Ad
related to: definition of partially ordered set in algebra 3 examples with solutionseducation.com has been visited by 100K+ users in the past month
Education.com is great and resourceful - MrsChettyLife
- Guided Lessons
Learn new concepts step-by-step
with colorful guided lessons.
- Lesson Plans
Engage your students with our
detailed lesson plans for K-8.
- Interactive Stories
Enchant young learners with
animated, educational stories.
- Activities & Crafts
Stay creative & active with indoor
& outdoor activities for kids.
- Guided Lessons
Search results
Results from the WOW.Com Content Network
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
Every partially ordered set can be viewed as a category in a natural way: there is a unique morphism from x to y if and only if x ≤ y. A monotone Galois connection is then nothing but a pair of adjoint functors between two categories that arise from partially ordered sets.
Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice.
A partially ordered group G is called integrally closed if for all elements a and b of G, if a n ≤ b for all natural n then a ≤ 1. [1]This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent. [2]
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
As Jourdan, Rampon & Jard (1994) observe, the problem of listing all cuts in a partially ordered set can be formulated as a special case of a simpler problem, of listing all maximal antichains in a different partially ordered set. If P is any partially ordered set, let Q be a partial order whose elements contain two copies of P: for each ...
Small finite examples: The three partially ordered sets on the left are trees (in blue); one branch of one of the trees is highlighted (in green). The partially ordered set on the right (in red) is not a tree because x 1 < x 3 and x 2 < x 3, but x 1 is not comparable to x 2 (dashed orange line).
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
Ad
related to: definition of partially ordered set in algebra 3 examples with solutionseducation.com has been visited by 100K+ users in the past month
Education.com is great and resourceful - MrsChettyLife