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Nonempty, upper-bounded, downward-closed subset
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion
Ideal_(order_theory)
Submodule of a mathematical ring
mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers
Ideal_(ring_theory)
Argument concerning political or social arrangements under favorable assumptions
philosophy, an ideal theory is a theory which specifies the optimal societal structure based on idealised assumptions and normative theory. It stems from
Ideal_theory_(politics)
Topics referred to by the same term
considered in abstract algebra Ideal, special subsets of a semigroup Ideal (order theory), special kind of lower sets of an order Ideal on a set, a collection
Ideal
Non-empty family of sets that is closed under finite unions and subsets
subsets representing "large" sets Ideal (order theory) – Nonempty, upper-bounded, downward-closed subset Ideal (ring theory) – Submodule of a mathematical
Ideal_on_a_set
Branch of mathematics
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing
Order_theory
Ideals in a Boolean algebra can be extended to prime ideals
appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article
Boolean_prime_ideal_theorem
Branch of number theory
factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions
Algebraic_number_theory
Conspiracy theory regarding a totalitarian world government
The New World Order (NWO) is a term often used in conspiracy theories which speculate about a secretly emerging totalitarian world government. The common
New World Order conspiracy theory
New_World_Order_conspiracy_theory
Set of a ring's prime ideals
more general notion of prime and maximal spectra of lattices, see Ideal (order theory) § Prime and maximal spectra. The idea of the spectrum of a ring
Spectrum_of_a_ring
Philosophical theory attributed to Plato
are various abstract ideals that exist even outside of human minds and that constitute the basis of reality. Thus, Plato's Theory of Forms is a type of
Theory_of_forms
Ring ideal generated by a single element of the ring
In mathematics, specifically ring theory, a principal ideal is an ideal I {\displaystyle I} in a ring R {\displaystyle R} that is generated by a single
Principal_ideal
In number theory, measure of non-unique factorization
{\displaystyle K} . The order of the group, which is finite, is called the class number of K {\displaystyle K} . The theory extends to Dedekind domains
Ideal_class_group
Submodule of fractions in abstract algebra
domains. Hence the theory of fractional ideals can be described for the rings of integers of number fields. In fact, class field theory is the study of such
Fractional_ideal
Mathematical model which approximates the behavior of real gases
both the Newtonian dynamics (as in "kinetic theory") and in quantum mechanics (as a "gas in a box"). The ideal gas model has also been used to model the
Ideal_gas
Mathematical set with an ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.
Partially_ordered_set
Psychological theory
The self-discrepancy theory states that individuals compare their "actual" self to internalized standards or the "ideal/ought self". Inconsistencies between
Self-discrepancy_theory
(mathematics) Upper set and lower set Ideal and filter Ultrafilter Completeness (order theory) Dense order Distributivity (order theory) Modular lattice Distributive
List_of_order_theory_topics
Existence of certain infima or suprema of a given poset
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset)
Completeness_(order_theory)
Concept in algebra
In ring theory, a branch of mathematics, the radical of an ideal I {\displaystyle I} of a commutative ring is another ideal defined by the property that
Radical_of_an_ideal
Branch of algebra
algebraic geometry. In commutative ring theory, numbers are often replaced by ideals, and the definition of the prime ideal tries to capture the essence of prime
Ring_theory
Term in the mathematical area of order theory
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted
Duality_(order_theory)
18th-century Bavarian secret society
Freemasonry, made him an ideal recruit. Knigge, for his own part, was flattered by the attention and drawn towards the order's stated aims of education
Illuminati
Equation of the state of a hypothetical ideal gas
the amount of substance, and R is the ideal gas constant. It can also be derived from the microscopic kinetic theory, as was achieved (independently) by
Ideal_gas_law
Theories in mathematical logic
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model
List_of_first-order_theories
Typological term
construction of abstract, hypothetical concepts. The "ideal type" is therefore a subjective element in social theory and research, and one of the subjective elements
Ideal_type
Type of topological space
of a space is purely a property of its lattice of open sets; cf. Ideal (order theory) § Prime and maximal spectra.) A T4 space is a T1 space X that is
Normal_space
Algebraic structure with addition and multiplication
(an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized)
Ring_(mathematics)
Algebraic integer which represents an ideal in a ring of integers
In number theory, an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed
Ideal_number
Order-preserving mathematical function
reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function
Monotonic_function
In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in
Conductor (class field theory)
Conductor_(class_field_theory)
Principles to describe the practical behavior of colors
Importantly, color theory relies upon objective standards in-order to be consistent in color mixing and presentation – i.e. to achieve the ideal color and effect
Color_theory
Special subset of a partially ordered set
appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special cases of filters include
Filter_(mathematics)
Generalization of vector spaces from fields to rings
generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into
Module_(mathematics)
Mathematical construct in computer algebra
defined for ideals in a polynomial ring R = K [ x 1 , … , x n ] {\displaystyle R=K[x_{1},\ldots ,x_{n}]} over a field K. Although the theory works for any
Gröbner_basis
Order whose elements are all comparable
maximal ideals. In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its opposite order. In
Total_order
Algebraic ring that need not have additive negative elements
Dedekind-complete. By definition, all first-order properties proven in the theory of the reals are also provable in the decidable theory of the real closed field. For
Semiring
1971 book by John Rawls
Rawlsianism” and its “ideal theory” against the actual history of racialized oppression in the modern era, and proposes that non-ideal theory is urgently needed
A_Theory_of_Justice
Study of objects of arithmetic interest over infinite towers of number fields
began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields. In the early
Iwasawa_theory
Generalized alphabetical order
lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences
Lexicographic_order
Limits ideals to be checked in order to determine the class number of a number field
In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number
Minkowski's_bound
Algebraic structure
commutative ring Divisibility (ring theory): nilpotent element, (ex. dual numbers) Ideals and modules: Radical of an ideal, Morita equivalence Ring homomorphisms:
Commutative_ring
Branch of mathematics
based on contemporary commutative algebra, including valuation theory and the theory of ideals. One of the goals was to give a rigorous framework for proving
Algebraic_geometry
Mathematical proposition equivalent to the axiom of choice
every proper ideal is contained in a maximal ideal and that every field has an algebraic closure. Zorn's lemma is equivalent to the well-ordering theorem and
Zorn's_lemma
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Glossary of terms used in branch of mathematics
Look up Appendix:Glossary of order theory in Wiktionary, the free dictionary. This is a glossary of some terms used in various branches of mathematics
Glossary_of_order_theory
In the theory of differential forms, a differential ideal I is an algebraic ideal in the ring of smooth differential forms on a smooth manifold, in other
Differential_ideal
Ring that is also a vector space or a module
maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)
Associative_algebra
In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let O ( x 1 , … , x n ) {\displaystyle
Jacobian_ideal
Mathematical property of subsets in order theory
theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of "subsequence". They are also important in order theory,
Cofinal_(mathematics)
Set whose pairs have minima and maxima
an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which
Lattice_(order)
Problem-solving tools
изобретательских задач, romanized: teoriya resheniya izobretatelskikh zadach, lit. 'theory of inventive problem solving') is a methodology which combines an organized
TRIZ
Subset of a preorder that contains all larger elements
set, or an order filter, while a lower set may also be called a downward closed set, down-set, decreasing set, semi-ideal, or order ideal. However, the
Upper_and_lower_sets
Algebraic structure
group theory. More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at
Semigroup
Branch of algebraic geometry
based on contemporary commutative algebra, including valuation theory and the theory of ideals by Oscar Zariski and others in the 1930s and 1940s. In 1949
Arithmetic_geometry
Aspect of algebraic number theory
prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory. The
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
Ideal generated by one-term polynomials
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. Let K {\displaystyle \mathbb
Monomial_ideal
theory Hodge theory Homology theory Homotopy theory Ideal theory Index theory Information theory Intersection theory Invariant theory Iwasawa theory K-theory
List_of_mathematical_theories
Well-quasi-ordering of finite trees
mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion). In 2004, the
Kruskal's_tree_theorem
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima
Distributivity_(order_theory)
Size of subsets in order theory
In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets
Cofinality
Concept in order theory
In mathematics, specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least
Join_and_meet
Commutative ring with no zero divisors other than zero
integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields An integral
Integral_domain
Topics referred to by the same term
proposition or set in intuitionistic type theory Type, a numeric property of an entire function with positive finite order Exponential type Type (biology), which
Type
Geometric system with a finite number of points
order 2. The Fano plane is called the projective plane of order 2 because it is unique (up to isomorphism). In general, the projective plane of order
Finite_geometry
Mathematical ordering of a partial order
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial
Linear_extension
Mathematical object
ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts of number theory
Ideal_lattice
Boolean analysis Boolean prime ideal theorem Compactness theorem Consensus theorem De Morgan's laws Duality (order theory) Laws of classical logic Peirce's
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Prime ideal that is an annihilator of a prime submodule
dividing the order of M. The group of order 2 is a quotient of the integers Z (considered as a free module over itself), but its associated prime ideal (2) is
Associated_prime
Theorem in order theory
Dushnik–Miller theorem is a result in order theory stating that every countably infinite linear order has a non-identity order embedding into itself. It is named
Dushnik–Miller_theorem
Mathematical structure in abstract algebra
such as ideal and subring, with the requirement to be *-invariant: x ∈ I ⇒ x* ∈ I and so on. *-rings are unrelated to star semirings in the theory of computation
*-algebra
Type of set in abstract algebra
number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal I {\displaystyle
Ideal_quotient
Period of history with a dramatic change in world political thought
comprising this new order.[citation needed] H. G. Wells wrote a book published in 1940 entitled The New World Order. It addressed the ideal of a world without
New_world_order_(politics)
Algebraic structure
in order to specifically exclude fields, while most other sources regard fields as trivial special cases of principal ideal domains. Principal ideal domains
Principal_ideal_domain
Branch of mathematics
concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure. These concepts have been used and
Geometry
Model of spatial distribution of organisms
reaching IFD. IFD theory can still be used to analyze foraging behaviors of animals, whether those behaviors support IFD, or violate it. The ideal free distribution
Ideal_free_distribution
On chains and antichains in partial orders
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of
Dilworth's_theorem
Secret society at Yale University, US
Skull and Bones (also known as The Order, Order 322 or The Brotherhood of Death) is an American undergraduate senior society at Yale University in New
Skull_and_Bones
identifying suitable topologies on partially ordered sets, as is done in order theory. Consider any topological space X {\displaystyle X} . The specialization
Specialization_preorder
Subset of incomparable elements
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are
Antichain
Characterizes the height of any finite partially ordered set
of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into
Mirsky's_theorem
Mathematics of varieties with integer coordinates
abelian varieties now includes Diophantine geometry along with class field theory, complex multiplication, local zeta-functions and L-functions. Paul Vojta
Diophantine_geometry
Leadership model formulated by James Scouller
leadership theories, Scouller highlighted certain limitations in relation to the development of a leader's skill and effectiveness: Trait theory: As Stogdill
Three levels of leadership model
Three_levels_of_leadership_model
Construction in order theory
Introduction to Lattices and Order (Second Edition), 2002, p. 18 Alexander Shen; Nikolai Konstantinovich Vereshchagin (2002). Basic Set Theory. American Mathematical
Product_order
Algebraic object with an ordered structure
Artin–Schreier theory of ordered fields and formally real fields. There are two equivalent common definitions of an ordered field. The definition of total order appeared
Ordered_field
Structure-preserving function between two rings
If I is an ideal of S then f−1(I) is an ideal of R. If R and S are commutative and P is a prime ideal of S then f−1(P) is a prime ideal of R. If R and
Ring_homomorphism
Psychological interpersonal process
Social Order. The Michelangelo phenomenon describes a three-step process where close partners shape each other so as to bring forth one another's ideal selves
Michelangelo_phenomenon
Shape in hyperbolic geometry
three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than
Ideal_polyhedron
1921 philosophical work by Ludwig Wittgenstein
lie outside of the metaphysical subject's world. In turn, a logically "ideal" language cannot supply meaning, it can only reflect the world, and so,
Tractatus Logico-Philosophicus
Tractatus_Logico-Philosophicus
Banach space with a compatible structure of a lattice
in functional analysis and order theory, a Banach lattice (X,‖·‖) is a complete normed vector space with a lattice order, ≤ {\displaystyle \leq } , such
Banach_lattice
Field (mathematics) generated by the square root of an integer
general theory of splitting of prime ideals in Galois extensions, this may be p {\displaystyle p} is inert ( p ) {\displaystyle (p)} is a prime ideal. The
Quadratic_field
Concept of the ideally slim female body
The thin ideal is the concept of the ideally slim female body. The common perception of this ideal is a woman who possesses a slender, feminine physique
Thin_ideal
Attempts to analyze the concept of truth
can be classified into verificationist theories, perspectivist or relativist theories, and pragmatic theories. Verificationism is based on verifying propositions
Epistemic_theories_of_truth
Equivalence of partially ordered sets
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism
Order_isomorphism
Algebra over a field where binary multiplication is not necessarily associative
that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd),
Non-associative_algebra
Studies linear representations of finite groups over fields of positive characteristic
on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly
Modular_representation_theory
Technical treatment of Boolean algebras
(mathematics) Free Boolean algebra Functional completeness Ideal (order theory) Lattice (order) Lindenbaum–Tarski algebra List of Boolean algebra topics
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Relative position of an argument in a binary operator
right identity. In ring theory, a subring which is invariant under any left multiplication in a ring is called a left ideal. Similarly, a right
Left_and_right_(algebra)
more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of
List of order structures in mathematics
List_of_order_structures_in_mathematics
Hypothetical wise ruler described by Plato
and Solomon were held up as examples of ideal rulers, with Plato's theory undergoing further distortions in order to meet the needs of Jewish philosophers
Philosopher_king
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IDEAL ORDER-THEORY
IDEAL ORDER-THEORY
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IDEAL ORDER-THEORY
IDEAL ORDER-THEORY
IDEAL ORDER-THEORY
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