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meet – nowhere dense
meet
A binary operation whose value on two elements a and b of a lattice is the greatest lower bound of a and b, denoted a  b.
 Cf. join.
meet-morphism
A function f on a lattice L is called a meet-morphism if for every a and b in L we have f(a b) = f(a) f(b). If f has an inverse that is also a meet-morphism, then f is called a meet-isomorphism. A meet-isomorphism from a lattice to itself is called a meet-automorphism.
Cf. meet, join-morphism.
module
A generalization of the notion of vector space. Specifically, if M is an Abelian group under addition, and R is a ring such that rm is in M whenever m is an element of M and r is an element of R, then M is called a left R-module (a right R-module is defined analogously), provided- r(m + n) = rm + rn,
- (r + s)m = rm + sm, and
- r(sm) = (rs)m
for all r, s in R and m, n in M. A module is cyclic if there is a generating element m of M such that every element of M is of the form rm for some r in R. A module is finitely generated if there are elements m1, m2, ..., mk in M such that every element of M is of the form r1m1 + r2m2 + ... + rkmk for some r1, r2, ..., rk in R.
monoid
See group.
monomorphism
A morphism f from X to Y is called a monomorphism when it is injective, that is, when to each element y of Y there corresponds at most one x in X such that f(x) = y.
Cf. epimorphism.
monotone function
Also called monotonic function. See order-preserving function.
morphism
A function from one set to another is called a morphism if it preserves some designated structural properties or operations on the domain set. Typically, the word morphism is not used by itself, but in combination with a prefix that indicates whether it is injective, surjective, etc.
Cf. automorphism, epimorphism, homeomorphsim, homomorphism, isomorphism, monomorphsim.
multiplication
A binary operation on numbers or quantities resulting in a product, usually but not always amounting to repeated addition. On the natural numbers multiplication is defined recursively by the Peano axioms, such that the product of two numbers n and m, denoted by n × m, is found by adding up m copies of n (or n copies of m).
Multiplication of most kinds of numbers is associative and commutative, but these properties sometimes fail, for example in the case of matrix multiplication.
When a product of more than two numbers or quantities is taken, the general product may be denoted by the capital Greek letter Pi, i.e., Pai denotes the product a1 × a2 × . . . × an.
multiply
To find the product of two numbers or quantities by multiplication.
mutually prime
Two integers are mutually prime if they have no common factors larger than 1 or -1.
natural base
See Euler number.
natural logarithm
A logarithm with base e, the Euler number. Often written “ln” rather than “log” to distinguish it from logarithms using other bases.
natural number
An element of the set N = {1, 2, 3, ...} consisting of all the “counting numbers.” When the number 0 is included, this set is sometimes called the whole numbers. In set theory, the natural numbers (incuding 0) are identified with the set w of finite ordinals. The natural numbers are a well-founded linear order with no largest member, and are countably infinite.
Cf. Peano axioms, rational number, real number.
Related MiniText: Number -- What Is How Many?
negation
If j is a statement, sentence, or formula of logic, then the negation of j, denoted by  j, is that formula which is true whenever j is false, and false whenever j is true.
negative
The negative of a number or quantity x is the number, denoted -x, which when added to x yields 0. That is, the negative of a number is its additive inverse.
non-denumerable
Uncountable.
norm
Analysis: A non-negative real-valued function “|| x ||” defined on a vector space, satisfying- || –x || = || x ||,
- || cx || = || c || × || x || for all scalars c, and
- || x + y || <= || x || + || y || (triangle inequality)
Statistics: Another term for the mode of a frequency distribution.
normal
A line intersecting a curve (or surface) perpendicular to the tangent line (or tangent plane) at the point of intersection. The normal to a surface expressed as a function of several variables xi is given by the gradient.
normal subgroup
See subgroup.
normed space
A vector space with a norm defined on it.
nowhere dense
Given a space X and a subset A of X, we say that A is nowhere dense if every open set of X contains an open subset that is disjoint from A. This is equivalent to saying that the complement of A is dense, or that A has empty interior.
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