0_Wizard
Hello, are you a first or second generation immigrant?
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Immigration to the United States
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Second-generation immigrants in the United States
Second-generation Americans in the United States are individuals born and raised in the United States who have at least one foreign born parent, or the very young foreign-born children of immigrants who are raised in the United States. Although there is some ambiguity in reference to the definition of second-generation Americans, this definition is cited by major research centers such as the United States Census Bureau and the Pew Research Center. Officially, second-generation Americans are U.S. citizens by birth. U.S. citizenship is guaranteed in the Fourteenth Amendment to the United States Constitution; but political debate over repealing this right has increased in recent years.
Immigrant generations
The term first-generation may refer either to a person who has immigrated to a new country and been naturalized, or to the children of such an immigrant. The term second-generation consequently may refer to either the children or the grandchildren of such an immigrant. The term first-generation, as it pertains to a person's nationality or residency in a country, has two incompatible meanings:  This ambiguity is captured and corroborated in "The Oxford English Dictionary"s definition of "generation":  There is no universal consensus on which of these meanings is always intended. The term 1.5 generation or 1.5G refers to individuals who immigrate to a new country before or during their early teens.
Immigrant paradox
The immigrant paradox is that recent immigrants often outperform more established immigrants and non-immigrants on a number of health-, education-, and conduct- or crime-related outcomes, despite the numerous barriers they face to successful social integration. According to the UN, the number of first-generation immigrants worldwide is 244 million. These large-scale population changes worldwide have led many scholars, across fields, to study the acculturation and adjustment of immigrants to their new homes. Specifically, researchers have examined immigrant experiences as they pertain to educational attainment, mental and physical health, and conduct/crime.
Academic achievement among different groups in Germany
Differences in academic achievement among different ethnic groups in Germany  To properly understand ethnic group differences in academic attainment in Germany, it must be understood that different ethnic groups in Germany have different histories of immigration. For example, one reason why the Italians are doing so poorly is that most of their ancestors arrived as unskilled Gastarbeiter. However socioeconomic factors cannot explain all the differences. The Spanish, for example, also arrived as Gastarbeiter, but soon started moving up the social ladder. This difference has been attributed to culture.
Morning Musume
, formerly simply and colloquially referred to as , is a Japanese female idol group, holding the second highest overall single sales (of a female group) on the Oricon charts as of February 2012, with the Oricon record of most top ten singles with an amount of 50, and they have sold over 21 million copies in Japan alone. Morning Musume was formed in 1997 by rock singer-songwriter turned record producer Tsunku, who later composed the vast majority of the group's songs over the decade. They are the lead group of Hello! Project that specialises in upbeat, pop-oriented music coupled with dance performance.
Immigration to Denmark
Denmark has seen a steady increase in immigration over the past thirty years, with the majority of new immigrants originating from non-Western countries. As of 2014, more than 8% of the population of Denmark consists of immigrants. The population of immigrants is approximately 476,059, excluding Danish born descendants of immigrants to Denmark. This recent shift in demographics has posed challenges to the nation as it attempts to address religious and cultural difference, employment gaps, education of both immigrants and their descendants, spatial segregation, crime rates, and language abilities.
Angerme
, formerly , is a Japanese idol girl group from Hello! Project, which originally consisted of four former Hello! Pro Egg (Hello! Project trainees) members that left Egg in 2010 and became full-time members of Hello! Project. In 2010, they won the Japan Record Award for Best New Artist. In 2011, five new girls were added to the group as part of the second generation, with two being former Egg members and three new girls. With 3 members graduating, S/mileage continued the rest of their career as a 6-member group. Since the formation of ANGERME, there has been 3 more generations, and 3 more graduations.
Marcus Lee Hansen
Marcus Lee Hansen (December 8, 1892 – May 11, 1938) was an American historian, who won the 1941 Pulitzer Prize for History for "The Atlantic Migration, 1607–1860" (1940). Hansen was born in Neenah, Wisconsin.He was one of eight children born to Danish immigrant Marcus Hansen (1851-1917) and Norwegian immigrant Gina O Lee Hansen (1854-1920). He received a BA from Central College, an MA from the University of Iowa, and a PhD from Harvard University, where he studied under Frederick Jackson Turner. He was Associate Professor of History (1928–30) and Professor of History (1930–38) at the University of Illinois at Urbana-Champaign.
Hello! Project
 In 1997, Japanese rock group Sharam Q, fronted by Tsunku, began auditions for a female vocalist. These auditions aired via the reality show Asayan, and resulted in Michiyo Heike being crowned winner. Tsunku, however, decided to give five of the runners-up, Yuko Nakazawa, Natsumi Abe, Kaori Iida, Asuka Fukuda and Aya Ishiguro, a chance to become a group in their own right. The girls were set the task of selling 50,000 copies of their demo single "Ai no Tane," in just five days. The girls sold the required number of copies in four days, and subsequently became an official group. Morning Musume's debut single, "Morning Coffee," was released on January 28, 1998 on the One Up Music label and charted at #6 on the Oricon weekly chart.
V. Trygve Jordahl
Verner Trygve "V.T." Jordahl (; October 26, 1898 – September 27, 1984) was an Evangelical Lutheran Church (ELC) District President; he served as a U.S. Army Chaplain, on the Luther College Board of Regents, and was the ELC Director of Service to Military Personnel. He was born October 26, 1898 in Norway Lake, Minnesota as a second generation immigrant from Norway. He retired in Sun City, Arizona. He died on September 27, 1984 in summer home in Decorah, Iowa. He received his Bachelor of Arts (B.A.) degree from Luther College in 1922, where he had also attended preparatory school. He then attended Luther Seminary in Saint Paul, Minnesota and received his Bachelor of Theology in 1925.
1_Apprentice
I an an bloodline of shoemakers
0_Wizard
Oh, glad it isn't the Jesus bloodline. Although they say it is a hypothetical sequence of lineage
The Jesus bloodline is a hypothetical sequence of lineal descendants of the historical Jesus, often by Mary Magdalene, usually portrayed as his wife.
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Jesus bloodline
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Jesus bloodline
The Jesus bloodline is a hypothetical sequence of lineal descendants of the historical Jesus, often by Mary Magdalene, usually portrayed as his wife. Differing and contradictory versions of a Jesus bloodline hypothesis have been proposed in numerous books by authors such as Louis Martin (1886), Donovan Joyce (1973), Andreas Faber-Kaiser (1977), Barbara Thiering (1992), Margaret Starbird (1993), and various websites. Dan Brown's novel "The Da Vinci Code" used the premise for its plot line. The 2007 documentary "The Lost Tomb of Jesus" proposed that evidence existed to show that Jesus was married to Mary Magdalene and that their son was named Judah, based upon inscriptions found on ossuaries discovered in Jerusalem in 1980.
The Holy Blood and the Holy Grail
The Holy Blood and the Holy Grail (retitled Holy Blood, Holy Grail in the United States) is a book by Michael Baigent, Richard Leigh and Henry Lincoln. The book was first published in 1982 by Jonathan Cape in London, as an unofficial follow-up to three BBC Two TV documentaries that were part of the "Chronicle" series. The paperback version was first published in 1983 by Corgi books. A sequel to the book, called " The Messianic Legacy", was originally published in 1986. The original work was reissued in an illustrated hardcover version with exclusive new material in 2005. One of the books that the authors claim influenced the project was "L'Or de Rennes" (later re-published as "Le Trésor Maudit"), a 1967 book by Gérard de Sède, with the collaboration of Pierre Plantard.
Priory of Sion
The Prieuré de Sion (French: [pʁijœʁe də sjɔ̃]), translated as Priory of Sion, is a fringe fraternal organisation, founded and dissolved in France in 1956 by Pierre Plantard as part of a hoax. In the 1960s, Plantard created a fictitious history for that organization, describing it as a secret society founded by Godfrey of Bouillon on Mount Zion in the Kingdom of Jerusalem in 1099, conflating it with a genuine historical monastic order, the Abbey of Our Lady of Mount Zion. In Plantard's version, the Priory was devoted to installing a secret bloodline of the Merovingian dynasty on the thrones of France and the rest of Europe.
Rosslyn Chapel
Rosslyn Chapel, formally known as the Collegiate Chapel of St Matthew, is a 15th century chapel located in the village of Roslin, Midlothian, Scotland. Rosslyn Chapel was founded on a small hill above Roslin Glen as a Catholic collegiate church (with between four and six ordained canons and two boy choristers) in the mid-15th century. The chapel was founded by William Sinclair, 1st Earl of Caithness of the Scoto-Norman Sinclair family. Rosslyn Chapel is the third Sinclair place of worship at Roslin, the first being in Roslin Castle and the second (whose crumbling buttresses can still be seen today) in what is now Roslin Cemetery.
Bloodline (2008 film)
Bloodline is a 2008 documentary film by Bruce Burgess, a filmmaker with an interest in paranormal claims, focused on the "Jesus bloodline" hypothesis and other elements of the book "The Holy Blood and the Holy Grail". The 2008 documentary was originally released in cinemas on 9 May. "Ben Hammott" relating to his alleged discoveries made in the vicinity of Rennes-le-Château since 1999; Burgess claims Ben Hammott would have found the treasure of Bérenger Saunière: a mummified corpse, which he believed were Mary Magdalene. In the film, Burgess interviews several people with alleged connections to the Priory of Sion, including a Gino Sandri and Nicolas Haywood.
The Da Vinci Code
The Da Vinci Code is a 2003 mystery thriller novel by Dan Brown. It follows symbologist Robert Langdon and cryptologist Sophie Neveu after a murder in the Louvre Museum in Paris causes them to become involved in a battle between the Priory of Sion and Opus Dei over the possibility of Jesus Christ having been a companion to Mary Magdalene. The title of the novel refers to the finding of the first murder victim in the Grand Gallery of the Louvre, naked and posed similar to Leonardo da Vinci's famous drawing, the "Vitruvian Man," with a cryptic message written beside his body and a pentagram drawn on his chest in his own blood.
Bloodline (TV series)
Bloodline is an American Netflix original thriller–drama web television series created by Todd A. Kessler, Glenn Kessler, and Daniel Zelman, and produced by Sony Pictures Television. The series premiered on February 9, 2015, in the Berlinale Special Galas section of the 65th Berlin International Film Festival, and the 13-episode first season premiered in its entirety, on Netflix, on March 20, 2015. On March 31, 2015, "Bloodline" was renewed for a 10-episode second season that debuted on May 27, 2016. On July 13, 2016, Netflix renewed "Bloodline" for a 10-episode third season, later confirmed to be the final season.
Genealogy of Jesus
The New Testament provides two accounts of the genealogy of Jesus, one in the Gospel of Matthew and another in the Gospel of Luke. Matthew's starts with Abraham, while Luke begins with Adam. The lists are identical between Abraham and David, but differ radically from that point. Traditional Christian scholars (starting with the historian Eusebius) have put forward various theories that seek to explain why the lineages are so different, such as that Matthew's account follows the lineage of Joseph, while Luke's follows the lineage of Mary. Some modern critical scholars like Marcus Borg and John Dominic Crossan claim both genealogies as inventions, to bring the Messianic claims into conformity with Jewish criteria.
Henry Lincoln
Henry Lincoln (born Henry Soskin; 1930) is a British author, television presenter, scriptwriter and former supporting actor. He co-wrote three "Doctor Who" multi-part serials in the 1960s, and — starting in the 1970s — inspired three Chronicle BBC Two documentaries on the alleged "mysteries" surrounding the French village of Rennes-le-Château (on which he was writer and presenter) — and later from the 1980s on co-authored and authored a series of books of which, the pseudohistorical "The Holy Blood and the Holy Grail" was the most popular, becoming the inspiration for Dan Brown's 2003 best-selling novel, "The Da Vinci Code".
First-order hold
The first-order hold (FOH) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator. For the FOH, the signal is reconstructed as a piecewise linear approximation to the original signal that was sampled. A mathematical model such as the FOH (or, more commonly, the zero-order hold) is necessary because, in the sampling and reconstruction theorem, a sequence of Dirac impulses, "x"("t"), representing the discrete samples, "x"("nT"), is low-pass filtered to recover the original signal that was sampled, "x"("t").
1_Apprentice
I wonder if some people who have taken an blood test would know thier bloodline
0_Wizard
Blood tests would be so crucial for differential diagnosis! They can eliminate life-threatening conditions!
Differential diagnostic procedures are used by physicians and other trained medical professionals to diagnose the specific disease in a patient, or, at least, to eliminate any imminently life-threatening conditions.
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Differential diagnosis
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Differential diagnosis
In medicine, a differential diagnosis is the distinguishing of a particular disease or condition from others that present similar clinical features. Differential diagnostic procedures are used by physicians and other trained medical professionals to diagnose the specific disease in a patient, or, at least, to eliminate any imminently life-threatening conditions. Often, each individual option of a possible disease is called a differential diagnosis (for example, acute bronchitis could be a differential diagnosis in the evaluation of a cough that ends up with a final diagnosis of common cold). More generally, a differential diagnostic procedure is a systematic diagnostic method used to identify the presence of a disease entity where multiple alternatives are possible.
Medical diagnosis
Medical diagnosis (abbreviated Dx or D) is the process of determining which disease or condition explains a person's symptoms and signs. It is most often referred to as diagnosis with the medical context being implicit. The information required for diagnosis is typically collected from a history and physical examination of the person seeking medical care. Often, one or more diagnostic procedures, such as diagnostic tests, are also done during the process. Sometimes Posthumous diagnosis is considered a kind of medical diagnosis. Diagnosis is often challenging, because many signs and symptoms are nonspecific.
Schizoaffective disorder
Schizoaffective disorder (SZA, SZD or SAD) is a mental disorder characterized by abnormal thought processes and deregulated emotions. The diagnosis is made when the person has features of both schizophrenia and a mood disorder—either bipolar disorder or depression—but does not strictly meet diagnostic criteria for either alone. The bipolar type is distinguished by symptoms of mania, hypomania, or mixed episode; the depressive type by symptoms of depression only. Common symptoms of the disorder include hallucinations, paranoid delusions, and disorganized speech and thinking. The onset of symptoms usually begins in young adulthood, currently with an uncertain lifetime prevalence because the disorder was redefined, but DSM-IV prevalence estimates have been less than one percent of the population, in the range of 0.5 to 0.8 percent.
Lyme disease
Lyme disease, also known as Lyme borreliosis, is an infectious disease caused by bacteria of the "Borrelia" type which is spread by ticks. The most common sign of infection is an expanding area of redness on the skin, known as erythema migrans, that begins at the site of a tick bite about a week after it has occurred. The rash is typically neither itchy nor painful. Approximately 25–50% of infected people do not develop a rash. Other early symptoms may include fever, headache and feeling tired. If untreated, symptoms may include loss of the ability to move one or both sides of the face, joint pains, severe headaches with neck stiffness, or heart palpitations, among others.
Pulmonary embolism
Pulmonary embolism (PE) is a blockage of an artery in the lungs by a substance that has moved from elsewhere in the body through the bloodstream (embolism). Symptoms of a PE may include shortness of breath, chest pain particularly upon breathing in, and coughing up blood. Symptoms of a blood clot in the leg may also be present such as a red, warm, swollen, and painful leg. Signs of a PE include low blood oxygen levels, rapid breathing, rapid heart rate, and sometimes a mild fever. Severe cases can lead to passing out, abnormally low blood pressure, and sudden death. PE usually results from a blood clot in the leg that travels to the lung.
Allergy
Allergies, also known as allergic diseases, are a number of conditions caused by hypersensitivity of the immune system to something in the environment that usually causes little or no problem in most people. These diseases include hay fever, food allergies, atopic dermatitis, allergic asthma, and anaphylaxis. Symptoms may include red eyes, an itchy rash, sneezing, a runny nose, shortness of breath, or swelling. Food intolerances and food poisoning are separate conditions. Common allergens include pollen and certain food. Metals and other substances may also cause problems. Food, insect stings, and medications are common causes of severe reactions.
Hepatitis
Hepatitis is inflammation of the liver tissue. Some people have no symptoms whereas others develop yellow discoloration of the skin and whites of the eyes, poor appetite, vomiting, tiredness, abdominal pain, or diarrhea. Hepatitis may be temporary (acute) or long term (chronic) depending on whether it lasts for less than or more than six months. Acute hepatitis can sometimes resolve on its own, progress to chronic hepatitis, or rarely result in acute liver failure. Over time the chronic form may progress to scarring of the liver, liver failure, or liver cancer. The most common cause worldwide is viruses.
Ovarian cancer
Ovarian cancer is a cancer that forms in or on an ovary. It results in abnormal cells that have the ability to invade or spread to other parts of the body. When this process begins, there may be no or only vague symptoms. Symptoms become more noticeable as the cancer progresses. These symptoms may include bloating, pelvic pain, abdominal swelling, and loss of appetite, among others. Common areas to which the cancer may spread include the lining of the abdomen, lymph nodes, lungs, and liver. The risk of ovarian cancer increases in women who have ovulated more over their lifetime. This includes those who have never had children, those who begin ovulation at a younger age and those who reach menopause at an older age.
Reference range
In health-related fields, a reference range or reference interval is the range of values for a physiologic measurement in healthy persons (for example, the amount of creatinine in the blood, or the partial pressure of oxygen). It is a basis for comparison (a frame of reference) for a physician or other health professional to interpret a set of test results for a particular patient. Some important reference ranges in medicine are reference ranges for blood tests and reference ranges for urine tests. The standard definition of a reference range (usually referred to if not otherwise specified) originates in what is most prevalent in a reference group taken from the general (i.e.
Leptospirosis
Leptospirosis is an infection caused by corkscrew-shaped bacteria called "Leptospira". Signs and symptoms can range from none to mild such as headaches, muscle pains, and fevers; to severe with bleeding from the lungs or meningitis. If the infection causes the person to turn yellow, have kidney failure and bleeding, it is then known as Weil's disease. If it also causes bleeding into the lungs then it is known as severe pulmonary hemorrhage syndrome. Up to 13 different genetic types of "Leptospira" may cause disease in humans. It is transmitted by both wild and domestic animals. The most common animals that spread the disease are rodents.
1_Apprentice
Oh I agree this is very finite
0_Wizard
But finite sets only have a finite number of elements. It is infinite, if you ask me
In mathematics, a finite set is a set that has a finite number of elements.
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Finite set
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Finite set
In mathematics, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (a non-negative integer) and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.
Boolean algebras canonically defined
Boolean algebra is a mathematically rich branch of abstract algebra. 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 (whose interpretation need not be numerical). Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under zero or more operations satisfying certain equations. Just as there are basic examples of groups, such as the group Z of integers and the permutation group "S" of permutations of "n" objects, there are also basic examples of Boolean algebra such as the following.
Rado graph
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The same graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 when one is a quadratic residue modulo the other.
Counting
Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term "enumeration" refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The "transfinite" cardinal numbers describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex.
Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number. Some authors use countable set to mean "countably infinite" alone. To avoid this ambiguity, the term "at most countable" may be used when finite sets are included and "countably infinite", "enumerable", or "denumerable" otherwise.
Dedekind-infinite set
In mathematics, a set "A" is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset "B" of "A" is equinumerous to "A". Explicitly, this means that there is a bijective function from "A" onto some proper subset "B" of "A". A set is Dedekind-finite if it is not Dedekind-infinite. Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite.
Representation theory of finite groups
The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Please note, that except for a few marked exceptions only finite groups will be considered in this article. We will also restrain to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero.
Paradoxes of set theory
This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable.
Adele ring
In mathematics, the adele ring (also adelic ring or ring of adeles) is defined in class field theory, a branch of algebraic number theory. It allows one to elegantly describe the Artin reciprocity law. The adele ring is a self-dual topological ring, which is built on a global field. It is the restricted product of all the completions of the global field and therefore contains all the completions of the global field. The , which is the quotient group of the group of units of the adele ring by the group of units of the global field, is a central object in class field theory. Notation: Throughout this article, formula_1 is a global field.
1_Apprentice
I belive this only an calculated idea of how many belive in the tracking and recording.
0_Wizard
Or we can just stick with Boolean algebra, and just have the values of 0 or 1
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 (whose interpretation need not be numerical).
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Boolean algebras canonically defined
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Boolean algebra
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values "true" and "false", usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction "and" denoted as ∧, the disjunction "or" denoted as ∨, and the negation "not" denoted as ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations.
Boolean algebras canonically defined
Boolean algebra is a mathematically rich branch of abstract algebra. 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 (whose interpretation need not be numerical). Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under zero or more operations satisfying certain equations. Just as there are basic examples of groups, such as the group Z of integers and the permutation group "S" of permutations of "n" objects, there are also basic examples of Boolean algebra such as the following.
Boolean algebra (structure)
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 (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨).
Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras. An interior algebra is an algebraic structure with the signature  where  is a Boolean algebra and postfix designates a unary operator, the interior operator, satisfying the identities:   "x" is called the interior of "x". The dual of the interior operator is the closure operator defined by "x" = (("x"′))′.
Boolean-valued model
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-valued models were introduced by Dana Scott, Robert M. Solovay, and Petr Vopěnka in the 1960s in order to help understand Paul Cohen's method of forcing. They are also related to Heyting algebra semantics in intuitionistic logic. Fix a complete Boolean algebra "B" and a first-order language "L"; the signature of "L" will consist of a collection of constant symbols, function symbols, and relation symbols.
Free Boolean algebra
In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that:  The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John is tall" and "Mary is rich". These generate a Boolean algebra with four atoms, namely:  Other elements of the Boolean algebra are then logical disjunctions of the atoms, such as "John is tall and Mary is not rich, or John is not tall and Mary is rich". In addition there is one more element, FALSE, which can be thought of as the empty disjunction; that is, the disjunction of no atoms.
Field of sets
In mathematics a field of sets is a pair formula_1 where formula_2 is a set and formula_3 is an algebra over formula_4 i.e., a non-empty subset of the power set of formula_2 closed under the intersection and union of pairs of sets and under complements of individual sets. In other words, formula_3 forms a subalgebra of the power set Boolean algebra of formula_2. (Many authors refer to formula_3 itself as a field of sets. The word "field" in "field of sets" is not used with the meaning of field from field theory.) Elements of formula_4 are called points and those of formula_3 are called complexes and are said to be the admissible sets of formula_2.
Heyting algebra
In mathematics, a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation "a" → "b" of "implication" such that "c" ∧ "a" ≤ "b" is equivalent to "c" ≤ "a" → "b". From a logical standpoint, "A" → "B" is by this definition the weakest proposition for which modus ponens, the inference rule "A" → "B", "A" ⊢ "B", is sound. Equivalently a Heyting algebra is a residuated lattice whose monoid operation "a"⋅"b" is "a" ∧ "b"; yet another definition is as a posetal cartesian closed category with all finite sums.
Boolean ring
In mathematics, a Boolean ring "R" is a ring for which "x" = "x" for all "x" in "R", such as the ring of integers modulo 2. That is, "R" consists only of idempotent elements. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference ("not" disjunction ∨, which would constitute a semiring). Boolean rings are named after the founder of Boolean algebra, George Boole. There are at least four different and incompatible systems of notation for Boolean rings and algebras. Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity.
Complete Boolean algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra "A" has an essentially unique completion, which is a complete Boolean algebra containing "A" such that every element is the supremum of some subset of "A". As a partially ordered set, this completion of "A" is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum.