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National FFA Organization
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National FFA Organization
Motto
Learning to Do,
Doing to Learn,
Earning to Live,
Living to Serve[1]
Formation
1928; 90 years ago (1928)
Type
Youth organization
Tax ID no.
54-0524844[3]
Legal status
501(c)(3) non-profit organization[2]
Purpose
To make a positive difference in the lives of students by developing their potential for premier leadership, personal growth, and career success through agricultural education.
Headquarters
U.S. Department of Education Offices of Vocational and Adult Education
1410 King Street,
Alexandria, VA 22314
6060 FFA Drive,
Indianapolis, Indiana 46278[3]
Region served
United States
Puerto Rico
United States Virgin Islands
Membership
649,355 (7,859 Chapters in 52 State Associations)
National FFA Advisor, Board Chair
Dr. Steve A. Brown[4]
Chief Executive Officer
Mark Poeschl[5]
Chief Operating Officer
Josh Bledsoe[6]
National FFA President
Breanna Holbert[7]
Subsidiaries
National FFA Foundation, Inc.[3]
Revenue (2015)
$27,832,251[3]
Expenses (2015)
$27,612,465[3]
Employees (2015)
232[3]
Volunteers (2015)
3,500[3]
Mission
To build partnerships with industry, education, government, foundations, and individuals to secure resources for the future of education, agriculture, and student leader development.[3]
Website
www.ffa.org
Formerly called
Future Farmers of America[8]
National FFA Organization is an American 501(c)(3) youth organization, specifically a career and technical student organization, based on middle and high school classes that promote and support agricultural education. It was founded in 1925 at Virginia Polytechnic Institute, by agriculture teachers, Henry C. Groseclose,[9] Walter Newman, Edmund Magill and Harry Sanders as Future Farmers of Virginia. In 1928, it became a nationwide organization known as Future Farmers of America. In 1988 the name was changed to the National FFA Organization, now commonly referred to as FFA, to recognize that the organization is for those with diverse interests in the food, fiber and natural resource industries, encompassing science, business and technology in addition to production agriculture.[10] Today FFA is among the largest youth organization in the United States, with 649,355 members in 7,859 chapters[11] throughout all 50 states, Puerto Rico, and the Virgin Islands. FFA is the largest of the career and technical student organizations in U.S. schools.
The organization holds a congressional charter under Title 36 of the United States Code.[12]
Contents
[hide]
1 Overview
2 FFA structure
3 Officers of FFA
3.1 Constitutional officers
3.2 Others may include
3.3 National officers
3.4 List of past national presidents
4 Official FFA dress
4.1 FFA jacket
4.2 Elements of FFA official dress
4.3 Awards and pins
5 Traditions
6 Career Development Events
6.1 Others
7 Supervised Agricultural Experience Programs
8 Notable alumni
8.1 In politics
8.2 In entertainment
8.3 In the arts
8.4 Other
9 Notable historic events and milestones
10 See also
11 References
12 External links
Overview[edit]
The National FFA Organization is a youth leadership organization that makes a positive difference in the lives of young people by developing their potential for premier leadership, personal growth and career success through agriculture education. The FFA Motto is Learning to do, Doing to learn, Earning to live, Living to serve.
FFA functions within the three-circle model of agricultural education as a student leadership organization that complements a student's classroom/laboratory instruction and supervised agricultural experience program.[13] FFA members can compete in Career Development Events (CDE) that cover job skills in everything from communications to mechanics. Some events allow students to compete as individuals, while others allow them to compete in teams.[14] These competitions can happen at a local or district level, state level, and on the national level.
Students are supervised by agricultural education teachers in cooperation with parents, employers and other adults who assist individuals in the development and achievement of educational and career goals.
Official colors of the FFA are National Blue and Corn Gold.[15]
FFA structure[edit]
United States Secretary of Agriculture Tom Vilsack (3rd from right) stands with five former National FFA Officers in 2011.
FFA is represented at several different levels across the nation; FFA administration may change from state to state. The basic levels are the national level, serving all of the United States of America, Puerto Rico and the U.S. Virgin Islands; the state level, serving an individual state association; and the chapter level, serving a school or set of schools in an area. By definition, there can be three types of chapters at the secondary level, they are middle for middle school, junior for ninth grade and senior which can be either tenth through twelfth or ninth through twelfth depending on the school. Other levels include districts, subdistricts, sections, regions, areas, and federations.
FFA was created to serve high school students, and has moved into middle schools where membership may begin as early as age 12, allowing members to become active earlier and stay active for longer. Each chapter is chartered as part of the state association and national organization. Collegiate chapters exist as well.
Most states hold FFA conventions at least once annually, where members gather to compete, be recognized for awards, attend leadership workshops, debate organizational issues in a delegate process, and more. Nationally, the National FFA Convention & Expo is held once a year in the fall. It was held in Kansas City, Missouri, from 1928 to 1998. The convention moved to Louisville, Kentucky, in 1999 before moving again in 2006 to Indianapolis, Indiana. Louisville again hosted the national convention from 2013 to 2015. FFA announced in July 2015 that the national convention would be hosted in Indianapolis each year from 2016 to 2024.[16]
Officers of FFA[edit]
As the FFA is a member-led organization intended to serve youth around the nation, it elects officers from its own diverse membership to certain levels of FFA. With these basic constitutional officers a basis of parliamentary procedure is completed during meetings and banquets; each officer during these procedures will stand and recite their office level and what it is stationed by, and why the particular position is stationed as so.Typical officer roles are:
Constitutional officers[edit]
President — Stationed by the rising sun, a token of a new era in agriculture.
Vice President — Stationed by the plow, the symbol of labor and tillage of the soil.
Secretary — Stationed by the ear of corn, to keep a record of all meetings and correspond with other secretaries wherever corn is grown and FFA members meet.
Treasurer — Stationed by the emblem of George Washington, to keep an accurate account of receipts and disbursements just as Washington kept his farm accounts.
Reporter — Stationed by the Flag, strives to inform the public in order that every man, woman, and child may know the FFA is a national organization.
Sentinel — Stationed by the door, to ensure the door is open to all and keep the meeting room comfortable as well as assist in maintaining order.
Advisor — Stationed by the owl, the time honored emblem of knowledge and wisdom
Others may include[edit]
Parliamentarian — Stationed by a copy of Robert's Rules of Order.
Historian — Stationed at the scroll or chapter scrapbook.
Student Advisor — Stationed by the owl or the key.
Chaplain — Stationed by the Bible or Book.
Executive Committee/Executive Board.
Each officer is an agricultural student and holds responsibilities needed to serve, and are elected each year by members at the respective levels.
National officers[edit]
President - Breanna Holbert, California
Secretary - Erica Baier, Iowa
Eastern Region Vice President - Gracie Furnish, Kentucky
Southern Region Vice President - Ian Bennett, Georgia
Central Region Vice President - Pipper Merritt, Oklahoma
Western Region Vice President - Bryce Cluff, Arizona
This allows officers to be elected to one of the four national regions of the FFA, as well as a President and a Secretary
List of past national presidents[edit]
2016-2017 - David Townsend, Delaware
2015–2016 - Taylor McNeel, Arkansas
2014–2015 - Andy Paul, Georgia
2013–2014 - Brian Walsh, Virginia
2012–2013 - Clay Sapp, Florida
2011–2012 - Ryan Best, New Mexico
2010–2011 - Riley Pagett, Oklahoma
2009–2010 - Levy Randolph, California
2008–2009 - Paul Moya, New Mexico
2007–2008 - Zach Kinne, Missouri
2006–2007 - Beau Williamson, California
2005–2006 - Travis Jett, Oklahoma
2004–2005 - Jackie Mundt, Wisconsin
2003–2004 - Javier Moreno, Puerto Rico
2002–2003 - Tim Hammerich, California
2001–2002 - Dane White, California
2000–2001 - Trent McKnight, Texas
1999–2000 - Chris Vitelli, Florida
1998–1999 - Lisa Ahrens, Iowa
1997–1998 - Hillary Smith, Georgia
1996–1997 - Corey Rosenbusch, Texas
1995–1996 - Seth Derner, Nebraska
1994–1995 - Corey Flournoy, Illinois
1993–1994 - Curtis Childers, Texas
1992–1993 - Travis Park, Indiana
1991–1992 - Lee Thurber, Nebraska
1990–1991 - Mark Timm, Indiana
Official FFA dress[edit]
Odessa Oldham, of Casper College in Wyoming, explained her role in the National FFA Organization to the United States Department of Agriculture, as part of its Native American Indian Heritage Month celebration in Washington D.C., in November 2011.
FFA jacket[edit]
Former FFA emblem in use until February 2015.
The most recognizable symbol of the organization is the blue corduroy FFA jacket that is worn by current FFA members. The back of the jacket features a large FFA emblem just underneath the name of each FFA member's state. The name of the local FFA chapter, district, region or area is embroidered below the emblem. The front of the FFA jacket features a smaller FFA emblem on the left chest and the FFA member's name and sometimes, if applicable, office and year on the right chest. FFA members are required to wear the FFA jacket as part of official dress while participating in all official organization activities.[17]
The FFA jacket was created to be worn by the Fredericktown Band of the Fredericktown FFA Chapter by Dr. Gus Lintner.[18] It was adopted in 1933.
The color of the jacket's corduroy has ranged from shades of blue to shades of purple through the years. In 2004, the National FFA Organization worked with a supplier in North Carolina to set a new standard for the blue corduroy by using samples from archived FFA jackets. The jacket's color standardization was accompanied by a restoration of the embroidered FFA emblems and fit corrections led by Clemson University's Apparel Research Center. The improved FFA jacket, produced in both Van Wert, Ohio and South Vietnam, was first made available in August 2005. Currently, all lettering, embroidery and finishing of FFA jackets is completed by Universal Lettering Company in Van Wert, Ohio.[19]
Elements of FFA official dress[edit]
FFA members are required to wear official FFA dress while participating in official organization activities. The National FFA Organization recently updated its bylaws, stating that members may choose to identify however they choose in regards to official dress, in order to include transgender as well as gender non-conforming individuals within the organization. For females, official dress consists of a black skirt (black slacks may be appropriate for traveling and outdoor activities), a white collared blouse, an official FFA blue scarf, black dress shoes with a closed heel and toe, black nylon hosiery, and an official FFA jacket zipped to the top. Male official dress includes black dress pants, a white dress shirt, an official FFA tie, black dress shoes with a closed heel and toe, black socks and an official FFA jacket zipped to the top.[20]
Awards and pins[edit]
FFA members earn metal pins that signify achievement within the organization. These pins can be placed on the front of the FFA jacket,[20] however, official guidelines state that no more than three pins may be worn at one time. The pins are to be placed beneath the FFA member's name on the right chest and can recognize the highest office, highest award, and highest organizational degree. Two exceptions exist within pin placement guidelines. When an FFA member earns a State FFA Degree or American FFA Degree, the award keys should be worn above the name on right chest or attached to the FFA jacket with a standard key chain, and no other pins should accompany these.[20]
Traditions[edit]
As an organization, the FFA has many traditions and trademarks identifying it as an agricultural education organization:[21]
The FFA Motto: Learning to Do, Doing to Learn, Earning to Live, Living to Serve.
The FFA Mission: The National FFA Organization is dedicated to making a positive difference in the lives of students by developing their potential for premier leadership, personal growth, and career success through agricultural education.[21]
The Official FFA Colors: National Blue and Corn Gold (worn on the Official FFA jackets).[22]
The FFA Emblem: Cross section of the ear of corn: represents unity. Where ever you live in the United States, corn is grown everywhere. Rising Sun: signifies progress. It represents that tomorrow will always bring a new day. Plow: signifies labor and tillage of the soil. It also shows the historic foundation of our country's strength. Eagle: symbolizes freedom and our ability to explore the new agriculture world. Owl: represents knowledge. Long recognized for wisdom and knowledge. Agriculture Education and FFA: symbolizes the combination of learning and leadership.
The FFA Creed: The creed was written by Erwin Milton "E.M." Tiffany of Lyndon, Kansas[23] and adopted at the 3rd National FFA Convention.[23] It was revised at the 38th and 63rd National FFA Conventions by the assembled delegate body. It is recited by new members to the organization to reflect their growing belief in agriculture and agricultural education. The FFA Creed also must be memorized and recited to earn the Greenhand Degree.
I believe in the future of agriculture, with a faith born not of words but of deeds – achievements won by the present and past generations of agriculturists; in the promise of better days through better ways, even as the better things we now enjoy have come to us from the struggles of former years.

I believe that to live and work on a good farm or to be engaged in other agricultural pursuits, is pleasant as well as challenging; for I know the joys and discomforts of agricultural life and hold an inborn fondness for those associations which, even in hours of discouragement I cannot deny.

I believe in leadership from ourselves and respect from others. I believe in my own ability to work efficiently and think clearly, with such knowledge and skill as I can secure, and in the ability of progressive agriculturists to serve our own and the public interest in producing and marketing the product of our toil.

I believe in less dependence on begging and more power in bargaining; in the life abundant and enough honest wealth to help make it so-for others as well as myself; in less need for charity and more of it when needed; in being happy myself and playing square with those whose happiness depends upon me.

I believe that American agriculture can and will hold true to the best traditions of our national life and that I can exert an influence in my home and community which will stand solid for my part in that inspiring task.
Career Development Events[edit]
FFA Career Development Event (CDEs) are contests that members compete in to test their skills learned through agricultural education instruction. They vary at the different levels of FFA, and some are contested only at certain levels and certain states, districts, areas or federations.[24]
At the national level, there are 24 Career Development Events:[24]
Ag Business
Ag Communications
Ag Issues Forum
Ag Mechanics
Ag Sales
Agronomy (Insects, seeds, etc.)
Dairy Cattle Evaluation
Dairy Cattle Handlers Activity
Dairy Foods
Environmental & Natural Resources
Extemporaneous Public Speaking
Floriculture
Food Science & Technology
Forestry
Horse Evaluation
Job Interview
Land Judging
Livestock Evaluation
Marketing Plan
Meats Evaluation & Technology
Nursery/Landscape
Parliamentary Procedure
Poultry Evaluation
Prepared Public Speaking
Rangeland Judging
Wildlife Management
Veterinarian Science
Quiz Bowl
Others[edit]
Examples of CDEs that can be available at the state level are:
Small Animal Care
Arboriculture
Aquaculture
Safe Equipment Operation
Opening/Closing Ceremonies
Wildlife
Website Development
Entomology
Tractor Operations
Greenhand CDE
F.A.R.M education
Envirothon
Demonstration
Supervised Agricultural Experience Programs[edit]
To be an active member in the National FFA Organization, a member must have an Supervised Agricultural Experience (SAE) project.[25] The projects involve hands-on application of concepts and principles learned in the agricultural education classroom, with guidelines for the SAE projects governed by the state FFA delegation. SAE programs are grouped into four different areas:
Exploratory – learning about the 'big picture' of agriculture and related careers
Research/Experimentation and Analysis – conducting research or analysis of information to discover new knowledge
Ownership/Entrepreneurship – planning and operation of an agriculture-related business
Placement – working either for pay or experience in an agricultural setting
Notable alumni[edit]
In politics[edit]
Harold Brubaker, North Carolina state representative, past National FFA Secretary from Pennsylvania
Jimmy Carter, Governor of Georgia, 39th President of the United States
Matt Lohr, Virginia state representative, past state FFA President and National FFA Vice President
Bruce Maloch, Arkansas state senator, past state FFA President and National FFA Secretary
Rick Perry, Governor of Texas, past district FFA officer
Jeff Miller (Florida politician), United States Representative from Florida, past state FFA Secretary
Sam Brownback, Governor of Kansas, past United States Senator & United States representative, past state FFA President and national FFA vice president
In entertainment[edit]
Trace Adkins, country music singer
Johnny Cash, country and rock & roll music singer/songwriter, actor
Chris Colfer, actor best known for Glee
Easton Corbin, country music singer/songwriter
Steve Doocy, network-television personality on the Fox News Channel and a best selling author. Co-host of Fox & Friends in the mornings.
Matthew Fox, actor best known for Lost and Party of Five
Brantley Gilbert, country music singer
Don Henley, music entertainer
Toby Keith, country music singer, actor
Lyle Lovett, country music singer
Tim McGraw, country music singer
Eddie Montgomery, country music singer, half of the duo Montgomery Gentry
Beyoncé Giselle Knowles, pop and R&B singer
Willie Nelson, country music singer
Jim Ross, WWE Hall of Famer, past state FFA Vice President
Orion Samuelson, radio broadcaster
Josh Shipp, motivational speaker & entertainer
Taylor Swift, country and pop music singer/songwriter, actress
Craig Morgan, country music singer/songwriter
In the arts[edit]
Jim Davis, creator of the Garfield comic strip
Jared Hess, director of Napoleon Dynamite
Nicholas Kristof, Pulitzer Prize Winner, past Oregon State Officer
John Mellencamp, recording artist, co-founder of Farm Aid
Other[edit]
Leonard J. Arrington, Church of Jesus Christ of Latter Day Saints Historian. "Dean of Mormon History", past National FFA Vice President
Howard Warren Buffett, grandson of famed American investor and philanthropist Warren Edward Buffett, past FFA chapter president
Bo Jackson, American athlete and a former multi-sport professional in American football (NFL) and Baseball (MLB)
Sterling Marlin, NASCAR driver
Brad Meester, American football center for the Jacksonville Jaguars of the National Football League, past FFA chapter president
Rebecca Podio, Miss Wyoming 2013
Bryant Reeves, retired American professional basketball player for the NBA's Vancouver Grizzlies
Carroll Shelby, American automotive designer, racing driver and entrepreneur
Chris Soules, The Bachelorette contestant and The Bachelor star, past Iowa FFA Member
Harry Stine, founder and owner of Stine Seed, past Iowa FFA Member
Mark Tauscher, American football Offensive Tackle for the Green Bay Packers of the National Football League
Notable historic events and milestones[edit]
Commemorative 25th anniversary Future Farmers of America postage stamp issued on October 13, 1953
1928: FFA is established
1929: National Blue and Corn Gold is adopted as official colors
1930: Official FFA creed adopted
1944: Future Farmers of America Foundation formed
1953: Future Farmers of America U.S. postage stamp issued by U.S. Postal Service
1965: African American New Farmers of America merged with FFA
1969: Girls are allowed to join FFA
1988: Official Name change from Future Farmers of America to FFA Organization
2006: National FFA Foundation receives first $1 million contribution from Ford Motor Company.
2009: FFA celebrates 40 years of women in the organization.
2010: FFA members earn a record 3,449 American FFA Degrees.
2011: National FFA Alumni Association celebrates its 40th anniversary.
2013: Official FFA Jacket celebrates 80th anniversary.
2015: National FFA celebrates 50 years since NFA merged with the FFA.
2017: National FFA elects first Woman African-American National President, Ms. Breanna Holbert from the State of California.
See also[edit]
4-H
National Junior Horticultural Association
References[edit]
Jump up ^ "FFA Mission and Motto". National FFA Organization.
Jump up ^ "Future Farmers of America State Associations & Local Chapter". Exempt Organizations Select Check. Internal Revenue Service. Retrieved 1 May 2017.
^ Jump up to: a b c d e f g h "Form 990: Return of Organization Exempt from Income Tax". Future Farmers of America. Guidestar. December 31, 2015.
Jump up ^ "Our Leadership". National FFA Organization. Retrieved 17 October 2014.
Jump up ^ "National FFA Organization and National FFA Foundation Name Mark Poeschl Chief Executive Officer". National FFA Organization. Retrieved 29 July 2016.
Jump up ^ "Our Leadership". National FFA Organization. Retrieved 1 May 2017.
Jump up ^ "David Townsend: Bio". National FFA Organization. Retrieved 1 May 2017.
Jump up ^ "Who We Are". National FFA Organization. Retrieved 17 October 2014.
Jump up ^ "Future Farmers to Meet Aug. 6-9: 1,000 Young Virginians Will Compete for Honors at Blacksburg". The Washington Post. 3 August 1929. p. 18.
Jump up ^ "FFA History". National FFA Organization. p. 2.
Jump up ^ "Statistics". National FFA Organization.
Jump up ^ "36 USC § 70901 – Organization". Federal Charter; Future Farmers of America is a federally chartered corporation. Legal Information Institute; Cornell Law School.
Jump up ^ "School-Based Agricultural Education" (PDF). National FFA Organization.
Jump up ^ "Career Development Events". National FFA Organization.
Jump up ^ http://www.FFA.org
Jump up ^ "National FFA Organization Announces National Convention and Expo Coming to Indianapolis 2016-2024". National FFA Organization. Retrieved 29 July 2016.
Jump up ^ "FFA Official Dress". National FFA Organization.
Jump up ^ Langstaff, Dave. "A Brief History of The FFA Jacket and The People Who Make It". Universal Lettering Company. Retrieved 2010-06-05.
Jump up ^ "FFA Official Dress PDF" (PDF). National FFA Organization.
^ Jump up to: a b c Cite error: The named reference FFA Official Dress was invoked but never defined (see the help page).
^ Jump up to: a b 105th U.S. Congress (August 12, 1998). "Public Law 105-225" (PDF). National FFA Organization. p. 6.
Jump up ^ "FFA Facts". Code of Ethics, Creed, Emblem, Mission, Motto, Name, Official Colors, Official Dress, Salute. Minnesota FFA Association. Retrieved 2010-06-05.
^ Jump up to: a b "The FFA Creed". National FFA Organization. Retrieved 2010-06-05. It was written in 1928
^ Jump up to: a b "FFA Career Development Events (CDE)". National FFA Organization. Retrieved 2013-03-11.
Jump up ^ "FFA Supervised Agricultural Experience (SAE)". National FFA Organization. Retrieved 2013-03-11.
External links[edit]
Wikimedia Commons has media related to National FFA Organization.
Official website
Archives of the National FFA Organization
Encyclopedia of Oklahoma History and Culture – Future Farmers of America
Retrieved from "https://en.wikipedia.org/w/index.php?title=National_FFA_Organization&oldid=810999530"
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For other uses, see Closure (disambiguation).
A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation. For example, the positive integers are closed under addition, but not under subtraction: 1 − 2 {\displaystyle 1-2} is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication (because 0 + 0 = 0 {\displaystyle 0+0=0} , 0 − 0 = 0 {\displaystyle 0-0=0} , and 0 × 0 = 0 {\displaystyle 0\times {0}=0} ).
Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually.
Contents
1 Basic properties
2 Closed sets
3 Examples
4 Closure operator
5 Binary relation closures
6 See also
7 Notes
8 References
Basic properties[edit]
A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure. Modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous; however in practice operations are often defined initially on a superset of the set in question and a closure proof is required to establish that the operation applied to pairs from that set only produces members of that set. For example, the set of even integers is closed under addition, but the set of odd integers is not.
When a set S is not closed under some operations, one can usually find the smallest set containing S that is closed. This smallest closed set is called the closure of S (with respect to these operations). For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An important example is that of topological closure. The notion of closure is generalized by Galois connection, and further by monads.
The set S must be a subset of a closed set in order for the closure operator to be defined. In the preceding example, it is important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not always defined.
The two uses of the word "closure" should not be confused. The former usage refers to the property of being closed, and the latter refers to the smallest closed set containing one that may not be closed. In short, the closure of a set satisfies a closure property.
Closed sets[edit]
A set is closed under an operation if the operation returns a member of the set when evaluated on members of the set. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known as the axiom of closure. For example, one may define a group as a set with a binary product operator obeying several axioms, including an axiom that the product of any two elements of the group is again an element. However the modern definition of an operation makes this axiom superfluous; an n-ary operation on S is just a subset of Sn+1. By its very definition, an operator on a set cannot have values outside the set.
Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarily imply closure on all subsets. Thus a subgroup of a group is a subset on which the binary product and the unary operation of inversion satisfy the closure axiom.
An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space is first-countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at least limits of nets). A set that is closed under this operation is usually just referred to as a closed set in the context of topology. Without any further qualification, the phrase usually means closed in this sense. Closed intervals like [1,2] = {x : 1 ≤ x ≤ 2} are closed in this sense.
A partially ordered set is downward closed (and also called a lower set) if for every element of the set all smaller elements are also in it; this applies for example for the real intervals (−∞, p) and (−∞, p], and for an ordinal number p represented as interval [ 0, p); every downward closed set of ordinal numbers is itself an ordinal number.
Upward closed and upper set are defined similarly.
Examples[edit]
In topology and related branches, the relevant operation is taking limits. The topological closure of a set is the corresponding closure operator. The Kuratowski closure axioms characterize this operator.
In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X and is closed under the operation of linear combination. This subset is a subspace.
In matroid theory, the closure of X is the largest superset of X that has the same rank as X.
In set theory, the transitive closure of a set.
In set theory, the transitive closure of a binary relation.
In algebra, the algebraic closure of a field.
In commutative algebra, closure operations for ideals, as integral closure and tight closure.
In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.
In formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language.
In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set.
In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection.
Closure operator[edit]
Main article: closure operator
Given an operation on a set X, one can define the closure C(S) of a subset S of X to be the smallest subset closed under that operation that contains S as a subset, if any such subsets exist. Consequently, C(S) is the intersection of all closed sets containing S. For example, the closure of a subset of a group is the subgroup generated by that set.
The closure of sets with respect to some operation defines a closure operator on the subsets of X. The closed sets can be determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural properties of all closure operations are: [1]
The closure is increasing or extensive: the closure of an object contains the object.
The closure is idempotent: the closure of the closure equals the closure.
The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).
An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closure of some object.
These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.
If X is contained in a set closed under the operation then every subset of X has a closure.
Binary relation closures[edit]
Consider first homogeneous relations R ⊆ A × A. If a relation S satisfies aSb ⇒ bSa, then it is a symmetric relation. An arbitrary homogeneous relation R may not be symmetric but it is always contained in some symmetric relation: R ⊆ S. The operation of finding the smallest such S corresponds to a closure operator called symmetric closure.
A transitive relation T satisfies aTb ∧ bTc ⇒ aTc. An arbitrary homogeneous relation R may not be transitive but it is always contained in some transitive relation: R ⊆ T. The operation of finding the smallest such T corresponds to a closure operator called transitive closure.
Among heterogeneous relations there are properties of difunctionality and contact which lead to difunctional closure and contact closure.[2] The presence of these closure operators in binary relations leads to topology since open-set axioms may be replaced by Kuratowski closure axioms. Thus each property P, symmetry, transitivity, difunctionality, or contact corresponds to a relational topology.[3]
In the theory of rewriting systems, one often uses more wordy notions such as the reflexive transitive closure R*—the smallest preorder containing R, or the reflexive transitive symmetric closure R≡—the smallest equivalence relation containing R, and therefore also known as the equivalence closure. When considering a particular term algebra, an equivalence relation that is compatible with all operations of the algebra [note 1] is called a congruence relation. The congruence closure of R is defined as the smallest congruence relation containing R.
For arbitrary P and R, the P closure of R need not exist. In the above examples, these exist because reflexivity, transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly defined as the intersection of all sets with property P containing R.[4]
Some important particular closures can be constructively obtained as follows:
clref(R) = R ∪ { ⟨x,x⟩ : x ∈ S } is the reflexive closure of R,
clsym(R) = R ∪ { ⟨y,x⟩ : ⟨x,y⟩ ∈ R } is its symmetric closure,
cltrn(R) = R ∪ { ⟨x1,xn⟩ : n >1 ∧ ⟨x1,x2⟩, ..., ⟨xn-1,xn⟩ ∈ R } is its transitive closure,
clemb,Σ(R) = R ∪ { ⟨f(x1,…,xi-1,xi,xi+1,…,xn), f(x1,…,xi-1,y,xi+1,…,xn)⟩ : ⟨xi,y⟩ ∈ R ∧ f ∈ Σ n-ary ∧ 1 ≤ i ≤ n ∧ x1,...,xn ∈ S } is its embedding closure with respect to a given set Σ of operations on S, each with a fixed arity.
The relation R is said to have closure under some clxxx, if R = clxxx(R); for example R is called symmetric if R = clsym(R).
Any of these four closures preserves symmetry, i.e., if R is symmetric, so is any clxxx(R). [note 2] Similarly, all four preserve reflexivity. Moreover, cltrn preserves closure under clemb,Σ for arbitrary Σ. As a consequence, the equivalence closure of an arbitrary binary relation R can be obtained as cltrn(clsym(clref(R))), and the congruence closure with respect to some Σ can be obtained as cltrn(clemb,Σ(clsym(clref(R)))). In the latter case, the nesting order does matter; e.g. if S is the set of terms over Σ = { a, b, c, f } and R = { ⟨a,b⟩, ⟨f(b),c⟩ }, then the pair ⟨f(a),c⟩ is contained in the congruence closure cltrn(clemb,Σ(clsym(clref(R)))) of R, but not in the relation clemb,Σ(cltrn(clsym(clref(R)))).
See also[edit]
Open set
Clopen set
Notes[edit]
Jump up ^ that is, such that e.g. xRy implies f(x,x2) R f(y,x2) and f(x1,x) R f(x1,y) for any binary operation f and arbitrary x1,x2 ∈ S
Jump up ^ formally: if R = clsym(R), then clxxx(R) = clsym(clxxx(R))
References[edit]
Mathematics portal
Jump up ^ Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications. 25. Am. Math. Soc. p. 111.
Jump up ^ Gunther Schmidt (2011) Relational Mathematics, pages 169 and 227, Encyclopedia of Mathematics and its Applications, vol. 132, Cambridge University Press ISBN 978-0-521-76268-7
Jump up ^ Gunter Schmidt and M. Winter (2018) Relational Topology, Lecture Notes in Mathematics vol. 2208, Springer Verlag, ISBN 978-3-319-74451-3
Jump up ^ Baader, Franz; Nipkow, Tobias (1998). Term Rewriting and All That. Cambridge University Press. pp. 8–9.
Weisstein, Eric W. "Algebraic Closure". MathWorld.
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