In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y that admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras. == Definition == In mathematics, a residuated lattice is an algebraic structure L = (L, ≤, •, I) such that (i) (L, ≤) is a lattice. (ii) (L, •, I) is a monoid. (iii) For all z there exists for every x a greatest y, and for every y a greatest x, such that x•y ≤ z (the residuation properties). In (iii), the "greatest y", being a function of z and x, is denoted x\z and called the right residual of z by x. Think of it as what remains of z on the right after "dividing" z on the left by x. Dually, the "greatest x" is denoted z/y and called the left residual of z by y. An equivalent, more formal statement of (iii) that uses these operations to name these greatest values is (iii)' for all x, y, z in L, y ≤ x\z ⇔ x•y ≤ z ⇔ x ≤ z/y. As suggested by the notation, the residuals are a form of quotient. More precisely, for a given x in L, the unary operations x• and x\ are respectively the lower and upper adjoints of a Galois connection on L, and dually for the two functions •y and /y. By the same reasoning that applies to any Galois connection, we have yet another definition of the residuals, namely, x•(x\y) ≤ y ≤ x\(x•y), and (y/x)•x ≤ y ≤ (y•x)/x, together with the requirement that x•y be monotone in x and y. (When axiomatized using (iii) or (iii)' monotonicity becomes a theorem and hence not required in the axiomatization.) These give a sense in which the functions x• and x\ are pseudoinverses or adjoints of each other, and likewise for •x and /x. This last definition is purely in terms of inequalities, noting that monotonicity can be axiomatized as x • y ≤ (x∨z) • y and similarly for the other operations and their arguments. Moreover, any inequality x ≤ y can be expressed equivalently as an equation, either x∧y = x or x∨y = y. This along with the equations axiomatizing lattices and monoids then yields a purely equational definition of residuated lattices, provided the requisite operations are adjoined to the signature (L, ≤, •, I) thereby expanding it to (L, ∧, ∨, •, I, /, \). When thus organized, residuated lattices form an equational class or variety, whose homomorphisms respect the residuals as well as the lattice and monoid operations. Note that distributivity x • (y ∨ z) = (x • y) ∨ (x • z) and x•0 = 0 are consequences of these axioms and so do not need to be made part of the definition. This necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. However distributivity of ∧ over ∨ is entailed when • and ∧ are the same operation, a special case of residuated lattices called a Heyting algebra. Alternative notations for x•y include x◦y, x;y (relation algebra), and x⊗y (linear logic). Alternatives for I include e and 1'. Alternative notations for the residuals are x → y for x\y and y ← x for y/x, suggested by the similarity between residuation and implication in logic, with the multiplication of the monoid understood as a form of conjunction that need not be commutative. When the monoid is commutative the two residuals coincide. When not commutative, the intuitive meaning of the monoid as conjunction and the residuals as implications can be understood as having a temporal quality: x•y means x and then y, x → y means had x (in the past) then y (now), and y ← x means if-ever x (in the future) then y (at that time), as illustrated by the natural language example at the end of the examples. == Examples == One of the original motivations for the study of residuated lattices was the lattice of (two-sided) ideals of a ring. Given a ring R, the ideals of R, denoted Id(R), forms a complete lattice with set intersection acting as the meet operation and "ideal addition" acting as the join operation. The monoid operation • is given by "ideal multiplication", and the element R of Id(R) acts as the identity for this operation. Given two ideals A and B in Id(R), the residuals are given by A / B := { r ∈ R ∣ r B ⊆ A } {\displaystyle A/B:=\{r\in R\mid rB\subseteq A\}} B ∖ A := { r ∈ R ∣ B r ⊆ A } {\displaystyle B\setminus A:=\{r\in R\mid Br\subseteq A\}} It is worth noting that {0}/B and B\{0} are respectively the left and right annihilators of B. This residuation is related to the conductor (or transporter) in commutative algebra written as (A:B)=A/B. One difference in usage is that B need not be an ideal of R: it may just be a subset. Boolean algebras and Heyting algebras are commutative residuated lattices in which x•y = x∧y (whence the unit I is the top element 1 of the algebra) and both residuals x\y and y/x are the same operation, namely implication x → y. The second example is quite general since Heyting algebras include all finite distributive lattices, as well as all chains or total orders, for example the unit interval [0,1] in the real line, or the integers and ± ∞ {\displaystyle \pm \infty } . The structure (Z, min, max, +, 0, −, −) (the integers with subtraction for both residuals) is a commutative residuated lattice such that the unit of the monoid is not the greatest element (indeed there is no least or greatest integer), and the multiplication of the monoid is not the meet operation of the lattice. In this example the inequalities are equalities because − (subtraction) is not merely the adjoint or pseudoinverse of + but the true inverse. Any totally ordered group under addition such as the rationals or the reals can be substituted for the integers in this example. The nonnegative portion of any of these examples is an example provided min and max are interchanged and − is replaced by monus, defined (in this case) so that x-y = 0 when x ≤ y and otherwise is ordinary subtraction. A more general class of examples is given by the Boolean algebra of all binary relations on a set X, namely the power set of X2, made a residuated lattice by taking the monoid multiplication • to be composition of relations and the monoid unit to be the identity relation I on X consisting of all pairs (x,x) for x in X. Given two relations R and S on X, the right residual R\S of S by R is the binary relation such that x(R\S)y holds just when for all z in X, zRx implies zSy (notice the connection with implication). The left residual is the mirror image of this: y(S/R)x holds just when for all z in X, xRz implies ySz. This can be illustrated with the binary relations < and > on {0,1} in which 0 < 1 and 1 > 0 are the only relationships that hold. Then x(>\<)y holds just when x = 1, while x()y holds just when y = 0, showing that residuation of < by > is different depending on whether we residuate on the right or the left. This difference is a consequence of the difference between <•> and >•<, where the only relationships that hold are 0(<•>)0 (since 0<1>0) and 1(>•<)1 (since 1>0<1). Had we chosen ≤ and ≥ instead of < and >, ≥\≤ and ≤/≥ would have been the same because ≤•≥ = ≥•≤, both of which always hold between all x and y (since x≤1≥y and x≥0≤y). The Boolean algebra 2Σ of all formal languages over an alphabet (set) Σ forms a residuated lattice whose monoid multiplication is language concatenation LM and whose monoid unit I is the language {ε} consisting of just the empty string ε. The right residual M\L consists of all words w over Σ such that Mw ⊆ L. The left residual L/M is the same with wM in place of Mw. The residuated lattice of all binary relations on X is finite just when X is finite, and commutative just when X has at most one element. When X is empty the algebra is the degenerate Boolean algebra in which 0 = 1 = I. The residuated lattice of all languages on Σ is commutative just when Σ has at most one letter. It is finite just when Σ is empty, consisting of the two languages 0 (the empty language {}) and the monoid unit I = {ε} = 1. The examples forming a Boolean algebra have special properties treated in the article on residuated Boolean algebras. == Residuated semilattice == A residuated semilattice is defined almost identically for residuated lattices, omitting just the meet operation ∧. Thus it is an algebraic structure L = (L, ∨, •, 1, /, \) satisfying all the residuated lattice equations as specified above except those containing an occurrence of the symbol ∧. The option of defining x ≤ y as x∧y = x is then not available, leaving on
Beauty.AI
Beauty.AI is a mobile beauty pageant for humans and a contest for programmers developing algorithms for evaluating human appearance. The mobile app and website created by Youth Laboratories that uses artificial intelligence technology to evaluate people's external appearance through certain algorithms, such as symmetry, facial blemishes, wrinkles, estimated age and age appearance, and comparisons to actors and models. The Beauty.AI 2.0 contest caused great concern over important ethical issues with deep neural networks such as age, race and gender bias and lead to the creation of the Diversity.AI think tank dedicated to developing new methods for uncovering and managing bias in artificially intelligent systems. Beauty.AI was also an attempt to find approaches on how machines can perceive human face through evaluating particular features, commonly associated with health and beauty. == Concept == The Beauty.AI app was created by Youth Laboratories, a company based out of Russia and Hong Kong that focuses on facial skin analytics. The bioinformation company Insilico Medicine assists in the Beauty.AI app by testing its deep learning techniques to the app. One goal of the app is to reduce the need for human and animal testing as well as improving people's overall health. Its first contest was started in December 2016, and the results were announced in August 2016. More than 60,000 people submitted entries into the contest. The mobile app uses artificial intelligence technology to inspect photographs for certain facial features in order to both determine a person's beauty through artificial means by multiple robots. Part of the Beauty.AI app's purpose is to collect visual and anecdotal data to improve its creator's Youth Laboratories skin analyst skills. == Accusations of racism == There were a total of 44 individuals from different age groups and genders judged as the most attractive, with 37 white entrants, six Asian entrants, and one dark-skinned entrant. The app has received criticism from social justice advocates and computer science professionals. However, Alex Zhavoronkov, PhD, chief science officer of Youth Laboratories and chief technology officer Konstantin Kiselev, both for Youth Laboratories, noted that a lack of data may have contributed to these results. Also, Kiselev added that another issue was that approximately 75% of entrants were white Europeans, whereas only 7% and 1% were from India and Africa, respectively. Kiselev stated that they would work on doing more and better outreach to these areas to improve in this area. Despite this, it was said by Dr. Zhavoronkov that the AI would discard photos of dark-skinned people if the lighting is too poor. Dr. Zhavoronkov vowed to weed out the issues for the next beauty pageant and to try to avoid a similar controversy in the future.
Storyful
Storyful (stylized as storyful.) is a social media intelligence company headquartered in Dublin, Ireland that is a subsidiary of News Corp, offering services such as social news monitoring, video licensing, and reputation risk management tools for corporate clients. The startup was launched as the first social media newswire, a content aggregator, verifying news sources and online content in Dublin in 2010 by Mark Little, a former journalist with RTÉ News. Storyful was acquired by News Corp in 2013 for USD$25 million. == Background == Mark Little, who had worked as a television journalist for RTÉ One, founded startup Storyful in Dublin, Ireland, in 2010, as a service that "verified news sources and online content". According to Nieman Lab, Storyful had a reputation for content aggregation as a social news agency—finding, verifying, distributing, licensing, and commercializing user-generated content, social media and online content from social networking services, including videos about stories in the news, such as the Syrian Civil War, Arab Spring protests, as well as "smaller viral moments". Storyful aimed to provide authority through its verification and monitoring tools while providing authenticity through user-generated content. On 20 December 2013 News Corp purchased Storyful for US$25 million and opened a New York office in the same building as Fox News' main studios. Little left Storyful in 2015 and Gavin Sheridan, Storyful's director of innovation left in 2014. News Corp CEO Robert Thomson said that through Storyful, News Corp would "define the opportunities that the digital landscape presents, rather than simply adapt to them." After the acquisition, the company expanded its service to include "commercial and creative work". After Murdoch acquired the company, from 2014 through to February 2018, losses "swelled", requiring a series of cash injections from News Corp. During that time the company expanded aggressively globally with a staff of about 200 worldwide up from about 30 in 2014. According to The Guardian, in 2016, journalists were encouraged by Storyful to use the social media monitoring software called Verify developed by Storyful. By installing Verify's web browser extension on their computers, Verify would inform the journalists when social media content had been "verified and cleared". The Guardian revealed that through the Verify plugin, dozens of staff in four offices had access to the journalists browsing activity without them knowing. This data allowed Storyful to actively monitor its own clients' activities on social media and to "turn it into an internal feed" at Storyful that "updates in real time". In November 2018, when a video circulated by Infowars' Paul Joseph Watson appeared to prove that CNN's Jim Acosta's contact with a White House intern was a physical blow, Storyful was able to prove that the 15-second-long clip had been doctored. According to a 21 January 2019 article in CNN Business, Rob McDonagh, the editor of Storyful's U.S. news team, had proven that one of the viral videos that served as catalysts in the January 2019 Lincoln Memorial confrontation at 18 January 2019 Indigenous Peoples March, was posted by a suspicious account, under the handle @2020fight. McDonagh's team validates videos and posts before adding them to their "digest", distinguishing true stories from those that are not. Storyful attempts to validate each post or video before including it in its digest. McDonagh reviewed previous content from @2020fight's account, and found it suspicious because it had a high follower count, a "highly polarized and yet inconsistent political messaging", an "unusually high rate of tweets", and "the use of someone else's image in the profile photo." reporter Donie O'Sullivan said that the @2020fight video that had been posted on 18 January, which had 2.5 million views by 22 January, was the one that "helped frame the news cycle". Currently the website offers a service by which video can be commercially brokered. == Services == Services include a newswire service—one of their "core pillars"—and social news monitoring. By February 2018, Storyful was developing "risk and reputation monitoring" services through which they would source and verify social news, fact-checking it and contextualising it for corporate clients. They were "developing tech tools" to "explore obscure or closed networks" for their intelligence team. can use to explore obscure or closed networks. They "track deviations in social conversations around brands and organisations and catch potential risks before they blow up. Like an alerts system." The company "released a re-booted version of its Newswire platform in 2018. According to FORA, Storyful was developing new tools to combat fake news online. == Clients == When Storyful was acquired by News Corp in 2013, the company already had the Wall Street Journal, the BBC, New York Times, YouTube, ITN and Channel 4 News as clients. By 2018 their clients included CNN, ABC News and Fox News, The New York Times, the Washington Post, in the United States, the Australian Broadcasting Corporation and all of News Corp’s own publications. Most of their "reputation-conscious corporate customers" clients prefer to not be named.
Protecting Our Kids from Social Media Addiction Act
Protecting Our Kids from Social Media Addiction Act also known as California SB 976 is a law that was enacted in September 2024 that is meant to address problematic social media usage among minors. The law prohibitions minors to have "addictive feeds" unless they have verifiable parental consent, minor's notifications are also restricted between 12 am to 6 am and during school hours between 8 am and 3 pm it also well requires minors to have default privacies settings and have social media companies to publicly disclose certain metrics about their users. The law was set to take effect in two steps the first being the restrictions on social media feeds, notifications, disclosures from social media companies and default settings which would have taken effect on January 1, 2025, and the age verification provision which would have taken effect on January 1, 2027. However, has faced legal challenges since its enactment delaying its enactment. == Legal Challenges == In November 2024 NetChoice a trade association representing many of the biggest social media companies such as YouTube, Facebook and Instagram sued the attorney general of California Rob Bonta hoping to get an injunction before the first set of the law's provisions would take effect in January of the next year. However, judge Edward Davila would only grant Netchoice's request as to the restrictions on notifications and public disclosures and would deny their request as to the rest of the law. The law was later fully enjoined temporarily by the District Court and Appellant Court pending appeal, and the case is now in the Ninth Circuit Court of Appeals and is pending a decision. === Social media platforms challenges to law === In November 2025 Meta, Google and TikTok filed lawsuits against the law arguing it violates the first amendment.
Group key
In cryptography, a group key is a cryptographic key that is shared between a group of users. Typically, group keys are distributed by sending them to individual users, either physically, or encrypted individually for each user using either that user's pre-distributed private key. A common use of group keys is to allow a group of users to decrypt a broadcast message that is intended for that entire group of users, and no one else. For example, in the Second World War, group keys (known as "iodoforms", a term invented by a classically educated non-chemist, and nothing to do with the chemical of the same name) were sent to groups of agents by the Special Operations Executive. These group keys allowed all the agents in a particular group to receive a single coded message. In present-day applications, group keys are commonly used in conditional access systems, where the key is the common key used to decrypt the broadcast signal, and the group in question is the group of all paying subscribers. In this case, the group key is typically distributed to the subscribers' receivers using a combination of a physically distributed secure cryptoprocessor in the form of a smartcard and encrypted over-the-air messages.
Connection string
In computing, a connection string is a string that specifies information about a data source and the means of connecting to it. It is passed in code to an underlying driver or provider in order to initiate the connection. Whilst commonly used for a database connection, the data source could also be a spreadsheet or text file. The connection string may include attributes such as the name of the driver, server and database, as well as security information such as user name and password. == Examples == This example shows a PostgreSQL connection string for connecting to wikipedia.com with SSL and a connection timeout of 180 seconds: DRIVER={PostgreSQL Unicode};SERVER=www.wikipedia.com;SSL=true;SSLMode=require;DATABASE=wiki;UID=wikiuser;Connect Timeout=180;PWD=ashiknoor Users of Oracle databases can specify connection strings: on the command line (as in: sqlplus scott/tiger@connection_string ) via environment variables ($TWO_TASK in Unix-like environments; %TWO_TASK% in Microsoft Windows environments) in local configuration files (such as the default $ORACLE_HOME/network/admin.tnsnames.ora) in LDAP-capable directory services
NYSERNet
NYSERNet, Inc. (New York State Education and Research Network), is a non-profit Internet service provider in New York State. It mainly provides Internet access to universities, colleges, museums, health care facilities, primary and secondary schools, and research institutions. == History == NYSERNet was founded in 1986 in Troy, New York. Its founders compared NYSERNet's network with the Erie Canal and considered it the next step in two centuries to draw the country together. NYSERNet's network reaches from Buffalo to New York City. Completed in 1987, it was the first statewide regional IP network in the United States.[1] Initial speed of 56 kbps was upgraded to T1 in 1989 and T3 in 1994. It was the original assignee of AS174 according to RFC1117. This ASN is used today by Cogent Communications for their global network.