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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 of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
A subset I of a partially ordered set is an ideal, if the following conditions hold: [1] [2]
While this is the most general way to define an ideal for arbitrary posets, it was originally defined for lattices only. In this case, the following equivalent definition can be given: a subset I of a lattice is an ideal if and only if it is a lower set that is closed under finite joins (suprema); that is, it is nonempty and for all x, y in I, the element of P is also in I. [3]
A weaker notion of order ideal is defined to be a subset of a poset P that satisfies the above conditions 1 and 2. In other words, an order ideal is simply a lower set. Similarly, an ideal can also be defined as a "directed lower set".
The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging with is a filter.
Frink ideals, pseudoideals and Doyle pseudoideals are different generalizations of the notion of a lattice ideal.
An ideal or filter is said to be proper if it is not equal to the whole set P. [3]
The smallest ideal that contains a given element p is a principal ideal and p is said to be a principal element of the ideal in this situation. The principal ideal for a principal p is thus given by ↓ p = {x∈P | x ≤ p}.
The above definitions of "ideal" and "order ideal" are the standard ones, [3] [4] [5] but there is some confusion in terminology. Sometimes the words and definitions such as "ideal", "order ideal", "Frink ideal", or "partial order ideal" mean one another. [6] [7]
An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows:
A subset I of a lattice is a prime ideal, if and only if
It is easily checked that this is indeed equivalent to stating that is a filter (which is then also prime, in the dual sense).
For a complete lattice the further notion of a completely prime ideal is meaningful. It is defined to be a proper ideal I with the additional property that, whenever the meet (infimum) of some arbitrary set A is in I, some element of A is also in I. So this is just a specific prime ideal that extends the above conditions to infinite meets.
The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within ZF (Zermelo–Fraenkel set theory without the axiom of choice). This issue is discussed in various prime ideal theorems, which are necessary for many applications that require prime ideals.
An ideal I is a maximal ideal if it is proper and there is no proper ideal J that is a strict superset of I. Likewise, a filter F is maximal if it is proper and there is no proper filter that is a strict superset.
When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
Maximal filters are sometimes called ultrafilters, but this terminology is often reserved for Boolean algebras, where a maximal filter (ideal) is a filter (ideal) that contains exactly one of the elements {a, ¬a}, for each element a of the Boolean algebra. In Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter.
There is another interesting notion of maximality of ideals: Consider an ideal I and a filter F such that I is disjoint from F. We are interested in an ideal M that is maximal among all ideals that contain I and are disjoint from F. In the case of distributive lattices such an M is always a prime ideal. A proof of this statement follows.
Assume the ideal M is maximal with respect to disjointness from the filter F. Suppose for a contradiction that M is not prime, i.e. there exists a pair of elements a and b such that a∧b in M but neither a nor b are in M. Consider the case that for all m in M, m∨a is not in F. One can construct an ideal N by taking the downward closure of the set of all binary joins of this form, i.e. N = { x | x ≤ m∨a for some m∈M}. It is readily checked that N is indeed an ideal disjoint from F which is strictly greater than M. But this contradicts the maximality of M and thus the assumption that M is not prime.
For the other case, assume that there is some m in M with m∨a in F. Now if any element n in M is such that n∨b is in F, one finds that (m∨n) ∨b and (m∨n) ∨a are both in F. But then their meet is in F and, by distributivity, (m∨n) ∨ (a∧b) is in F too. On the other hand, this finite join of elements of M is clearly in M, such that the assumed existence of n contradicts the disjointness of the two sets. Hence all elements n of M have a join with b that is not in F. Consequently one can apply the above construction with b in place of a to obtain an ideal that is strictly greater than M while being disjoint from F. This finishes the proof.
However, in general it is not clear whether there exists any ideal M that is maximal in this sense. Yet, if we assume the axiom of choice in our set theory, then the existence of M for every disjoint filter–ideal-pair can be shown. In the special case that the considered order is a Boolean algebra, this theorem is called the Boolean prime ideal theorem. It is strictly weaker than the axiom of choice and it turns out that nothing more is needed for many order-theoretic applications of ideals.
The construction of ideals and filters is an important tool in many applications of order theory.
Ideals were introduced by Marshall H. Stone first for Boolean algebras, [8] where the name was derived from the ring ideals of abstract algebra. He adopted this terminology because, using the isomorphism of the categories of Boolean algebras and of Boolean rings, the two notions do indeed coincide.
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.
In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal.
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. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.
In 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, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.
In the mathematical field of order theory, an ultrafilter on a given partially ordered set is a certain subset of namely a maximal filter on that is, a proper filter on that cannot be enlarged to a bigger proper filter on
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a conditionally complete lattice. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals, or distributive lattices and maximal ideals. This article focuses on prime ideal theorems from order theory.
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.
This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:
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 and a unique infimum. 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.
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist.
In mathematics, the Fréchet filter, also called the cofinite filter, on a set is a certain collection of subsets of . A subset of belongs to the Fréchet filter if and only if the complement of in is finite. Any such set is said to be cofinite in , which is why it is alternatively called the cofinite filter on .
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra.
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.
In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal, and the union of any two elements of the ideal must also be in the ideal.
This is a glossary of set theory.
In the mathematical field of set theory, an ultrafilter on a set is a maximal filter on the set In other words, it is a collection of subsets of that satisfies the definition of a filter on and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of that is also a filter. Equivalently, an ultrafilter on the set can also be characterized as a filter on with the property that for every subset of either or its complement belongs to the ultrafilter.
