Semantic integration is the process of interrelating information from diverse sources, for example calendars and to do lists, email archives, presence information (physical, psychological, and social), documents of all sorts, contacts (including social graphs), search results, and advertising and marketing relevance derived from them. In this regard, semantics focuses on the organization of and action upon information by acting as an intermediary between heterogeneous data sources, which may conflict not only by structure but also context or value. == Applications and methods == In enterprise application integration (EAI), semantic integration can facilitate or even automate the communication between computer systems using metadata publishing. Metadata publishing potentially offers the ability to automatically link ontologies. One approach to (semi-)automated ontology mapping requires the definition of a semantic distance or its inverse, semantic similarity and appropriate rules. Other approaches include so-called lexical methods, as well as methodologies that rely on exploiting the structures of the ontologies. For explicitly stating similarity/equality, there exist special properties or relationships in most ontology languages. OWL, for example has "owl:equivalentClass", "owl:equivalentProperty" and "owl:sameAs". Eventually system designs may see the advent of composable architectures where published semantic-based interfaces are joined together to enable new and meaningful capabilities. These could predominately be described by means of design-time declarative specifications, that could ultimately be rendered and executed at run-time. Semantic integration can also be used to facilitate design-time activities of interface design and mapping. In this model, semantics are only explicitly applied to design and the run-time systems work at the syntax level. This "early semantic binding" approach can improve overall system performance while retaining the benefits of semantic driven design. == Semantic integration situations == From the industry use case, it has been observed that the semantic mappings were performed only within the scope of the ontology class or the datatype property. These identified semantic integrations are (1) integration of ontology class instances into another ontology class without any constraint, (2) integration of selected instances in one ontology class into another ontology class by the range constraint of the property value and (3) integration of ontology class instances into another ontology class with the value transformation of the instance property. Each of them requires a particular mapping relationship, which is respectively: (1) equivalent or subsumption mapping relationship, (2) conditional mapping relationship that constraints the value of property (data range) and (3) transformation mapping relationship that transforms the value of property (unit transformation). Each identified mapping relationship can be defined as either (1) direct mapping type, (2) data range mapping type or (3) unit transformation mapping type. == KG vs. RDB approaches == In the case of integrating supplemental data source, KG(Knowledge graph) formally represents the meaning involved in information by describing concepts, relationships between things, and categories of things. These embedded semantics with the data offer significant advantages such as reasoning over data and dealing with heterogeneous data sources. The rules can be applied on KG more efficiently using graph query. For example, the graph query does the data inference through the connected relations, instead of repeated full search of the tables in relational database. KG facilitates the integration of new heterogeneous data by just adding new relationships between existing information and new entities. This facilitation is emphasized for the integration with existing popular linked open data source such as Wikidata.org. SQL query is tightly coupled and rigidly constrained by datatype within the specific database and can join tables and extract data from tables, and the result is generally a table, and a query can join tables by any columns which match by datatype. SPARQL query is the standard query language and protocol for Linked Open Data on the web and loosely coupled with the database so that it facilitates the reusability and can extract data through the relations free from the datatype, and not only extract but also generate additional knowledge graph with more sophisticated operations(logic: transitive/symmetric/inverseOf/functional). The inference based query (query on the existing asserted facts without the generation of new facts by logic) can be fast comparing to the reasoning based query (query on the existing plus the generated/discovered facts based on logic). The information integration of heterogeneous data sources in traditional database is intricate, which requires the redesign of the database table such as changing the structure and/or addition of new data. In the case of semantic query, SPARQL query reflects the relationships between entities in a way that aligned with human's understanding of the domain, so the semantic intention of the query can be seen on the query itself. Unlike SPARQL, SQL query, which reflects the specific structure of the database and derived from matching the relevant primary and foreign keys of tables, loses the semantics of the query by missing the relationships between entities. Below is the example that compares SPARQL and SQL queries for medications that treats "TB of vertebra". SELECT ?medication WHERE { ?diagnosis a example:Diagnosis . ?diagnosis example:name “TB of vertebra” . ?medication example:canTreat ?diagnosis . } SELECT DRUG.medID FROM DIAGNOSIS, DRUG, DRUG_DIAGNOSIS WHERE DIAGNOSIS.diagnosisID=DRUG_DIAGNOSIS.diagnosisID AND DRUG.medID=DRUG_DIAGNOSIS.medID AND DIAGNOSIS.name=”TB of vertebra” == Examples == The Pacific Symposium on Biocomputing has been a venue for the popularization of the ontology mapping task in the biomedical domain, and a number of papers on the subject can be found in its proceedings.
ConEmu
ConEmu (short for Console emulator) is a free and open-source tabbed terminal emulator for Windows. ConEmu presents multiple consoles and simple GUI applications as one customizable GUI window with tabs and a status bar. It also provides emulation for ANSI escape codes for color, bypassing the capabilities of the standard Windows Console Host to provide 256 and 24-bit color in Windows. The program has a large range of customization, including custom color palettes for the standard 16 colors, hotkeys, transparency, an auto-hideable mode (similar to the way Quake originally displayed its developer console). Initially, the program was created as a companion to Far Manager, bringing some features common for graphical file managers to this console application (thumbnails and tiles, drag and drop with other windows, true color interface, and others). As of 2012, ConEmu could be used with any other Win32 console application or simple GUI tool (such as Notepad, PuTTY or DOSBox). ConEmu doesn't provide any shell itself, but rather allows using any other shell. It does provide a limited macro language, to control the hosted applications startup.
2024 Bilderberg Conference
The 2024 Bilderberg Conference was held between May 30–June 2, 2024 in Madrid, Spain at the Eurostars Suites Mirasierra hotel. The 2024 meeting was the 70th edition of the event. A Bilderberg Group press release stated that there were 131 participants from around 25 countries. Established in 1954 by Prince Bernhard of the Netherlands, Bilderberg conferences (or meetings) are an annual private gathering of the European and North American political and business elite. Events are attended by between 120 and 150 people each year invited by the Bilderberg Group's steering committee; including prominent politicians, CEOs, national security experts, academics and journalists. Several US presidents have attended the meetings before winning a presidential election. These politicians include Bill Clinton and Barack Obama. Bilderberg conferences operate under the Chatham House Rule, meaning that participants are sworn to secrecy and cannot disclose the identity or affiliation of any particular speaker. == Agenda == The key topics for discussion were announced on the Bilderberg website shortly before the meeting. These topics included: == Participants == A list of 131 participants was published on the Bilderberg website. This list may not be complete, as a source connected to the Bilderberg group told The Daily Telegraph in 2013 that some attendees do not have their names publicized. King Felipe VI of Spain was reported to have attended the meeting despite his name not being on the list.
TuVox
TuVox is a company that produces VXML-based telephone speech-recognition applications to replace DTMF touch-tone systems for their clients. == History == TuVox was founded in 2001 by Steven S. Pollock and Ashok Khosla, formerly of Apple Computer Corporation and Claris Corporation. Since then, TuVox has grown to over 150 employees and has US offices in Cupertino, California and Boca Raton, Florida as well as international offices in London, Vancouver and Sydney. In 2005, TuVox acquired the customers and hosting facilities of Net-By-Tel. In 2007, the company raised $20m for its speech recognition, and phone menu software. On July 22, 2010, West Interactive — a subsidiary of West Corporation — announced its acquisition of TuVox. == Customers == TuVox clients include: 1-800-Flowers.com, AMC Entertainment, American Airlines, British Airways, M&T Bank, Canon Inc., Gateway, Inc., Motorola, Progress Energy Inc., Telecom New Zealand, Time, Inc., BECU, Virgin America and USAA.
Removal of Sam Altman from OpenAI
On November 17, 2023, OpenAI's board of directors ousted co-founder and chief executive Sam Altman. In an official post on the company's website, it was stated that "the board no longer has confidence in his ability to continue leading OpenAI". The removal was predicated by employee concerns about his handling of artificial intelligence safety, and allegations of abusive behavior. Altman was reinstated on November 22 after pressure from employees and investors. The removal and subsequent reinstatement caused widespread reactions, including impacts felt in the financial markets and technology sector. Microsoft, a partner of OpenAI, received little notice of the removal and experienced a drop in the share price of its stock. The removal also promoted interest in investigations from regulatory agencies. == Background == === OpenAI === OpenAI is an artificial intelligence firm founded in December 2015 as a non-profit entity. The for-profit division of the organization released ChatGPT in November 2022, contributing to a resurgence in generative artificial intelligence funding. The board of directors of the controlling non-profit formerly comprised chief scientist Ilya Sutskever, as well as Adam D'Angelo, chief executive of Quora, entrepreneur Tasha McCauley, and Helen Toner, strategy director for the Center for Security and Emerging Technology. As of October 2023, the company is valued at US$80 billion and was set to bring in US$1 billion in revenue. Altman has described OpenAI's relationship with Microsoft as the "best bromance in tech". OpenAI is uniquely structured, an intentional decision to avoid investor control. A board of directors controls the non-profit OpenAI, Inc. The non-profit owns and controls a for-profit company itself controlling a capped-profit company, OpenAI Global, LLC and a holding company owned by employees and other investors. The holding company is the majority owner of OpenAI Global, LLC.; Microsoft owns a minority stake in the capped-profit company. OpenAI's bylaws, enacted in January 2016, allow a majority of its board of directors to remove any director without prior warning or a formal meeting with written consent. === Sam Altman === Sam Altman is a co-founder of OpenAI and its former chief executive; Altman took over the company following co-chair Elon Musk's resignation in 2018. Under Altman, OpenAI has shifted to becoming a for-profit entity. Altman is credited with convincing Microsoft chief executive Satya Nadella with investing US$10 billion in cash and computing credits into OpenAI and leading several tender offer transactions that tripled the company's valuation. Altman testified before the United States Congress speaking critically of artificial intelligence and appeared at the 2023 AI Safety Summit. In the days leading up to his removal, Altman made several public appearances, announcing the GPT-4 Turbo platform at OpenAI's DevDay conference, attending APEC United States 2023, and speaking at an event related to Burning Man. == Events leading up to the removal == The resignation of LinkedIn co-founder Reid Hoffman, venture capitalist Shivon Zilis, and former Republican representative Will Hurd from the board allowed the remaining members to remove Altman. According to Kara Swisher and The Wall Street Journal, Sutskever was instrumental in Altman's removal. Disagreements over the safety of artificial intelligence divided employees prior to Altman's removal. The release of ChatGPT created divisions with OpenAI as a for-profit company without considerations for the safety of artificial intelligence and a non-profit cautious of artificial intelligence's capabilities; in a staff email sent in 2019 and obtained by The Atlantic, Altman referred to these divisions as "tribes". Prior to his removal, Altman was seeking billions from Middle Eastern sovereign wealth funds to develop an artificial intelligence chip to compete with Nvidia and courted SoftBank chairman Masayoshi Son to develop artificial intelligence hardware with former Apple designer Jony Ive. Sutskever and his allies opposed these efforts, viewing them as unjustly using the OpenAI name. Altman reduced Sutskever's role in October 2023, furthering divisions; Sutskever successfully appealed to several members of the board. Swisher and The Verge reporter Alex Heath stated that opposition to Altman's profit-driven strategy culminated in the DevDay conference in which Altman announced custom ChatGPT instances. According to Axios, the removal was driven by growing discontent and distrust with Altman. On November 22, 2023, reports emerged suggesting that Sam Altman's dismissal from OpenAI might be linked to his alleged mishandling of a significant breakthrough in the organization's secretive project codenamed Q. According to sources within OpenAI, Q is aimed at developing AI capabilities in logical and mathematical reasoning, and reportedly involves performing math on the level of grade-school students. Concerns about Altman's response to this development, specifically regarding the potential safety implications of the discovery, were reportedly raised to the company's board shortly before his firing. A report from The Washington Post in December stated that OpenAI's board of directors were concerned over Altman's allegedly abusive behavior; the complaints were purportedly a major factor in his removal. The Post previously reported that Altman's alleged pattern of deception and subversiveness that ostensibly resulted in his removal from Y Combinator ultimately resulted in the board's decision to remove him. In April 2026, an investigative report from The New Yorker found that Sutskever and others, in response to the board's request, had compiled an approximately 70-page-long annotated dossier consisting of internal communications, documents, and photos. The dossier claimed that Altman "exhibits a consistent pattern of [...] Lying", and that Altman misrepresented information to the company's senior management and board, particularly regarding safety issues. == Removal == On November 17, 2023, at approximately noon PST, OpenAI's board of directors ousted Altman effective immediately following a "deliberative review process". The board concluded that Altman was not "consistently candid in his communications". Altman was informed of his removal five to ten minutes before it occurred on a Google Meet while watching the Las Vegas Grand Prix. Within thirty minutes, Sutskever invited OpenAI chairman and president Greg Brockman to a Google Meet to inform him of Altman's removal. According to an internal memo obtained by Axios, the removal was not due to "malfeasance", and OpenAI chief executive Emmett Shear denied accusations that the removal was due to disagreements. The board publicly announced Altman's removal thirty minutes later. Chief Technology Officer Mira Murati was immediately appointed to interim chief executive officer. Hours after Altman's removal, Brockman resigned as chairman, joined by director of research Jakub Pachocki and researchers Aleksander Mądry and Szymon Sidor. During an all-hands meeting, Sutskever defended the ouster and denied accusations of a hostile takeover. An OpenAI representative requested former board member Will Hurd's presence. == Reinstatement == According to The New Yorker, Altman retreated to his San Francisco home and enlisted the help of communications consultant Chris Lehane and Airbnb chief executive Brian Chesky, as well as former staff and a legal team, to plan his reinstatement. Lehane encouraged Altman to engage on social media, while Chesky sent a journalist negative information about the board. Altman told interim CEO Murati that his team was conducting opposition research on her and the individuals responsible for his removal; Altman later stated he did not remember saying this. Altman insisted multiple times that all board members who supported his removal should resign. Tiger Global Management and Sequoia Capital had attempted to reinstate Altman, according to The Information; Bloomberg News reported that Microsoft and Thrive Capital were seeking Altman's reinstatement. On November 18, The Verge reported that OpenAI's board of directors discussed reinstating Altman. The board agreed in principle to resign and to allow Altman to return, but missed the deadline. According to The Verge, Altman was ambivalent about returning and would seek significant changes to the company, including replacing the board. A list of directors had been prepared by investors in the event that the board steps down, and purportedly included former Salesforce executive Bret Taylor. According to chief strategy officer Jason Kwon, OpenAI was optimistic it could return Altman, Brockman, and other employees. On November 19, Altman and Brockman appeared at OpenAI's headquarters to negotiate, mediated by Nadella. According to Bloomberg News, Murati, Kwon, and chief operating officer Brad Lightcap were pushing for a new board of direc
Lose It!
Lose It! is an American health and wellness mobile app developed by FitNow, Inc. The app generates calorie budgets for users by tracking weight, exercise, food and calorie intake, and personal goals, primarily to assist them in achieving weight loss. == History == Lose It! was developed in Boston and debuted in 2008. The app and its associated company were founded by J.J. Allaire, Charles Teague and Paul Dicristina. Prior to founding Lose It!, Teague and Allaire had founded the online research tool Onfolio, which was acquired by Microsoft in 2006. The Lose It! app was originally released as an iOS app before being released as a website in 2010 and an Android app in 2011. In 2015, Lose It! announced plans to release the app internationally. Lose It! was also available as an app for Apple Watch at its launch in 2015. The app’s “Snap It” feature, which allows users to approximate calorie counts by taking pictures of their daily meals and snacks, was released in beta in 2016. Snap It was named an Innovation Awards Honoree at the 2017 Consumer Electronics Show in Las Vegas. In 2020, Patrick Wetherille, one of the company’s earliest employees, was appointed chief executive officer. == App == Lose It! is weight loss app. The app allows users to set goals such as increasing strength, overall health/maintenance, and weight loss. It provides users recommended calorie budgets based on data such as their current weight and their desired weight. Lose It! also tracks data such as exercise/activity level and food consumption and allows users to track calories consumed by scanning barcodes for food products then retrieving calorie information for products. The app can also estimate the amount of calories in a food products. Lose It! has integration features connecting it to other apps such as Fitbit and Runkeeper. It also has social features such as joining groups and sharing progress with friends. The Premium version of the app allows users to track foods according to specific diets like keto, heart healthy or Mediterranean.
Residuated Boolean algebra
In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ {\displaystyle \Sigma } under concatenation, the set of all binary relations on a given set X {\displaystyle X} under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example. == Definition == A residuated Boolean algebra is an algebraic structure ( L , ∧ , ∨ , ¬ , 0 , 1 , ∙ , I , / , ∖ ) {\displaystyle (L,\wedge ,\vee ,\neg ,0,1,\bullet ,\mathbf {I} ,/,\backslash )} such that An equivalent signature better suited to the relation algebra application is ( L , ∧ , ∨ , ¬ , 0 , 1 , ∙ , I , ▹ , ◃ ) {\displaystyle (L,\wedge ,\vee ,\neg ,0,1,\bullet ,\mathbf {I} ,\triangleright ,\triangleleft )} where the unary operations x ∖ {\displaystyle x\backslash } and x ▹ {\displaystyle x\triangleright } are intertranslatable in the manner of De Morgan's laws via x ∖ y = ¬ ( x ▹ ¬ y ) {\displaystyle x\backslash y=\neg (x\triangleright \neg y)} , x ▹ y = ¬ ( x ∖ ¬ y ) {\displaystyle x\triangleright y=\neg (x\backslash \neg y)} , and dually / y {\displaystyle /y} and ◃ y {\displaystyle \triangleleft y} as x / y = ¬ ( ¬ x ◃ y ) {\displaystyle x/y=\neg (\neg x\triangleleft y)} , x ◃ y = ¬ ( ¬ x / y ) {\displaystyle x\triangleleft y=\neg (\neg x/y)} , with the residuation axioms in the residuated lattice article reorganized accordingly (replacing z {\displaystyle z} by ¬ z {\displaystyle \neg z} ) to read ( x ▹ z ) ∧ y = 0 ⇔ ( x ∙ y ) ∧ z = 0 ⇔ ( z ◃ y ) ∧ x = 0 {\displaystyle (x\triangleright z)\wedge y=0\ \Leftrightarrow \ (x\bullet y)\wedge z=0\ \Leftrightarrow \ (z\triangleleft y)\wedge x=0} This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy. Since residuated lattices and Boolean algebras are each definable with finitely many equations, so are residuated Boolean algebras, whence they form a finitely axiomatizable variety. == Examples == Any Boolean algebra, with the monoid multiplication ∙ {\displaystyle \bullet } taken to be conjunction and both residuals taken to be material implication x → y {\displaystyle x\to y} . Of the remaining 15 binary Boolean operations that might be considered in place of conjunction for the monoid multiplication, only five meet the monotonicity requirement, namely 0 , 1 , x , y {\displaystyle 0,1,x,y} and x ∨ y {\displaystyle x\vee y} . Setting y = z = 0 {\displaystyle y=z=0} in the residuation axiom y ≤ x ∖ z ⇔ x ∙ y ≤ z {\displaystyle y\leq x\backslash z\ \Leftrightarrow \ x\bullet y\leq z} , we have 0 ≤ x ∖ 0 ⇔ x ∙ 0 ≤ 0 {\displaystyle 0\leq x\backslash 0\ \Leftrightarrow \ x\bullet 0\leq 0} , which is falsified by taking x = 1 {\displaystyle x=1} when x ∙ y = 1 {\displaystyle x\bullet y=1} , x {\displaystyle x} , or x ∨ y {\displaystyle x\vee y} . The dual argument for z / y {\displaystyle z/y} rules out x ∙ y = y {\displaystyle x\bullet y=y} . This just leaves x ∙ y = 0 {\displaystyle x\bullet y=0} (a constant binary operation independent of x {\displaystyle x} and y {\displaystyle y} ), which satisfies almost all the axioms when the residuals are both taken to be the constant operation x / y = x ∖ y = 1 {\displaystyle x/y=x\backslash y=1} . The axiom it fails is x ∙ I = x = I ∙ x {\displaystyle x\bullet \mathbf {I} =x=\mathbf {I} \bullet x} , for want of a suitable value for I {\displaystyle \mathbf {I} } . Hence conjunction is the only binary Boolean operation making the monoid multiplication that of a residuated Boolean algebra. The power set 2 X 2 {\displaystyle 2^{X^{2}}} made a Boolean algebra as usual with ∩ {\displaystyle \cap } , ∪ {\displaystyle \cup } and complement relative to X 2 {\displaystyle X^{2}} , and made a monoid with relational composition. The monoid unit I {\displaystyle \mathbf {I} } is the identity relation { ( x , x ) | x ∈ X } {\displaystyle \{(x,x)|x\in X\}} . The right residual R ∖ S {\displaystyle R\backslash S} is defined by x ( R ∖ S ) y ⇔ ∀ z ∈ X , z R x ⇒ z S y {\displaystyle x(R\backslash S)y\ \Leftrightarrow \ \forall z\in X,zRx\Rightarrow zSy} . Dually the left residual S / R {\displaystyle S/R} is defined by y ( S / R ) x ⇔ ∀ z ∈ X , x R z ⇒ y S z {\displaystyle y(S/R)x\ \Leftrightarrow \ \forall z\in X,xRz\Rightarrow ySz} . The power set 2 Σ ∗ {\displaystyle 2^{\Sigma ^{}}} made a Boolean algebra as for Example 2, but with language concatenation for the monoid. Here the set Σ {\displaystyle \Sigma } is used as an alphabet while Σ ∗ {\displaystyle \Sigma ^{}} denotes the set of all finite (including empty) words over that alphabet. The concatenation L M {\displaystyle LM} of languages L {\displaystyle L} and M {\displaystyle M} consists of all words u v {\displaystyle uv} such that u ∈ L {\displaystyle u\in L} and v ∈ M {\displaystyle v\in M} . The monoid unit is the language { ε } {\displaystyle \{\varepsilon \}} consisting of just the empty word ε {\displaystyle \varepsilon } . The right residual M ∖ L {\displaystyle M\backslash L} consists of all words w {\displaystyle w} over Σ {\displaystyle \Sigma } such that M w ⊆ L {\displaystyle Mw\subseteq L} . The left residual L / M {\displaystyle L/M} is the same with w M {\displaystyle wM} in place of M w {\displaystyle Mw} . == Conjugacy == The De Morgan duals ▹ {\displaystyle \triangleright } and ◃ {\displaystyle \triangleleft } of residuation arise as follows. Among residuated lattices, Boolean algebras are special by virtue of having a complementation operation ¬ {\displaystyle \neg } . This permits an alternative expression of the three inequalities y ≤ x ∖ z ⇔ x ∙ y ≤ z ⇔ x ≤ z / y {\displaystyle y\leq x\backslash z\ \Leftrightarrow \ x\bullet y\leq z\ \Leftrightarrow \ x\leq z/y} in the axiomatization of the two residuals in terms of disjointness, via the equivalence x ≤ y ⇔ x ∧ ¬ y = 0 {\displaystyle x\leq y\ \Leftrightarrow \ x\wedge \neg y=0} . Abbreviating x ∧ y = 0 {\displaystyle x\wedge y=0} to x # y {\displaystyle x\#y} as the expression of their disjointness, and substituting ¬ z {\displaystyle \neg z} for z {\displaystyle z} in the axioms, they become with a little Boolean manipulation ¬ ( x ∖ ¬ z ) # y ⇔ x ∙ y # z ⇔ ¬ ( ¬ z / y ) # x {\displaystyle \neg (x\backslash \neg z)\#y\ \Leftrightarrow \ x\bullet y\#z\ \Leftrightarrow \ \neg (\neg z/y)\#x} Now ¬ ( x ∖ ¬ z ) {\displaystyle \neg (x\backslash \neg z)} is reminiscent of De Morgan duality, suggesting that x ∖ {\displaystyle x\backslash } be thought of as a unary operation f {\displaystyle f} , defined by f ( y ) = x ∖ y {\displaystyle f(y)=x\backslash y} , that has a De Morgan dual ¬ f ( ¬ y ) {\displaystyle \neg f(\neg y)} , analogous to ∀ x ϕ ( x ) = ¬ ∃ x ¬ ϕ ( x ) {\displaystyle \forall x\phi (x)=\neg \exists x\neg \phi (x)} . Denoting this dual operation as x ▹ {\displaystyle x\triangleright } , we define x ▹ z {\displaystyle x\triangleright z} as ¬ x ∖ ¬ z {\displaystyle \neg x\backslash \neg z} . Similarly we define another operation z ◃ y {\displaystyle z\triangleleft y} as ¬ ( ¬ z / y ) {\displaystyle \neg (\neg z/y)} . By analogy with x ∖ {\displaystyle x\backslash } as the residual operation associated with the operation x ∙ {\displaystyle x\bullet } , we refer to x ▹ {\displaystyle x\triangleright } as the conjugate operation, or simply conjugate, of x ∙ {\displaystyle x\bullet } . Likewise ◃ y {\displaystyle \triangleleft y} is the conjugate of ∙ y {\displaystyle \bullet y} . Unlike residuals, conjugacy is an equivalence relation between operations: if f {\displaystyle f} is the conjugate of g {\displaystyle g} then g {\displaystyle g} is also the conjugate of f {\displaystyle f} , i.e. the conjugate of the conjugate of f {\displaystyle f} is f {\displaystyle f} . Another advantage of conjugacy is that it becomes unnecessary to speak of right and left conjugates, that distinction now being inherited from the difference between x ∙ {\displaystyle x\bullet } and ∙ x {\displaystyle \bullet x} , which have as their respective conjugates x ▹ {\displaystyle x\triangleright } and ◃ x {\displaystyle \triangleleft x} . (But this advantage accrues also to residuals when x ∖ {\displaystyle x\backslash } is taken to be the residual operation to x ∙ {\displaystyle x\bullet } .) All this yields (along with the Boolean algebra and monoid axioms) the following equivalent axiomatization of a residuated Boolean algebra. y # x ▹ z ⇔ x ∙ y # z ⇔ x # z ◃ y {\displaystyle y\#x\triangleright z\ \Leftrightarrow \ x\bullet y\#z\ \Leftrightarrow \ x\#z\triangleleft y} With this signature it remains the case that this axiomatization can be expressed as