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lattice – meet-morphism
lattice
A partially ordered set in which each pair of elements has a least upper bound, called the join, and a greatest lower bound, called the meet. The join of two elements x and y is denoted by x  y, and the meet by x  y.
 A lattice may be equivalently defined as a set with an algebra having two binary operations (meet) and (join) defined on it, satisfying- x x = x = x x, and
- x (x y) = x = x (x y)
A lattice is called complete if every subset of the lattice has a least upper bound and a greatest lower bound.
least upper bound
An upper bound which is less than or equal to every other upper bound.
Cf. greatest lower bound.
least upper bound axiom
“Any subset of the real numbers which has an upper bound has a least upper bound.” This axiom, together with the field axioms, completely characterizes the set of real numbers.
Cf. supremum.
Lebesgue measure
The unique measure m on the real line, generated by the outer measure whose value on intervals is the length of the intervals, is called Lebesgue measure.
liar paradox
According to a legend found in many ancient sources (including a letter of St. Paul), there was a certain Cretan named Epimenides who proclaimed, “All Cretans are liars.” This statement involves a contradiction on its face, since if all Cretans are liars and Epimenides is a liar, then the statement must be false, that is, not all Cretans are liars.
There are many versions of the liar paradox, and many puzzles based upon it. The simplest form is, “This statement is false,” a statement that’s true if it’s false and false if it’s true.
limit point
Given a sequence of points ai , i = 1, 2, 3, ... , a point L is called a limit point of the sequence if every neighborhood of L contains all but finitely many of the ai .
If X is a metric space, then L is a limit point of X if a sequence from X may be chosen so that L is the limit point of that sequence. Limit points are not in general unique.
See also: accumulation point, perfect set.
linear order
A set X is said to be a linear order if there is a relation “<” on its elements such that for all distinct x and y in X, either x < y or y < x, but not both. Sometimes also called a total order.
Cf. partial order, dense linear order.
locally compact
A topological space is locally compact if every point of the space has a neighborhood whose closure is compact.
locally integrable
A measurable function on a finite-dimensional real space with range in the complex numbers is said to be locally integrable if for all compact sets K we have:
The space of all such functions is denoted L1LOC.
Cf. Lebesgue integral.
logic
In the broadest sense, the term “logic” refers to the reasoning processes or forms of argument people use to reach valid conclusions from one or more premises (assumptions). More specifically, any particular logic is a system of inference, by which one reasons from premises to conclusions, or by which one validates one’s conclusions. An incorrect inference, i.e., one which violates the rules of logic, is called a fallacy, and conclusions reached by means of a fallacy are called invalid. (By contrast, when the premises of an argument are false but the argument is formally valid, then the conclusion is called unsound.)
Ordinary logic falls into two broad types; deductive and inductive. Reasoning from general premises to a specific conclusion is deductive. For example, knowing that heavy clouds and a brisk breeze often signal an oncoming storm, one could conclude deductively that it might rain this afternoon. Reasoning from specific premises to a general conclusion, on the other hand, is inductive. For example, observing that every raven one sees is black, one might conclude inductively that all ravens are black. Deductive conclusions are necessarily true if the premises are true and the logic used formally valid (free of fallacy). Inductive conclusions are never certain, but are only more or less reliable. In mathematics only deductive reasoning is used (but see the entry for mathematical induction).
Mathematicians use symbolic logic, that is, logic that can be reduced to a purely syntactical system which disregards the semantic content of any of the symbols in use. The two primary forms of symbolic logic are the propositional calculus and the predicate calculus. The latter augments the former with existential and universal quantifiers (permitting sentences that begin with “there exists ___ such that” and “for all ___”). When quantification is permitted only over elements of the universe of discourse, the logic is called “first-order.” When quantification over classes of objects and/or over predicates is permitted, it is called second-order logic.
Cf. axiom, fuzzy logic, equational logic, syllogism, Hempel's Ravens Paradox.
lower bound
A lower bound of an ordered set is an element which is less than or equal to every element in the set.
Cf. greatest lower bound.
l.u.b
Abbreviation for least upper bound.
Mahlo cardinal
A cardinal k is called Mahlo if it is inaccessible and {a < k:a is inaccessible} is stationary in k.
maximal filter
See filter.
measurable cardinal
If there exists a non-principal k-complete ultrafilter on a set of size k, then k is called a measurable cardinal.
measurable function
Given measure spaces (X, M, m) and (Y, N, n), a function f from X to Y is said to be measurable if the inverse image of every measurable set in Y is measurable in X.
measurable space
A set X together with a s-algebra of sets M defined on it is called a measurable space, usually denoted (X, M). The elements of M are called measurable sets.
Cf. measure.
measure
Given a set X and a s-algebra of sets M defined on X, a measure on X is a positive real-valued function m whose domain is M, and which satisfies: - the measure of the empty set is zero.
- (Countable additivity) For any countable, disjoint sequence of sets in the domain of m, the measure of the union of the sequence is equal to the sum of the measures of the sets in the sequence.
If the measure of every set in M is finite, then m is called a finite measure. If X can be expressed as a union of sets in M all of whose measures are finite, then m is called a s-finite measure. If for every set E in the domain of m whose measure is infinite there is a subset F of E which has finite measure, then m is called a semi-finite measure. If for any set E in M which has zero measure all the subsets of E are also in M (and therefore have zero measure), then m is called a complete measure.
Cf. measure space, signed measure, outer measure, Lebesgue measure, Borel measure.
measure space
A set X together with a s-algebra of sets M defined on it and a measure m defined on M is called a measure space, usually denoted by (X, M, m).
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.
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