Upper setIn mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set is a subset with the following property: if s is in S and if x in X is larger than s (that is, if ), then x is in S. In other words, this means that any x element of X that is to some element of S is necessarily also an element of S. The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is to some element of S is necessarily also an element of S.
Discrete valuationIn mathematics, a discrete valuation is an integer valuation on a field K; that is, a function: satisfying the conditions: for all . Note that often the trivial valuation which takes on only the values is explicitly excluded. A field with a non-trivial discrete valuation is called a discrete valuation field. To every field with discrete valuation we can associate the subring of , which is a discrete valuation ring. Conversely, the valuation on a discrete valuation ring can be extended in a unique way to a discrete valuation on the quotient field ; the associated discrete valuation ring is just .
Ordered vector spaceIn mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Given a vector space over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied implies implies If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure.
Graded posetIn mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy the following two properties: The rank function is compatible with the ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and The rank is consistent with the covering relation of the ordering, meaning that for all x and y, if y covers x then ρ(y) = ρ(x) + 1.
Rank of a partitionIn mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition.
Distributive latticeIn mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.
Boolean domainIn mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1}, or The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain.
Poincaré conjectureIn the mathematical field of geometric topology, the Poincaré conjecture (UKˈpwæ̃kæreɪ, USˌpwæ̃kɑːˈreɪ, pwɛ̃kaʁe) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere.
Partition typeThe partition type (or partition ID) in a partition's entry in the partition table inside a master boot record (MBR) is a byte value intended to specify the the partition contains or to flag special access methods used to access these partitions (e.g. special CHS mappings, LBA access, logical mapped geometries, special driver access, hidden partitions, secured or encrypted file systems, etc.). Lists of assigned partition types to be used in the partition table in the MBR were originally maintained by IBM and Microsoft internally.
Fence (mathematics)In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations: or A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences. A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century.
