In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") 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 If is an arbitrary set, its power set ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on are usually called . An ultrafilter on a set may be considered as a finitely additive measure on . In this view, every subset of is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not. Ultrafilters have many applications in set theory, model theory, topology and combinatorics. In order theory, an ultrafilter is a subset of a partially ordered set that is maximal among all proper filters. This implies that any filter that properly contains an ultrafilter has to be equal to the whole poset. Formally, if is a set, partially ordered by then a subset is called a filter on if is nonempty, for every there exists some element such that and and for every and implies that is in too; a proper subset of is called an ultrafilter on if is a filter on and there is no proper filter on that properly extends (that is, such that is a proper subset of ). Every ultrafilter falls into exactly one of two categories: principal or free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form for some (but not all) elements of the given poset. In this case is called the of the ultrafilter. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter. For ultrafilters on a powerset a principal ultrafilter consists of all subsets of that contain a given element Each ultrafilter on that is also a principal filter is of this form. Therefore, an ultrafilter on is principal if and only if it contains a finite set.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related lectures (1)
Correlation Inequalities
Explores correlation inequalities and their applications in various contexts.
Related publications (7)

Fourier non-uniqueness sets from totally real number fields

Martin Peter Stoller

Let K be a totally real number field of degree n >= 2. The inverse different of K gives rise to a lattice in Rn. We prove that the space of Schwartz Fourier eigenfunctions on R-n which vanish on the "component-wise square root" of this lattice, is infinite ...
EUROPEAN MATHEMATICAL SOC-EMS2022

Extending Boolean Methods for Scalable Logic Synthesis

Giovanni De Micheli, Mathias Soeken, Pierre-Emmanuel Julien Marc Gaillardon, Luca Gaetano Amarù, Eleonora Testa

In recent years, Boolean methods in logic synthesis have been drawing the attention of EDA researchers due to the continuous push to advance quality of results. Boolean methods require high computational cost, as they rely on complete functional properties ...
2020

Improvements to Boolean resynthesis

Giovanni De Micheli, Mathias Soeken, Luca Gaetano Amarù

In electronic design automation Boolean resynthesis techniques are increasingly used to improve the quality of results where algebraic methods hit local minima Boolean methods rely on complete functional properties of a logic circuit, preferably including ...
IEEE2018
Show more
Related people (1)
Related concepts (17)
Ultrafilter on a set
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. (In the above, by definition a filter on a set does not contain the empty set.) 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.
Compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent. The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces, hence the theorem's name.
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this.
Show more

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.