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Subsumption Lattice
A subsumption lattice is a mathematical structure used in the theoretical background of automated theorem proving and other symbolic computation applications. Definition A Term (logic), term ''t''1 is said to ''subsume'' a term ''t''2 if a Substitution (logic), substitution ''σ'' exists such that ''σ'' applied to ''t''1 yields ''t''2. In this case, ''t''1 is also called ''more general than'' ''t''2, and ''t''2 is called ''more specific than'' ''t''1, or ''an instance of'' ''t''1. The set of all (first-order) terms over a given Signature (logic), signature can be made a Lattice (order), lattice over the partial ordering relation "''... is more specific than ...''" as follows: * consider two terms equal if they differ only in their variable naming, * add an artificial minimal element Ω (the ''overspecified term''), which is considered to be more specific than any other term. This lattice is called the subsumption lattice. Two terms are said to be unifiable if their meet differs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N5 Terms
N5 or N-5 may refer to: Science and technology * N5, the minimal non-modular and non-distributive lattice in mathematical order theory *N5, abbreviation for the 5 nanometer semiconductor technology process node Roads Other uses * N°5, a shortening for Number Five, see Number Five (other) * London Buses route N5 * Nexus 5, an Android smartphone * N5, a postcode district in the N postcode area, North London, England * SP&S Class N-5, a steam locomotives class, used by the Spokane, Portland and Seattle Railway * USS ''N-5'' (SS-57), a 1917 N-class coastal defense submarine of the United States Navy * The first level in the Japanese-Language Proficiency Test * "N5" (song), by Lali, 2022 See also *N05 (other) *Pentazenium (N5+), a pentanitrogen cation in chemistry * pentazolium cation (N5+), a cation that is made up of five nitrogen atoms, in chemistry. *pentazolate In chemistry, a pentazolate is a compound that contains a ''cyclo''-N5− ion, the anio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain (order Theory)
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gordon D
Gordon may refer to: People * Gordon (given name), a masculine given name, including list of persons and fictional characters * Gordon (surname), the surname * Gordon (slave), escaped to a Union Army camp during the U.S. Civil War * Clan Gordon, aka the House of Gordon, a Scottish clan Education * Gordon State College, a public college in Barnesville, Georgia * Gordon College (Massachusetts), a Christian college in Wenham, Massachusetts * Gordon College (Pakistan), a Christian college in Rawalpindi, Pakistan * Gordon College (Philippines), a public university in Subic, Zambales * Gordon College of Education, a public college in Haifa, Israel Places Australia *Gordon, Australian Capital Territory * Gordon, New South Wales * Gordon, South Australia *Gordon, Victoria *Gordon River, Tasmania * Gordon River (Western Australia) Canada *Gordon Parish, New Brunswick *Gordon/Barrie Island, municipality in Ontario *Gordon River (Chochocouane River), a river in Quebec Scotland *Gordo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term (logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constructed from constant symbols, variables and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, is a term built from the constant 1, the variable , and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of . Besides in logic, terms play important roles in universal algebra, and rewriting systems. Formal definition Given a set ''V'' of variable symbols, a set ''C'' of constant symbols and sets ''F''''n'' of ''n''-ary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. Definition As in the case of arbitrary lattices, one can choose to consider a distributive lattice ''L'' either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient. A lattice (''L'',∨,∧) is distributive if the following additional identity holds for all ''x'', ''y'', and ''z'' in ''L'': : ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). Viewing lattices as partiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N5 Linear Terms
N5 or N-5 may refer to: Science and technology * N5, the minimal non-modular and non-distributive lattice in mathematical order theory *N5, abbreviation for the 5 nanometer semiconductor technology process node Roads Other uses * N°5, a shortening for Number Five, see Number Five (other) * London Buses route N5 * Nexus 5, an Android smartphone * N5, a postcode district in the N postcode area, North London, England * SP&S Class N-5, a steam locomotives class, used by the Spokane, Portland and Seattle Railway * USS ''N-5'' (SS-57), a 1917 N-class coastal defense submarine of the United States Navy * The first level in the Japanese-Language Proficiency Test * "N5" (song), by Lali, 2022 See also *N05 (other) *Pentazenium (N5+), a pentanitrogen cation in chemistry * pentazolium cation (N5+), a cation that is made up of five nitrogen atoms, in chemistry. *pentazolate In chemistry, a pentazolate is a compound that contains a ''cyclo''-N5− ion, the anio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M3 Linear Terms
M3, M-3 or M03 may refer to: Computing and electronics * Intel m3, a brand of microprocessors * M.3 (aka NF1/NGSFF), a specification for internally mounted expansion cards * Leica M3, a landmark 35mm rangefinder camera * Modula-3 (M3), a programming language * M3, a British peak programme meter standard used for measuring the volume of audio broadcasts * m3, a macro processor for the AP-3 minicomputer, the predecessor to m4 * M3, a surface-mount version of the 1N4003 general-purpose silicon rectifier diode * M3 (email client), an unreleased email client for the Vivaldi browser Entertainment * M3, a comic book created by Vicente Alcazar * M3 adapter, a Game Boy Advance movie player * M3 (Canadian TV channel), a music and entertainment television channel * M3 (Hungarian TV channel), a Hungarian television channel * '' M3: Malay Mo Ma-develop'', a 2010 Philippine TV series * M3 Music Card, a 2007 flash-based MP3 player * M3 Perfect, M3 Simply and M3 Real, Nintendo DS and 3DS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalizations Of Abc
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest subset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set equipped with one or several methods for producing elements of from other elements of . Operations and (partial) multivariate function are examples of such methods. If is a topological space, the limit of a sequence of element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. Definition As in the case of arbitrary lattices, one can choose to consider a distributive lattice ''L'' either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient. A lattice (''L'',∨,∧) is distributive if the following additional identity holds for all ''x'', ''y'', and ''z'' in ''L'': : ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). Viewing lattices as partiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition, ;Modular law: implies where are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice. In a not necessarily modular lattice, there may s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ground Term
In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables. In first-order logic with identity, the sentence Q(a) \lor P(b) is a ground formula, with a and b being constant symbols. A ground expression is a ground term or ground formula. Examples Consider the following expressions in first order logic over a signature containing the constant symbols 0 and 1 for the numbers 0 and 1, respectively, a unary function symbol s for the successor function and a binary function symbol + for addition. * s(0), s(s(0)), s(s(s(0))), \ldots are ground terms; * 0 + 1, \; 0 + 1 + 1, \ldots are ground terms; * 0+s(0), \; s(0)+ s(0), \; s(0)+s(s(0))+0 are ground terms; * x + s(1) and s(x) are terms, but not ground terms; * s(0) = 1 and 0 + 0 = 0 are ground formulae. Formal definitions What follows is a formal definition for first-order languages. Let a first-order languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
