elementary set theory - difference between join and union - Mathematics Stack Exchange
Andbecauseofthesesubstructures,theMV-algebra meet and join operations ⊗ and ⊕ are sometimes called bold meet and join to distinguish them from the lattice. Join is a lattice-theoretic concept that need not have anything to do with . join: unique supremum, or least upper bound; meet: unique infimum. Results of Meet and Join Operations of C3 and C4 Abstraction Level 1 2 3 4 meet join ⊥ C C† C3 C4 C3|4 C † level 1), the resulting value of the meet.
Antimatroid - Wikipedia
This class gives rise to a broad range of practical examples. The set of compact elements of an arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property which distinguishes arithmetic lattices from algebraic latticesfor which the compacts do only form a join-semilattice.
Both of these classes of complete lattices are studied in domain theory.
Further examples of lattices are given for each of the additional properties discussed below. Counter-examples[ edit ] Pic.
Most partial ordered sets are not lattices, including the following.
In particular the two-element discrete poset is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility 12 and 18 do not divide each other.
Likewise the pair 12, 18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility 2 and 3 do not divide each other.
Morphisms of lattices[ edit ] Pic. The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition.
meet and join operations on a lattice : math
Thus f is a homomorphism of the two underlying semilattices. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism usually called just "lattice homomorphism" f between two bounded lattices L and M should also have the following property: In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins.
For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set. Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see preservation of limits. The converse is not true: Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism.
Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a category. Sublattices[ edit ] A sublattice of a lattice L is a nonempty subset of L that is a lattice with the same meet and join operations as L.
- Join and meet
- Lattice (order)
Sx is meet-irreducible, meaning that it is not the meet of any two larger lattice elements: That is, the lattice has unique meet-irreducible decompositions. An interval is atomistic if every element in it is the join of atoms the minimal elements above the bottom element xand it is Boolean if it is isomorphic to the lattice of all subsets of a finite set.
For an antimatroid, every interval that is atomistic is also boolean. Translating this condition into the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element may be added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: A semimodular and meet-semidistributive lattice is called a join-distributive lattice.
These three characterizations are equivalent: Any antimatroid gives rise to a finite join-distributive lattice, and any finite join-distributive lattice comes from an antimatroid in this way. This representation of any finite join-distributive lattice as an accessible family of sets closed under unions that is, as an antimatroid may be viewed as an analogue of Birkhoff's representation theorem under which any finite distributive lattice has a representation as a family of sets closed under unions and intersections.
DBMS - Joins
Supersolvable antimatroids[ edit ] Motivated by a problem of defining partial orders on the elements of a Coxeter groupArmstrong studied antimatroids which are also supersolvable lattices. A supersolvable antimatroid is defined by a totally ordered collection of elements, and a family of sets of these elements.
The family must include the empty set. As Armstrong observes, any family of sets of this type forms an antimatroid.