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

Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module.

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

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 an evaluation e of variables by elements of M < sub > w </ sub >, we

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

Given an SVD of M, as described above, the following two relations hold:

Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

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 category C, an idempotent of C is an endomorphism

Given an unramified finite extension L / K of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.

Given an element x of a Lie algebra, one defines the adjoint action of x on as the endomorphism with

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 Boolean ring R, for x and y in R we can define

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 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 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 a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).

Given two groups A and H there exist two variations of the wreath product: the unrestricted wreath product A Wr H ( also written A ≀ H ) and the restricted wreath product A wr H. Given a set Ω with an H-action there exists a generalisation of the wreath product which is denoted by A Wr < sub > Ω </ sub > H or A wr < sub > Ω </ sub > H respectively.

Given that the logit is not generally interpreted and that the inverse of the natural logarithm, the exponential function of the logit is generally interpreted instead, it is also helpful to examine this function ( denoted: ).

; Degree of an extension: Given an extension E / F, the field E can be considered as a vector space over the field F, and the dimension of this vector space is the degree of the extension, denoted by: F.

Given a map f on X, then its germ at x is usually denoted < sub > x </ sub >.

Another partition of the given interval, Q, is defined as a refinement of the partition, P, when it contains all the points of P and possibly some other points as well ; the partition Q is said to be “ finer ” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, re-numbered in order.

Given a field extension K / F, the field K can be considered as a vector space over the field F. The dimension of this vector space is the degree of the extension and is denoted by: F.

Given a pattern T, the number of other patterns may have Kolmogorov complexity no larger than that of T is denoted by φ ( T ).

Given any vector in a vector space, the scalar product with another vector, denoted or, is a scalar.

Given two real numbers, say x and y, with we define an uncountably infinite family of open sets denoted by S < sub > x, y </ sub > as follows:

Given some mereotopology X, adding C8 to X results in what Casati and Varzi call the Whiteheadian extension of X, denoted WX.

Given a volume form ω on M, one can define the divergence of a vector field X as the unique scalar-valued function, denoted by div X, satisfying

Given a binary relation R between fixed strings in the alphabet, called rewrite rules, denoted by, an SRS extends the rewriting relation to all strings in which the left-and right-hand side of the rules appear as substrings, that is, where s, t, u, and v are strings.

The cylindrification operation takes a relation as input of arity k and outputs a relation of arity k + 1 as follows: Given a relation R of arity K, its cylindrification denoted by c ( R ), is the following set

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.