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	Finitary - Wikipedia, the free encyclopedia
		In mathematics or logic, a finitary operation is one, like those of arithmetic, that take a number of input values to produce an output.
		An operation such as taking an integral of a function, in calculus, is defined in such a way as to depend on all the values of the function (infinitely many of them, in general), and is so not prima facie finitary.
		In the logic proposed for quantum mechanics, depending on the use of subspaces of Hilbert space as propositions, operations such as taking the intersection of subspaces are used; this in general cannot be considered a finitary operation.
	en.wikipedia.org /wiki/Finitary (522 words)

	Operation (mathematics) - Wikipedia, the free encyclopedia
		Other operations only involve a single value, for example negation (changing the sign of a number), inversion (dividing one by the number) and taking a square or square root.
		An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.
		are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k (the number of arguments) is called the type or arity of the operation.
	en.wikipedia.org /wiki/Operation_(mathematics) (550 words)

	PlanetMath: closure of sets closed under a finitary operation
		It similarly follows that the closure of a normal subgroup of a topological group is a normal subgroup.
		Cross-references: binary operation, complex, real, convex sets, scalar, addition, vector, topological vector space, vector subspace, right ideal, left ideal, ring, topological ring, subring, automorphisms, characteristic, characteristic subgroup, group, maps, unary, normal subgroup, group operation, topological group, subgroup, closed under, continuous, topological space
		This is version 15 of closure of sets closed under a finitary operation, born on 2007-05-09, modified 2007-05-11.
	planetmath.org /encyclopedia/ClosureOfSetsClosedUnderAFinitaryOperation.html (0 words)

	Finitary (Site not responding. Last check: 2007-11-01)
		In mathematics or logic a finitary operation is one like those of arithmetic that take a number of input to produce an output.
		In the logic proposed for quantum mechanics depending on the use of subspaces Hilbert space as propositions operations such as taking the intersection of subspaces are used; this in cannot be considered a finitary operation.
		The emphasis on finitary methods has historical In general infinitary logic in which the conditions limiting logical to the finitary ones stands aside from usual development for example in model theory.
	www.freeglossary.com /Finitary_operation (469 words)

	Wikipedia: Exponentiation
		If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition".
		When one has several operations around, any of which might be repeated using exponentiation, it's common to indicate which operation is being repeated by placing its symbol in the superscript.
		If the base of the exponentiation operation is itself a set, then by default we assume the operation to be the Cartesian product.
	www.factbook.org /wikipedia/en/e/ex/exponentiation.html (1235 words)

	Finitary - Definition, explanation
		A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finite set of axioms.
		In general infinitary logic, in which the conditions limiting logical operations to the finitary ones, stands aside from the usual development, for example in model theory.
		The aim itself was proved impossible by Kurt Gödel in 1931, with his Incompleteness Theorem, but the general mathematical trend is to use a finitary approach, arguing that this avoids considering mathematical objects that cannot be fully defined.
	www.calsky.com /lexikon/en/txt/f/fi/finitary.php (0 words)

	Springer Online Reference Works
		A related theory of polyadic algebras, due to P.R. Halmos [a5], emphasizes operators corresponding to the logical operation of substitution of terms for variables.
		This gives rise to a categorical duality between Boolean algebras with operators and (topological) relational structures that is developed and applied in [a1], [a3].
		A survey of the theory of Boolean algebras with operators is given in [a10].
	eom.springer.de /b/b110750.htm (0 words)

	boolean operation \| English \| Dictionary & Translation by Babylon
		In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer.
		In the case where k = 0, the "function" is simply a constant element of B.More generally, a function of the form f : X → B, where X is an arbitrary set, is a boolean-valued function.
		Any operation in which each of the operands and the result take one of two values.
	www.babylon.com /definition/boolean_operation (0 words)

	Hilbert's Program (Stanford Encyclopedia of Philosophy)
		To count as a finitary consistency proof, the operation itself must be acceptable from the finitist standpoint, and the proofs required must use only finitarily acceptable principles.
		Another interesting analysis of finitary proof, which, however, does not provide as detailed a philosophical justification, was proposed by Kreisel (1960).
		Whether such methods would be considered finitary according to the original conception of finitism or constitute an extension of the original finitist viewpoint is a matter of debate.
	plato.stanford.edu /entries/hilbert-program (0 words)

	Free object (Site not responding. Last check: 2007-11-01)
		It is part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations); but on the other hand it has a clean formulation in terms of category theory (in yet more abstract terms).
		As that example suggests, free objects look like constructions from syntax; and we can reverse that to some extent by saying that major uses of syntax can be explained and characterised as free objects, in a way that makes apparently heavy 'punctuation' explicable (and more memorable).
		In general, the setting for a free object is like this: a category C of algebraic structures (sets plus operations, obeying some laws) has a functor F to Sets, the category of sets and functions, that simply ignores the operations.
	bopedia.com /en/wikipedia/f/fr/free_object.html (424 words)

	Exponentiation
		A quandle[?] is an algebraic structure in which these laws of conjugation play a central role.
		The above algebraic treatment of exponentiation builds a finitary operation[?] out of a binary operation.
		In more general contexts, one may be able to define an infinitary operation[?] directly on an indexed set[?].
	www.ebroadcast.com.au /lookup/encyclopedia/ex/Exponent.html (1209 words)

	Mislove N00014-91-J-1692
		An operational model is meant to show how programs act on an abstract machine, and it usually is given by a set of transition rules which describe the "next steps" that a process can undertake.
		The results just described applied to any finitary language (i.e., they do not assume the language is a free or an initial algebra), but the model for the language of closed terms produced by the theory does not satisfy any equations, even if the original language satisfied them.
		For example, even if the sequential composition operation ";" in the denotational model for the finitary model were assumed to be associative, the same would not be true of this operation in the denotational model for the language of closed terms.
	www.math.tulane.edu /~mwm/ftp/eoyl95.html (2603 words)

	Encyclopedia :: encyclopedia : Closed-form solution (Site not responding. Last check: 2007-11-01)
		The classic example involves the two roots of a quadratic equation, which can be expressed in closed form in terms of addition and subtraction, multiplication and division, and square root extraction.
		When no closed-form solutions exist – as is the case for fifth-order or higher polynomial equations, for example – such equations have to be solved numerically, typically by using some root-finding algorithm.
		For example, many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well-known.
	www.hallencyclopedia.com /Closed-form_solution (209 words)

	Exponentiation (Site not responding. Last check: 2007-11-01)
		When one has several operations around any which might be repeated using exponentiation it common to indicate which operation is being by placing its symbol in the superscript.
		In general contexts one may be able to an infinitary operation directly on an indexed set.
		If the base of the exponentiation operation itself a set then by default we the operation to be the Cartesian product.
	www.freeglossary.com /Raised_to_the_power (1464 words)

	On an idea of Peter Dybjer's (Site not responding. Last check: 2007-11-01)
		Erik Palmgren's next-universe operator, (the one under which the superuniverse is closed) and strips out all the closure conditions pertaining to quantifiers, given sets, logical constants, the identity type etc.
		It is a closure operator, in the sense that it is inflationary, monotone, and idempotent.
		Perhaps one can start with the `two to the _' operation, which in the case of linear orderings is the lexicographic order of chains which are descending in the exponent order.
	homepages.inf.ed.ac.uk /v1phanc1/dybjer.html (3402 words)

	LINEAR NUMBERS 0
		As operations defined by transfinite induction through two inductive rules that are already satisfied by the usual operations in the natural numbers.
		in the Hessenberg natural operations is isomorphic with the free semiring of α many generators in the category of abelian semirings ;or isomorphic with the algebra of polynomial symbols of α inderminates of the type of algebra of semirings with constants the natural numbers.
		Thus the Hessenberg natural operations should be coined as the standard abelian operations in the ordinal numbers, for all practical algebraic purposes.There are already extended applications of this.
	www.softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/FreeHB.htm (1746 words)

	Articles - Exponent (Site not responding. Last check: 2007-11-01)
		The next generalized operation after exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function.
		Exponentiation is a basic mathematical operation that is used pervasively in other fields as well, including physics, chemistry, biology, computer science and economics, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
		In this very general situation, we can define x for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn´t matter in which order we perform the multiplications.
	www.winacea.com /articles/Exponent (2642 words)

	Practical Foundations of Mathematics
		The method extends to congruences for finitary algebraic theories, and is used to impose laws (Example 4.6.3(j), Section 7.4).
		By finitary we really do mean that the arity k has to be finitely enumerated (Definition 6.6.2(a)).
		Colimits by duality Although this way of constructing coequalisers is not in general available for infinitary operations, most of those of interest (meets, joins, limits and colimits) are defined by universal properties.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s56.html (0 words)

	finitary - OneLook Dictionary Search
		We found one dictionary with English definitions that includes the word finitary:
		Tip: Click on the first link on a line below to go directly to a page where "finitary" is defined.
		Phrases that include finitary: finitary boolean function, finitary operation, finitary relation
	www.onelook.com /?ls=a&w=finitary (79 words)

	[No title]
		Abstract: The finitely observable, or finitary, part of bisimulation is a key tool in establishing full abstraction results for denotational semantics for process algebras with respect to bisimulation-based preorders.
		Complementation is an important operation because it is fundamental for treating the logical connective ``not'' in decision procedures for monadic secondorder logics.
		While it is impossible to define an operator in the lambda-calculus which encodes all closed lambda-expressions, it is possible to construct restricted versions of such an encoding operator.
	www.brics.dk /BRICS/BRICS/RS/95/Abs/BRICS-RS-95-Abs.txt (7280 words)

	[No title]
		A wide variety of algorithms and architectures operate continuously in time, producing streams of data, for example: systolic arrays, data-flow machines, neural networks and cellular automata.
		A variety of algorithms and architectures operate continuously in time, producing streams of data, for example: systolic arrays, data-flow machines, neural networks and cellular automata.
		\newpage \sheadrun{6.2}{Definability of operations by theories} \Defs (1) Suppose $A'$ is a \Sigp-expansion of $A$.
	www.cas.mcmaster.ca /~zucker/Pubs/Brno/text (3699 words)

	CSCI320: The Z Formal Specification Language
		data_refinement::=`the process of showing that one set of operations is implementd by another set on a different state space`.
		non_deterministic::=`an operation in an abstract data type where there may be more than one possible state after for each single state befor it`.
		operation_refinement::=`the process of showing that one operation is implemented by another with the same state space`.
	www.csci.csusb.edu /dick/cs320/comp.spec.Z.html (670 words)

	Exponentiation
		In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition.
		(The next operation after exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function.
		When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript.
	www.askfactmaster.com /Exponentiation (1823 words)

	Exponentiation
		Exponentiation is a basic mathematical tool that is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
		If the base of the exponentiation operation is a set, then by default we assume the operation to be the Cartesian product.
		The next generalized operation after multiplication and exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function.
	www.1bx.com /en/Power_(mathematics).htm (2724 words)

	Zero Saga
		If the result of dividing by zero (which is a meaningless operation in the first place) by calculator was infinity, then calculator would give you "infinity" not the word "error", or "E", this is so, because only a limited number of letters such as E or the word Error could be formed.
		However, in division operation, this is not a necessary condition, such as in speed, expressed, e.g., as kilometers/hour.
		operations on numbers can be viewed as the results of movements in certain directions, either to the left or to the right.
	home.ubalt.edu /ntsbarsh/Business-stat/opre/ZERO.HTM (18483 words)

	\bf RESIDUATED POSETS
		Since the operations of meet and join are obviously isotone in both components, and since the nullary operation, e, is vacuously isotone, we see that lattice-ordered groups are ordered algebras.
		Identities 1-3 imply that the operation · is isotone in both components, that the operation \ is isotone in its second component, and that the operation / is isotone in its first component.
		Closure operators are very well-studied maps; they arise in a natural way from adjunctions and play an important role in topology (hence the terminology associated with them).
	www.mtsu.edu /~jhart/NRESLAT.html (5847 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		In this paper, we investigate subclasses of flow event structures which are both suited for the process algebraic composition operators, and for action refinement as a means of regarding processes on different levels of abstraction.
		Since the model is category-based, we can define operations on D-schedules and processes via universal constructions that depend little on the choice of D. Also, given a suitable choice of process structure and process morphism, the constructions used for process operations and schedule operations are remarkably similar.
		Although none of the operations is particularly oriented to nets it is nevertheless possible to use them to express processes constructed as a net of subprocesses, and more generally as a system consisting of components.
	boole.stanford.edu /pub/ABSTRACTS (6512 words)

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