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Heap (mathematics)
In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set ''H'' with a ternary operation denoted ,y,z\in H that satisfies a modified associativity property: \forall a,b,c,d,e \in H \quad a,b,cd,e] = ,[d,c,be= [a,b,[c,d,e">,c,b">,[d,c,b<_a>e.html" ;"title=",c,b.html" ;"title=",[d,c,b">,[d,c,be">,c,b.html" ;"title=",[d,c,b">,[d,c,be= [a,b,[c,d,e. A biunitary element ''h'' of a semiheap satisfies [''h'',''h'',''k''] = ''k'' = [''k'',''h'',''h''] for every ''k'' in ''H''. A heap is a semiheap in which every element is biunitary. It can be thought of as a group_(mathematics), group with the identity element "forgotten". The term ''heap'' is derived from груда, Russian for "heap", "pile", or "stack". Anton Sushkevich used the term in his ''Theory of Generalized Groups'' (1937) which influenced Viktor Wagner, promulgator of semiheaps, heaps, and generalized heaps.C.D. Hollings & M.V. Lawson (2017) ''Wagner's Theory of Generalised Heap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heterogeneous Relation
In mathematics, a binary relation associates some elements of one set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the " divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations, and especially homogeneous relations, are used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices and is denoted as . Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of functions, composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. John Mackintosh Howie, a prominent semigroup theorist, described this work as "so all-pervading that, on encountering a new semigroup, almost the first question one asks is 'What are the Green relations like?'" (Howie 2002). The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for Group (mathematics), groups, but in this case tell us nothing useful, because groups always have divisibility. Instead of working directly with a semigroup ''S'', it is convenient to define Green's relations over the monoid ''S''1. (''S''1 is "''S'' with an identity adjoined if necessary"; if ''S'' is not already a monoid, a new element is adjoined and defined to be an identity.) This ensures that principal id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup Ideal
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup'' o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Semigroup
In group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that and , i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries. (The convention followed in this article will be that of writing a function on the right of its argument, e.g. ''x'' ''f'' rather than ''f''(''x''), and composing functions from left to right—a convention often observed in semigroup theory.) Origins Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a Set (mathematics), set: a partial function, partial transformation ''α'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain of . If equals , that is, if is defined on every element in , then is said to be a total function. In other words, a partial function is a binary relation over two sets that associates to every element of the first set ''at most'' one element of the second set; it is thus a univalent relation. This generalizes the concept of a (total) function by not requiring ''every'' element of the first set to be associated to an element of the second set. A partial function is often used when its exact domain of definition is not known, or is difficult to specify. However, even when the exact domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in calculus, where, for example, the quotien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
