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Band (algebra)
In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by . The lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard. Semilattices, left-zero bands, right-zero bands, rectangular bands, normal bands, left-regular bands, right-regular bands and regular bands are specific subclasses of bands that lie near the bottom of this lattice and which are of particular interest; they are briefly described below. Varieties of bands A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct products. Each variety of bands can be defined by a single defining identity. Semilattices Semilattices are exactly commutative bands; that is, they are the bands satisfying the equation * for all and . Bands induce a preorder that may be defined as x \le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Of Categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the dual (category theory), opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Classes Of Semigroups
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ''ab'' = ''ba'' for all elements ''a'' and ''b'' in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nowhere Commutative Semigroup
In mathematics, a nowhere commutative semigroup is a semigroup ''S'' such that, for all ''a'' and ''b'' in ''S'', if ''ab'' = ''ba'' then ''a'' = ''b''. A. H. Clifford, G. B. Preston (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). American Mathematical Society (p.26). A semigroup ''S'' is nowhere commutative if and only if any two elements of ''S'' are inverses of each other. Characterization of nowhere commutative semigroups Nowhere commutative semigroups can be characterized in several different ways. If ''S'' is a semigroup then the following statements are equivalent: *''S'' is nowhere commutative. *''S'' is a rectangular band (in the sense in which the term is used by John Howie ). *For all ''a'' and ''b'' in ''S'', ''aba'' = ''a''. *For all ''a'', ''b'' and ''c'' in ''S'', ''a''2 = ''a'' and ''abc'' = ''ac''. Even though, by definition, the rectangular bands are concrete semigroups, they have the defect that their definition is formulated not in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Ring
In mathematics, a Boolean ring is a ring for which for all in , that is, a ring that consists of only idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet , and ring addition to exclusive disjunction or symmetric difference (not disjunction , which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole. Notation There are at least four different and incompatible systems of notation for Boolean rings and algebras: * In commutative algebra the standard notation is to use for the ring sum of and , and use for their product. * In logic, a common notation is to use for the meet (same as the ring product) and use for the join, given in terms of ring notation (given just above) by . * In set theory and logic it is also common to use f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Lattice
In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, 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 (set theory), union and intersection (set theory), intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to order isomorphism, isomorphism—given as such a lattice of sets. Definition 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. Both views and their mutual correspondence are discussed in the article on lattice (order), lattices. In the present situation, the algebraic description appears to be more convenient. A lattice (''L'',∨,∧) is distributive if the following additional identity holds for all ''x'', ''y'', and ''z'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum ( join) and an infimum ( meet). A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only ''pairs'' of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete. Complete lattices appear in many applications in mathematics and computer science. Both order theory and universal algebra study them as a special class of lattices. Complete lattices must not be confused with complete partial orders (CPOs), a more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales). Formal definition A ''complete lattice'' is a partially ordered set (''L'', ≤) such that every subset ''A'' of ''L'' has both a greatest lower bound (the infimum, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countability
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (order)
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 (mathematics), join) and a unique infimum (also called a greatest lower bound or meet (mathematics), meet). An example is given by the power set of a set, partially ordered by Subset, inclusion, for which the supremum is the Union (set theory), union and the infimum is the Intersection (set theory), 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 Identity (mathematics), identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilatti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is Reflexive relation, reflexive, antisymmetric relation, antisymmetric, and Transitive relation, transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bands
Bands may refer to: * Bands (song), song by American rapper Comethazine * Bands (neckwear), form of formal neckwear * Bands (Italian Army irregulars) Bands () was an Italian military term for Irregular military, irregular forces, composed of natives, with Italian officers and NCOs in command. These units were employed by the Italian Army as auxiliaries to the Regular army, regular national and ..., Italian military term for irregular forces See also * Band (other) {{dab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Sets
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However some a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
