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Algebraic Lattice
In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions of compactness in mathematics.) Formal definition In a partially ordered set (''P'',≤) an element ''c'' is called ''compact'' (or ''finite'') if it satisfies one of the following equivalent conditions: * For every directed subset ''D'' of ''P'', if ''D'' has a supremum sup ''D'' and ''c'' ≤ sup ''D'' then ''c'' ≤ ''d'' for some element ''d'' of ''D''. * For every ideal ''I'' of ''P'', if ''I'' has a supremum sup ''I'' and ''c'' ≤ sup ''I'' then ''c'' is an element of ''I''. If the poset ''P'' additionally is a join-semila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Least Element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Definitions Let (P, \leq) be a preordered set and let S \subseteq P. An element g \in P is said to be if g \in S and if it also satisfies: :s \leq g for all s \in S. By switching the side of the relation that s is on in the above definition, the definition of a least element of S is obtained. Explicitly, an element l \in P is said to be if l \in S and if it also satisfies: :l \leq s for all s \in S. If (P, \leq) is also a partially ordered set then S can have at most one greatest element and it can have at most one least element. Whenever a greatest element of S exists and is unique then this element is called greatest element of S. The terminology least element of S is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Universal Algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of studythis is the subject of group theory and ring theory in universal algebra, the object of study is the possible types of algebraic structures and their relationships. Basic idea In universal algebra, an (or algebraic structure) is a set ''A'' together with a collection of operations on ''A''. Arity An ''n''- ary operation on ''A'' is a function that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a '' constant'', often denoted by a letter like ''a''. A 1-ary operation (or '' unary operation'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Domain Theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to topology. Motivation and intuition The primary motivation for the study of domains, which was initiated by Dana Scott in the late 1960s, was the search for a denotational semantics of the lambda calculus. In this formalism, one considers "functions" specified by certain terms in the language. In a purely syntactic way, one can go from simple functions to functions that take other functions as their input arguments. Using again just the syntactic transformations available in this formalism, one can obtai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Directed Complete Partial Order
In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central role in theoretical computer science: in denotational semantics and domain theory. Definitions The term complete partial order, abbreviated cpo, has several possible meanings depending on context. A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. (A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset.) In the literature, dcpos sometimes also appear under the label up-complete poset. A pointed directed-complete partial order (pointed dcpo, sometimes abbreviated cppo), is a dcpo with a least element (usually denoted \bot). Formulated differently, a pointed dcpo has a supremum for every directed ''or empty'' s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Join-prime
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' structures all admi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
