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Modular Pair
In the branch of mathematics called order theory, a modular lattice is a lattice (order), lattice that satisfies the following self-duality (order theory), dual condition, ;Modular law: implies where are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety (universal algebra), variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the Second isomorphism theorem, 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2d Modular Lattice
D, or d, is the fourth Letter (alphabet), letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Letter names, ''dee'' (pronounced ), plural ''dees''. History The Semitic languages, Semitic letter Daleth, Dāleth may have developed from the logogram for a fish or a door. There are many different Egyptian hieroglyphs that might have inspired this. In Semitic, Ancient Greek and Latin, the letter represented ; in the Etruscan alphabet the letter was archaic but still retained. The equivalent Greek letter is delta, Delta (letter), Δ. The lower case, minuscule (lower-case) form of 'd' consists of a lower-story left Typeface anatomy#Strokes, bowl and a Typeface anatomy#Strokes, stem ascender. It most likely developed by gradual variations on the upper case, majuscule (capital) form 'D', and is now composed as a stem with a full Typeface ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sublattice
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). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' structures all admi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ascending Chain Condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler. Definition A partially ordered set (poset) ''P'' is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence : a_1 < a_2 < a_3 < \cdots of elements of ''P'' exists. Equivalently, every weakly ascending sequence : of elements of ''P'' eventually stabilizes, meaning that there exists a pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
