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Ordered Monoid
In mathematics, an ordered semigroup is a semigroup (''S'',•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that ''x'' ≤ ''y'' implies z•x ≤ z•y and x•z ≤ y•z for all ''x'', ''y'', ''z'' in ''S''. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order that makes them ordered semigroups. The terms ''posemigroup'', ''pogroup'' and ''pomonoid'' are sometimes used, where "po" is an abbreviation for "partially ordered". The positive integers, the nonnegative integers and the integers form respectively a posemigroup, a pomonoid, and a pogroup under addition and the natural ordering. Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=". A morphism or homomorphism of posemigroups is a semigroup homomorphism that ''preserves'' the order (equivalently, that is monotonically increasing). Category-theoretic interpret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then ''a'' + ''g'' ≤ ''b'' + ''g'' and ''g'' +'' a'' ≤ ''g'' +'' b''. An element ''x'' of ''G'' is called positive if 0 ≤ ''x''. The set of elements 0 ≤ ''x'' is often denoted with ''G''+, and is called the positive cone of ''G''. By translation invariance, we have ''a'' ≤ ''b'' if and only if 0 ≤ -''a'' + ''b''. So we can reduce the partial order to a monadic property: if and only if For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially orderable group if and only if there exists a subset ''H'' (which is ''G''+) of ''G'' such that: * 0 ∈ ''H'' * if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a'' + ''b'' ∈ ''H'' * if ''a'' ∈ ''H'' then -''x'' + ''a'' + '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automata ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonnegative Integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup Homomorphism
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonically Increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an abstractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
