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slovo definícia algebra (encz) algebra,algebra algebra (czen) algebra,algebra Algebra (gcide) Mathematics \Math`e*mat"ics\, n. [F. math['e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See Mathematic, and -ics.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations. [1913 Webster] Note: Mathematics embraces three departments, namely: 1. Arithmetic. 2. Geometry, including Trigonometry and Conic Sections. 3. Analysis, in which letters are used, including Algebra, Analytical Geometry, and Calculus. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations. [1913 Webster] Algebra (gcide) Algebra \Al"ge*bra\, n. [LL. algebra, fr. Ar. al-jebr reduction of parts to a whole, or fractions to whole numbers, fr. jabara to bind together, consolidate; al-jebr w'almuq[=a]balah reduction and comparison (by equations): cf. F. alg[`e]bre, It. & Sp. algebra.] 1. (Math.) That branch of mathematics which treats of the relations and properties of quantity by means of letters and other symbols. It is applicable to those relations that are true of every kind of magnitude. [1913 Webster] 2. A treatise on this science. [1913 Webster] Algebraic algebra (wn) algebra n 1: the mathematics of generalized arithmetical operations algebra (foldoc) algebra 1. A loose term for an {algebraic structure}. 2. A vector space that is also a ring, where the vector space and the ring share the same addition operation and are related in certain other ways. An example algebra is the set of 2x2 matrices with {real numbers} as entries, with the usual operations of addition and matrix multiplication, and the usual scalar multiplication. Another example is the set of all polynomials with real coefficients, with the usual operations. In more detail, we have: (1) an underlying set, (2) a field of scalars, (3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a vector space, (4) an operation of addition of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a vector space or a ring, (5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring. This whole thing constitutes an `algebra' iff: (1) it is a vector space if you discard item (5) and (2) it is a ring if you discard (2) and (3) and (3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB). In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra. Note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example. Another example (an example of a Banach algebra) is the set of all bounded linear operators on a Hilbert space, with the usual norm. The multiplication is the operation of composition of operators, and the addition and scalar multiplication are just what you would expect. Two other examples are tensor algebras and {Clifford algebras}. [I. N. Herstein, "Topics in Algebra"]. (1999-07-14)

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Zdeněk Brož linear algebra (encz) linear algebra, n: matrix algebra (encz) matrix algebra, n: vector algebra (encz) vector algebra, n: algebra (czen) algebra,algebra algebra logiky (czen) algebra logiky,algebra of logic algebraicky (czen) algebraicky,algebraic algebraicky,algebraical algebraicky,algebraically[mat.] algebraická rovnice (czen) algebraická rovnice,algebraic equationn: [mat.] algebraické číslo (czen) algebraické číslo,algebraic number algebraik (czen) algebraik,algebraistn: Zdeněk Brož boolovská algebra (czen) boolovská algebra,algebra of logic high speed algebraic logic (czen) High Speed Algebraic Logic,HSAL[zkr.] [voj.] Zdeněk Brož a automatický překlad součet algebraicky (czen) součet algebraicky,algebraic sum Algebraic (gcide) Algebraic \Al`ge*bra"ic\, Algebraical \Al`ge*bra"ic*al\, a. 1. Of or pertaining to algebra; using algebra; according to the laws of algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings; algebraic geometry. [1913 Webster] 2. progressing by constant multiplicatory factors; -- of a series of numbers. Contrasted to arithmetical. algebraic progression Syn: algebraic [PJC] Algebraic curve, a curve such that the equation which expresses the relation between the co["o]rdinates of its points involves only the ordinary operations of algebra; -- opposed to a transcendental curve. [1913 Webster] Algebraic curve (gcide) Algebraic \Al`ge*bra"ic\, Algebraical \Al`ge*bra"ic*al\, a. 1. Of or pertaining to algebra; using algebra; according to the laws of algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings; algebraic geometry. [1913 Webster] 2. progressing by constant multiplicatory factors; -- of a series of numbers. Contrasted to arithmetical. algebraic progression Syn: algebraic [PJC] Algebraic curve, a curve such that the equation which expresses the relation between the co["o]rdinates of its points involves only the ordinary operations of algebra; -- opposed to a transcendental curve. [1913 Webster] Algebraic function (gcide) Function \Func"tion\, n. [L. functio, fr. fungi to perform, execute, akin to Skr. bhuj to enjoy, have the use of: cf. F. fonction. Cf. Defunct.] 1. The act of executing or performing any duty, office, or calling; performance. "In the function of his public calling." --Swift. [1913 Webster] 2. (Physiol.) The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body. [1913 Webster] 3. The natural or assigned action of any power or faculty, as of the soul, or of the intellect; the exertion of an energy of some determinate kind. [1913 Webster] As the mind opens, and its functions spread. --Pope. [1913 Webster] 4. The course of action which peculiarly pertains to any public officer in church or state; the activity appropriate to any business or profession. [1913 Webster] Tradesmen . . . going about their functions. --Shak. [1913 Webster] The malady which made him incapable of performing his regal functions. --Macaulay. [1913 Webster] 5. (Math.) A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x^2, 3^x, Log. x, and Sin. x, are all functions of x. [1913 Webster] 6. (Eccl.) A religious ceremony, esp. one particularly impressive and elaborate. Every solemn `function' performed with the requirements of the liturgy. --Card. Wiseman. [Webster 1913 Suppl.] 7. A public or social ceremony or gathering; a festivity or entertainment, esp. one somewhat formal. This function, which is our chief social event. --W. D. Howells. [Webster 1913 Suppl.] Algebraic function, a quantity whose connection with the variable is expressed by an equation that involves only the algebraic operations of addition, subtraction, multiplication, division, raising to a given power, and extracting a given root; -- opposed to transcendental function. Arbitrary function. See under Arbitrary. Calculus of functions. See under Calculus. Carnot's function (Thermo-dynamics), a relation between the amount of heat given off by a source of heat, and the work which can be done by it. It is approximately equal to the mechanical equivalent of the thermal unit divided by the number expressing the temperature in degrees of the air thermometer, reckoned from its zero of expansion. Circular functions. See Inverse trigonometrical functions (below). -- Continuous function, a quantity that has no interruption in the continuity of its real values, as the variable changes between any specified limits. Discontinuous function. See under Discontinuous. Elliptic functions, a large and important class of functions, so called because one of the forms expresses the relation of the arc of an ellipse to the straight lines connected therewith. Explicit function, a quantity directly expressed in terms of the independently varying quantity; thus, in the equations y = 6x^2, y = 10 -x^3, the quantity y is an explicit function of x. Implicit function, a quantity whose relation to the variable is expressed indirectly by an equation; thus, y in the equation x^2 + y^2 = 100 is an implicit function of x. Inverse trigonometrical functions, or Circular functions, the lengths of arcs relative to the sines, tangents, etc. Thus, AB is the arc whose sine is BD, and (if the length of BD is x) is written sin ^-1x, and so of the other lines. See Trigonometrical function (below). Other transcendental functions are the exponential functions, the elliptic functions, the gamma functions, the theta functions, etc. One-valued function, a quantity that has one, and only one, value for each value of the variable. -- {Transcendental functions}, a quantity whose connection with the variable cannot be expressed by algebraic operations; thus, y in the equation y = 10^x is a transcendental function of x. See Algebraic function (above). -- {Trigonometrical function}, a quantity whose relation to the variable is the same as that of a certain straight line drawn in a circle whose radius is unity, to the length of a corresponding are of the circle. Let AB be an arc in a circle, whose radius OA is unity let AC be a quadrant, and let OC, DB, and AF be drawnpependicular to OA, and EB and CG parallel to OA, and let OB be produced to G and F. E Then BD is the sine of the arc AB; OD or EB is the cosine, AF is the tangent, CG is the cotangent, OF is the secant OG is the cosecant, AD is the versed sine, and CE is the coversed sine of the are AB. If the length of AB be represented by x (OA being unity) then the lengths of Functions. these lines (OA being unity) are the trigonometrical functions of x, and are written sin x, cos x, tan x (or tang x), cot x, sec x, cosec x, versin x, coversin x. These quantities are also considered as functions of the angle BOA. Function Algebraic sum (gcide) Sum \Sum\, n. [OE. summe, somme, OF. sume, some, F. somme, L. summa, fr. summus highest, a superlative from sub under. See Sub-, and cf. Supreme.] 1. The aggregate of two or more numbers, magnitudes, quantities, or particulars; the amount or whole of any number of individuals or particulars added together; as, the sum of 5 and 7 is 12. [1913 Webster] Take ye the sum of all the congregation. --Num. i. 2. [1913 Webster] Note: Sum is now commonly applied to an aggregate of numbers, and number to an aggregate of persons or things. [1913 Webster] 2. A quantity of money or currency; any amount, indefinitely; as, a sum of money; a small sum, or a large sum. "The sum of forty pound." --Chaucer. [1913 Webster] With a great sum obtained I this freedom. --Acts xxii. 28. [1913 Webster] 3. The principal points or thoughts when viewed together; the amount; the substance; compendium; as, this is the sum of all the evidence in the case; this is the sum and substance of his objections. [1913 Webster] 4. Height; completion; utmost degree. [1913 Webster] Thus have I told thee all my state, and brought My story to the sum of earthly bliss. --Milton. [1913 Webster] 5. (Arith.) A problem to be solved, or an example to be wrought out. --Macaulay. [1913 Webster] A sum in arithmetic wherein a flaw discovered at a particular point is ipso facto fatal to the whole. --Gladstone. [1913 Webster] A large sheet of paper . . . covered with long sums. --Dickens. [1913 Webster] Algebraic sum, as distinguished from arithmetical sum, the aggregate of two or more numbers or quantities taken with regard to their signs, as + or -, according to the rules of addition in algebra; thus, the algebraic sum of -2, 8, and -1 is 5. In sum, in short; in brief. [Obs.] "In sum, the gospel . . . prescribes every virtue to our conduct, and forbids every sin." --Rogers. [1913 Webster] Algebraical (gcide) Algebraic \Al`ge*bra"ic\, Algebraical \Al`ge*bra"ic*al\, a. 1. Of or pertaining to algebra; using algebra; according to the laws of algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings; algebraic geometry. [1913 Webster] 2. progressing by constant multiplicatory factors; -- of a series of numbers. Contrasted to arithmetical. algebraic progression Syn: algebraic [PJC] Algebraic curve, a curve such that the equation which expresses the relation between the co["o]rdinates of its points involves only the ordinary operations of algebra; -- opposed to a transcendental curve. [1913 Webster] Algebraically (gcide) Algebraically \Al`ge*bra"ic*al*ly\, adv. By algebraic process. [1913 Webster] Algebraist (gcide) Algebraist \Al"ge*bra`ist\, n. One versed in algebra. [1913 Webster] Algebraize (gcide) Algebraize \Al"ge*bra*ize\, v. t. To perform by algebra; to reduce to algebraic form. [1913 Webster] Graphic algebra (gcide) Graphic \Graph"ic\ (gr[a^]f"[i^]k), Graphical \Graph"ic*al\ (gr[a^]f"[i^]*kal), a. [L. graphicus, Gr. grafiko`s, fr. gra`fein to write; cf. F. graphique. See Graft.] 1. Of or pertaining to the arts of painting and drawing; of or pertaining to graphics; as, graphic art work. [WordNet sense 2] [1913 Webster + WordNet 1.5] 2. Of or pertaining to the art of writing. [1913 Webster] 3. Written or engraved; formed of letters or lines. [1913 Webster] The finger of God hath left an inscription upon all his works, not graphical, or composed of letters. --Sir T. Browne. [1913 Webster] 4. Having the faculty of clear, detailed, and impressive description; as, a graphic writer. [1913 Webster] 5. Well delineated; clearly and vividly described; characterized by, clear, detailed, and impressive description; vivid; evoking lifelike images within the mind; as graphic details of the President's sexual misbehavior; a graphic description of the accident; graphic images of violence. [WordNet sense 5] Syn: lifelike, pictorial, vivid. [1913 Webster + WordNet 1.5] 6. Hence: describing nudity or sexual activity in explicit detail; as, a novel with graphic sex scenes. [WordNet 1.5] 7. relating to or presented by a graph[2]; as, a graphic presentation of the data. [WordNet sense 3] Syn: graphical. [WordNet 1.5] Graphic algebra, a branch of algebra in which, the properties of equations are treated by the use of curves and straight lines. Graphic arts, a name given to those fine arts which pertain to the representation on a fiat surface of natural objects; as distinguished from music, etc., and also from sculpture. Graphic formula. (Chem.) See under Formula. Graphic granite. See under Granite. Graphic method, the method of scientific analysis or investigation, in which the relations or laws involved in tabular numbers are represented to the eye by means of curves or other figures; as the daily changes of weather by means of curves, the abscissas of which represent the hours of the day, and the ordinates the corresponding degrees of temperature. Graphical statics (Math.), a branch of statics, in which the magnitude, direction, and position of forces are represented by straight lines Graphic tellurium. See Sylvanite. [1913 Webster] Multiple algebra (gcide) Multiple \Mul"ti*ple\, a. [Cf. F. multiple, and E. quadruple, and multiply.] Containing more than once, or more than one; consisting of more than one; manifold; repeated many times; having several, or many, parts. [1913 Webster] Law of multiple proportion (Chem.), the generalization that when the same elements unite in more than one proportion, forming two or more different compounds, the higher proportions of the elements in such compounds are simple multiples of the lowest proportion, or the proportions are connected by some simple common factor; thus, iron and oxygen unite in the proportions FeO, Fe2O3, Fe3O4, in which compounds, considering the oxygen, 3 and 4 are simple multiplies of 1. Called also the Law of Dalton or Dalton's Law, from its discoverer. Multiple algebra, a branch of advanced mathematics that treats of operations upon units compounded of two or more unlike units. Multiple conjugation (Biol.), a coalescence of many cells (as where an indefinite number of amoeboid cells flow together into a single mass) from which conjugation proper and even fertilization may have been evolved. Multiple fruits. (Bot.) See Collective fruit, under Collective. Multiple star (Astron.), several stars in close proximity, which appear to form a single system. [1913 Webster] algebra (wn) algebra n 1: the mathematics of generalized arithmetical operations algebraic (wn) algebraic adj 1: of or relating to algebra; "algebraic geometry" [syn: algebraic, algebraical] algebraic language (wn) algebraic language n 1: an algorithmic language having statements that resemble algebraic expressions algebraic number (wn) algebraic number n 1: root of an algebraic equation with rational coefficients algebraical (wn) algebraical adj 1: of or relating to algebra; "algebraic geometry" [syn: algebraic, algebraical] algebraically (wn) algebraically adv 1: in an algebraic manner; "algebraically determined" algebraist (wn) algebraist n 1: a mathematician whose specialty is algebra boolean algebra (wn) Boolean algebra n 1: a system of symbolic logic devised by George Boole; used in computers [syn: Boolean logic, Boolean algebra] linear algebra (wn) linear algebra n 1: the part of algebra that deals with the theory of linear equations and linear transformation matrix algebra (wn) matrix algebra n 1: the part of algebra that deals with the theory of matrices vector algebra (wn) vector algebra n 1: the part of algebra that deals with the theory of vectors and vector spaces algebra (foldoc) algebra 1. A loose term for an {algebraic structure}. 2. A vector space that is also a ring, where the vector space and the ring share the same addition operation and are related in certain other ways. An example algebra is the set of 2x2 matrices with {real numbers} as entries, with the usual operations of addition and matrix multiplication, and the usual scalar multiplication. Another example is the set of all polynomials with real coefficients, with the usual operations. In more detail, we have: (1) an underlying set, (2) a field of scalars, (3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a vector space, (4) an operation of addition of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a vector space or a ring, (5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring. This whole thing constitutes an `algebra' iff: (1) it is a vector space if you discard item (5) and (2) it is a ring if you discard (2) and (3) and (3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB). In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra. Note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example. Another example (an example of a Banach algebra) is the set of all bounded linear operators on a Hilbert space, with the usual norm. The multiplication is the operation of composition of operators, and the addition and scalar multiplication are just what you would expect. Two other examples are tensor algebras and {Clifford algebras}. [I. N. Herstein, "Topics in Algebra"]. (1999-07-14) algebra of communicating processes (foldoc) Algebra of Communicating Processes ACP (ACP) Compare CCS. ["Algebra of Communicating Processes with Abstraction", J.A. Bergstra & J.W. Klop, Theor Comp Sci 37(1):77-121 1985]. [Summary?] (1994-11-08) algebraic (foldoc) ALGEBRAIC An early system on MIT's Whirlwind. [CACM 2(5):16 (May 1959)]. (1995-01-24) algebraic In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements. If the set of compact elements is countable it is called omega-algebraic. [Significance?] (1995-04-25) algebraic compiler and translator (foldoc) Algebraic Compiler and Translator ACT 1 (ACT 1) A language and compiler for the {Royal McBee} LGP-30, designed around 1959, apparently by Clay S. Boswell, Jr, and programmed by Mel Kaye. (http://ed-thelen.org/comp-hist/lgp-30-man.html) (2008-08-04) algebraic data type (foldoc) algebraic data type sum of products type (Or "sum of products type") In {functional programming}, new types can be defined, each of which has one or more constructors. Such a type is known as an algebraic data type. E.g. in Haskell we can define a new type, "Tree": data Tree = Empty | Leaf Int | Node Tree Tree with constructors "Empty", "Leaf" and "Node". The constructors can be used much like functions in that they can be (partially) applied to arguments of the appropriate type. For example, the Leaf constructor has the functional type Int -> Tree. A constructor application cannot be reduced (evaluated) like a function application though since it is already in {normal form}. Functions which operate on algebraic data types can be defined using pattern matching: depth :: Tree -> Int depth Empty = 0 depth (Leaf n) = 1 depth (Node l r) = 1 + max (depth l) (depth r) The most common algebraic data type is the list which has constructors Nil and Cons, written in Haskell using the special syntax "[]" for Nil and infix ":" for Cons. Special cases of algebraic types are product types (only one constructor) and enumeration types (many constructors with no arguments). Algebraic types are one kind of {constructed type} (i.e. a type formed by combining other types). An algebraic data type may also be an abstract data type (ADT) if it is exported from a module without its constructors. Objects of such a type can only be manipulated using functions defined in the same module as the type itself. In set theory the equivalent of an algebraic data type is a discriminated union - a set whose elements consist of a tag (equivalent to a constructor) and an object of a type corresponding to the tag (equivalent to the constructor arguments). (1994-11-23) algebraic interpretive dialogue (foldoc) Algebraic Interpretive Dialogue AID (AID) A version of Joss II for the PDP-10. ["AID (Algebraic Interpretive Dialogue)", DEC manual, 1968]. (1995-04-12) algebraic logic functional language (foldoc) Algebraic Logic Functional language ALF (ALF) A language by Rudolf Opalla which combines functional programming and logic programming techniques. ALF is based on Horn clause logic with equality which consists of predicates and Horn clauses for {logic programming}, and functions and equations for {functional programming}. Any functional expression can be used in a goal literal and arbitrary predicates can occur in conditions of equations. ALF uses narrowing and rewriting. ALF includes a compiler to Warren Abstract Machine code and run-time support. (ftp://ftp.germany.eu.net/pub/programming/languages/LogicFunctional). ["The Implementation of the Functional-Logic Language ALF", M. Hanus and A. Schwab]. (1992-10-08) algebraic manipulation package (foldoc) Algebraic Manipulation Package (AMP) A symbolic mathematics program written in Modula-2, seen on CompuServe. (1994-10-19) algebraic specification language (foldoc) Algebraic Specification Language 1. (ASL) ["Structured Algebraic Specifications: A Kernel Language", M. Wirsing, Theor Comput Sci 42, pp.123-249, Elsevier 1986]. 2. (ASF) A language for equational specification of abstract data types. ["Algebraic Specification", J.A. Bergstra et al, A-W 1989]. (1995-12-13) algebraic structure (foldoc) algebraic structure Any formal mathematical system consisting of a set of objects and operations on those objects. Examples are Boolean algebra, numerical algebra, set algebra and matrix algebra. [Is this the most common name for this concept?] (1997-02-25) banach algebra (foldoc) Banach algebra An algebra in which the vector space is a Banach space. (1997-02-25) boeing airplane company algebraic interpreter coding (foldoc) Boeing Airplane Company Algebraic Interpreter Coding BACAIC (BACAIC) A pre-Fortran system on the IBM 701 and IBM 650. (1995-02-08) boolean algebra (foldoc) Boolean algebra (After the logician George Boole) 1. Commonly, and especially in computer science and digital electronics, this term is used to mean two-valued logic. 2. This is in stark contrast with the definition used by pure mathematicians who in the 1960s introduced "Boolean-valued models" into logic precisely because a "Boolean-valued model" is an interpretation of a theory that allows more than two possible truth values! Strangely, a Boolean algebra (in the mathematical sense) is not strictly an algebra, but is in fact a lattice. A Boolean algebra is sometimes defined as a "complemented distributive lattice". Boole's work which inspired the mathematical definition concerned algebras of sets, involving the operations of intersection, union and complement on sets. Such algebras obey the following identities where the operators ^, V, - and constants 1 and 0 can be thought of either as set intersection, union, complement, universal, empty; or as two-valued logic AND, OR, NOT, TRUE, FALSE; or any other conforming system. a ^ b = b ^ a a V b = b V a (commutative laws) (a ^ b) ^ c = a ^ (b ^ c) (a V b) V c = a V (b V c) (associative laws) a ^ (b V c) = (a ^ b) V (a ^ c) a V (b ^ c) = (a V b) ^ (a V c) (distributive laws) a ^ a = a a V a = a (idempotence laws) --a = a -(a ^ b) = (-a) V (-b) -(a V b) = (-a) ^ (-b) (de Morgan's laws) a ^ -a = 0 a V -a = 1 a ^ 1 = a a V 0 = a a ^ 0 = 0 a V 1 = 1 -1 = 0 -0 = 1 There are several common alternative notations for the "-" or logical complement operator. If a and b are elements of a Boolean algebra, we define a information algebra (foldoc) Information Algebra Theoretical formalism for DP, never resulted in a language. Language Structure Group of CODASYL, ca. 1962. Sammet 1969, 709. international algebraic language (foldoc) ALGOL 58 IAL International Algebraic Language An early version of ALGOL 60, originally known as "IAL". Michigan Algorithm Decoder (MAD), developed in 1959, was based on IAL. ["Preliminary report - International Algebraic Language", CACM 1(12):8, 1958]. [Details? Relationship to ALGOL 60?] (1999-12-10)