 221–228. arXiv:math/0510449. Bibcode:2005math.....10449S. doi:10.1214/07-ba209. ISSN 1936-0975. Archived from the original (PDF) on 2015-09-03.

• • Shahbaba, Babak; Neal, Radford M. (2006). "Gene function classification using Bayesian models with hierarchy-based priors". BMC Bioinformatics. London, England: BioMed Central. 7: 448. arXiv:q-bio/0605015. Bibcode:2006q.bio.....5015S. doi:10.1186/1471-2105-7-448. ISSN 1471-2105. PMC 1618412. PMID 17038174.

External links

• 模板:Commons category-inline

• Principles and annotated bibliography of hierarchy theory

• Summary of the Principles of Hierarchy Theory — S.N. Salthe

Category:Patterns

分类: 模式

Category:Structure

类别: 结构

Category:Political culture

类别: 政治文化

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