1.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers

2.
Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics

3.
Lattice (order)
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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of an ordered set in which every two elements have a unique supremum and a unique infimum. An example is given by the numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple. 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 and these lattice-like structures all admit order-theoretic as well as algebraic descriptions. If is an ordered set, and S ⊆ L is an arbitrary subset. A set may have many upper bounds, or none at all, an upper bound u of S is said to be its least upper bound, or join, or supremum, if u ≤ x for each upper bound x of S. A set need not have a least upper bound, but it cannot have more than one, dually, l ∈ L is said to be a lower bound of S if l ≤ s for each s ∈ S. A lower bound l of S is said to be its greatest lower bound, or meet, or infimum, a set may have many lower bounds, or none at all, but can have at most one greatest lower bound. A partially ordered set is called a join-semilattice and a meet-semilattice if each two-element subset ⊆ L has a join and a meet, denoted by a ∨ b, is called a lattice if it is both a join- and a meet-semilattice. This definition makes ∨ and ∧ binary operations, both operations are monotone with respect to the order, a1 ≤ a2 and b1 ≤ b2 implies that a1 ∨ b1 ≤ a2 ∨ b2 and a1 ∧ b1 ≤ a2 ∧ b2. It follows by an argument that every non-empty finite subset of a lattice has a least upper bound. With additional assumptions, further conclusions may be possible, see Completeness for more discussion of this subject, a bounded lattice is a lattice that additionally has a greatest element 1 and a least element 0, which satisfy 0 ≤ x ≤1 for every x in L. The greatest and least element is called the maximum and minimum, or the top and bottom element. A partially ordered set is a lattice if and only if every finite set of elements has a join. Taking B to be the empty set, ⋁ = ∨ = ∨0 = ⋁ A and ⋀ = ∧ = ∧1 = ⋀ A which is consistent with the fact that A ∪ ∅ = A. A lattice element y is said to another element x, if y > x. Here, y > x means x ≤ y and x ≠ y

4.
Atom (order theory)
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In the mathematical field of order theory, an element a of a partially ordered set with least element 0 is an atom if 0 < a and there is no x such that 0 < x < a. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, let <, denote the cover relation in a partially ordered set. A partially ordered set with a least element 0 is atomic if every element b >0 has an atom a below it, every finite partially ordered set with 0 is atomic, but the set of nonnegative real numbers is not atomic. A partially ordered set is relatively atomic if for all a < b there is an element c such that a <, c ≤ b or, equivalently, every relatively atomic partially ordered set with a least element is atomic. A partially ordered set with least element 0 is called atomistic if every element is the least upper bound of a set of atoms, every finite poset is relatively atomic, but the linear order with three elements is not atomistic. Atoms in partially ordered sets are abstract generalizations of singletons in set theory, atomicity provides an abstract generalization in the context of order theory of the ability to select an element from a non-empty set. The terms coatom, coatomic, and coatomistic are defined dually, introduction to Lattices and Order, Cambridge University Press, ISBN 978-0-521-78451-1 Atom