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Antihomomorphism
In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inverse of an antiautomorphism is also an antiautomorphism. Definition Informally, an antihomomorphism is a map that switches the order of multiplication. Formally, an antihomomorphism between structures X and Y is a homomorphism \phi\colon X \to Y^, where Y^ equals Y as a set, but has its multiplication reversed to that defined on Y. Denoting the (generally non-commutative) multiplication on Y by \cdot, the multiplication on Y^, denoted by *, is defined by x*y := y \cdot x. The object Y^ is called the opposite object to Y (respectively, opposite group, opposite algebra, opposite category etc.). This definition is equivalent to that of a homomorphism \phi\colon X^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
