A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x, y ∈ N then x · y ∈ N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N = N *, where the superscript * is the Kleene star: the set is closed under composition or concatenation of its elements.

Equivalently, a preordered set P can be defined as a category with objects the elements of P, and each hom-set having at most one element ( one for objects which are related, zero otherwise ).

Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.

Equivalently, a family of closed sets forms a base for the closed sets if for each closed set A and each point x not in A there exists an element of F containing A but not containing x.

Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination ( under the group operation ) of finitely many elements of the subset and their inverses.

Elements of are known as homogeneous elements of degree n. An ideal or other subset ⊂ A is homogeneous if for every element a ∈, the homogeneous parts of a are also contained in Equivalently, an ideal is homogeneous if it is generated by homogeneous elements.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Equivalently, any non-zero element of W has at least eight non-zero coordinates.

Equivalently, a group is complete if the conjugation map ( sending an element g to conjugation by g ) is an isomorphism: 1-to-1 corresponds to centerless, onto corresponds to no outer automorphisms.

A subset S of a topological vector space, or more generally topological abelian group, X is totally bounded if and only if, given any neighbourhood E of the identity ( zero ) element of X, there exists a finite cover of S by subsets of X each of which is a translate of a subset of E. ( In other words, a " size " here is a neighbourhood of the identity element, and a subset is of size E if it is translate of a subset of E .) Equivalently, S is totally bounded if and only if, given any E as before, there exist elements a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > of X such that S is contained in the union of the n translates of E by the points a < sub > i </ sub >.

Equivalently, they count the number of different equivalence relations with precisely k equivalence classes that can be defined on an n element set.

With this definition, it is necessary to consider the direction of p ( pointed clockwise or counter-clockwise ) to figure out the sign of L. Equivalently:

Equivalently, a set is countable if it has the same cardinality as some subset of the set of natural numbers.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

Equivalently, the dyne is defined as " the force required to accelerate a mass of one gram at a rate of one centimetre per second squared ":

Equivalently, the DFT is often thought of as a matched filter: when looking for a frequency of + 1, one correlates the incoming signal with a frequency of − 1.

Equivalently, inverting is an involution.

Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable.

Equivalently, it is the lowest wage at which workers may sell their labor.

Equivalently, a perfect number is a number that is half the sum of all of its positive divisors ( including itself ) i. e. σ < sub > 1 </ sub >( n ) = 2n.

Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.

Equivalently, one can define a profinite group to be a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups.

Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors ( the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice ).

Equivalently, an ideal of R is a sub-R-bimodule of R.

Equivalently, a right ideal of is a right-submodule of.

Equivalently, a left ideal of is a left-submodule of.

The row rank of a matrix A is the maximum number of linearly independent row vectors of A. Equivalently, the column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A.

Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.

The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprime ideal in R. The Baer radical is the lower radical of the class of nilpotent rings.

A powerful number is a positive integer m such that for every prime number p dividing m, p < sup > 2 </ sup > also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a < sup > 2 </ sup > b < sup > 3 </ sup >, where a and b are positive integers.

Equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with.

Equivalently, a problem is # P-complete if and only if it is in # P, and for any non-deterministic Turing machine (" NP machine "), the problem of computing its number of accepting paths can be reduced to this problem.

Equivalently, a set is bounded if it is contained in some open ball of finite radius.

Equivalently they will be equal if their coordinates are equal.

Equivalently, if a particle travels in a closed loop, the net work done ( the sum of the force acting along the path multiplied by the distance travelled ) by a conservative force is zero.

b. Equivalently, if a ≠ b, then f ( a ) ≠ f ( b ).

Equivalently, a set is closed if and only if it contains all of its limit points.

Equivalently, M is a maximal submodule if and only if the quotient module A / M is a simple module.

More formally, a set R X Y is called a ( combinatorial ) rectangle if whenever R and R then R. Equivalently, R can also be viewed as a submatrix of the input matrix A such that R = M N where M X and N Y.

Equivalently, if X is a locally compact metric space, then ƒ is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X.

Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals ; that is, given any chain:

Equivalently, a density operator ρ is a pure state if and only if