Relatively complemented lattice
- Relatively complemented lattice
In mathematics, a relatively complemented lattice is a lattice "L" in which for all "a", "b", "c" in "L" with "a" ≤ "b" ≤ "c" there is some "x" in "L" such that "x" ∨ "b" = "c" and "x" ∧ "b" = "a". An element "x" with this property is a complement of "b" relative to the interval ["a","c"] .
Two particular cases are frequently seen:
*If "A" and "B" are sets with
::
:then the complement of "A" relative to "B" (the "interval" involved is from the empty set to "B") is
::
*If the lattice is a Boolean algebra, then the complement of "b" relative to the interval ["a", "c"] is "a" ∨ (~ "b") ∧ "c". (In general, the expression "x" ∨ "y" ∧ "z" is ambiguous in Boolean algebra. But the fact that "a" ≤ "b" "c" removes the ambiguity in this case.) In the usual interpretation of Boolean algebra as a model of propositional logic, if "a" is a sufficient condition for "b" and "c" is a necessary condition for "b", the complement of "b" relative to the interval ["a", "c"] is the unique (up to logical equivalence) proposition "d" such that
:*"a" is sufficient for "d" and "c" is necessary for "d", and:*"d" becomes equivalent to [not "b"] if one learns that "a" is false and "c" is true.
Wikimedia Foundation.
2010.
Look at other dictionaries:
Complemented lattice — Hasse diagram of a complemented lattice A point and a line of the Fano plane are complements, when In the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a … Wikipedia
Congruence lattice problem — In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most … Wikipedia
Map of lattices — The concept of a lattice arises in order theory, a branch of mathematics. The Hasse diagram below depicts the inclusion relationships among some important subclasses of lattices. Proofs of the relationships in the map 1. A boolean algebra is a… … Wikipedia
Outline of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… … Wikipedia
List of order topics — This is a list of order topics, by Wikipedia page.An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value, optimization (mathematics), domain theory.Basic… … Wikipedia
Relative — can refer to: *Kinship, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be relatives Physics*Relativity as a concept in physics (for example Albert Einstein s… … Wikipedia
List of order theory topics — Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is less than or precedes another. An alphabetical list of many… … Wikipedia
Universe (mathematics) — In mathematical logic, the universe of a structure (or model ) is its domain.In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set – not to be… … Wikipedia
List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia
Complement (mathematics) — Complement has a variety of uses in mathematics:* complement, an operation that transforms an integer into its additive inverse, useful for subtracting numbers when only addition is possible, or is easier * complement, a system for working with… … Wikipedia
