Lattice (order)Set whose pairs have minima and maximaY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Semimodular latticeIn the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition: Semimodular lawa ∧ b <: a implies b <: a ∨ b.The notation a <: b means that bcoversa, i.e. a < b and there is no element c such that a < c < b. An atomistic semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids.Module (mathematics)In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.
Vector spaceAlgebraic structure in linear algebraVector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars.
Variety (universal algebra)In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products.
Duality (order theory)In the mathematical area of order theory, every partially ordered setP gives rise to a dual (or opposite) partially ordered set which is often denoted by P or P. This dual order P is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in Pif and only ify ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set.
Universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of study—this is the subject of group theory and ring theory—in universal algebra, the object of study is the possible types of algebraic structures and their relationships.
Join and meetY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.Order theoryOrder theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". Background and motivationOrders are everywhere in mathematics and related fields like computer science.
AlgebraBranch of mathematicsElementary algebra studies which values solve equations formed using arithmetical operations.Abstract algebra studies algebraic structures, such as the ring of integers given by the set of integers(Z){\displaystyle (\mathbb {Z} )} together with operations of addition (+{\displaystyle +}) and multiplication (×{\displaystyle \times }).Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems.
Isomorphism theoremsIn mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.Partially ordered setMathematical set with an orderingY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.*As an Amazon Associate I earn from qualifying purchases.