Given a Boolean algebra B, we turn B into a Boolean ring by using the symmetric difference as addition and the meet operation as multiplication.

Given two interior algebras A and B, a map f: A → B is an interior algebra homomorphism if and only if f is a homomorphism between the underlying Boolean algebras of A and B, that also preserves interiors and closures.

Given a topological space X = 〈 X, T 〉 one can form the power set Boolean algebra of X:

Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra ( which form a base for a topology ).

Given a set X, one can form the free Boolean algebra A generated by this set and then take its completion B.

Given how these structures interact / collide with each other and with static signals, they can be used to store information as states and implement different Boolean functions.

Given a complete Boolean algebra B there is a Boolean-valued model denoted by V < sup > B </ sup >, which is the Boolean-valued analogue of the von Neumann universe V. ( Strictly speaking, V < sup > B </ sup > is a proper class, so we need to reinterpret what it means to be a model appropriately.

Given a forcing poset P, there is a corresponding complete Boolean algebra B, often obtained as the collection of regular open subsets of P, where the topology on P is generated by cones ( sets of the form

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given a sample of wood, the variation of the tree ring growths provides not only a match by year, it can also match location because the climate across a continent is not consistent.

Given a ring R and a proper ideal I of R ( that is I ≠ R ), I is a maximal ideal of R if any of the following equivalent conditions hold:

Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

; Factor ring or quotient ring: Given a ring R and an ideal I of R, the factor ring is the ring formed by the set R / I of cosets

Given a ring R and a subset S, one wants to construct some ring R * and ring homomorphism from R to R *, such that the image of S consists of units ( invertible elements ) in R *.

Given a *- ring, there is also the map.

Given a module M over a ring R, an R endomorphism f of M is called an involution if f < sup > 2 </ sup > is the identity homomorphism on M.

Given a ring R and an R-module M, a composition series for M is a series of submodules

Given a module A over a ring R, and a submodule B of A, the quotient space A / B is defined by the equivalence relation

Given an integral domain, let be an element of, the polynomial ring with coefficients in.

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

: Given: a function f: A R from some set A to the real numbers

# Given u in W and a scalar c in R, if u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) again, then cu = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, c0 ) = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, 0 ).

Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

: Given two sets, A and T, of equal size, together with a weight function C: A × T → R. Find a bijection f: A → T such that the cost function:

Given a small ball or marbles, he would invent games and play by himself for hours.

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given the absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Love and Erlandson continued to pursue with the band, and released the single " Be A Man "— an outtake from the Celebrity Skin sessions — for the soundtrack of the Oliver Stone film Any Given Sunday ( 1999 ).

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given more time to prepare for trial, Kidd likely would have been able to find the deposition Palmer gave when he was captured in Rhode Island two years earlier.

Given their speed and maneuverability, clippers frequently mounted cannon or carronades and were used for piracy, privateering, smuggling, or interdiction service.

Given Chrétien's known tendency to create new stories and characters, being the first to mention the hero Lancelot and his love affair with Queen Guinevere for example, the name might also be entirely invented.

Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.

Given a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given the limited success of the Charity School, however, Wheelock intended his new College as one primarily for whites.

Given a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given the magnitude of the disaster, reconstruction and economic recovery will remain the primary focus of the Salvadoran Government for some time to come.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given the wide disparity of concepts and methods across domains, 4GL limitations lead to recognition of the need for the 5GL.

Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").