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, 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, the boundary of a set is the intersection of its closure with the closure of its complement.

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

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

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, 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 ( and with no need to arbitrarily choose two points ) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points.

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.

Equivalently, a set is recursively enumerable if and only if it is the range of some computable function.

Equivalently X is arithmetical if X is or for some integer n. A set X is arithmetical in a set Y, denoted, if X is definable a some formula in the language of Peano arithmetic extended by a predicate for membership in Y. Equivalently, X is arithmetical in Y if X is in or for some integer n. A synonym for is: X is arithmetically reducible to Y.

If X is an affine algebraic set ( irreducible or not ) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some Equivalently, it can be checked that:

Equivalently, every dependent set contains a finite dependent set.

Equivalently, a function is convex if its epigraph ( the set of points on or above the graph of the function ) is a convex set.

Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its image.

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, 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, 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, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, ... of elements of X such that x < sub > n + 1 </ sub > R x < sub > n </ sub > for every natural number n.

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

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, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal.

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