SUMMARY / RELATED TOPICS

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity, they occur in several branches of mathematics. For example, the functions from a set into itself form a monoid with respect to function composition. More in category theory, the morphisms of an object to itself form a monoid, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for concurrent computing. In theoretical computer science, the study of monoids is fundamental for automata theory, formal language theory. See Semigroup for the history of the subject, some other general properties of monoids. Suppose that S is a set and • is some binary operation S × S → S S with • is a monoid if it satisfies the following two axioms: Associativity For all a, b and c in S, the equation • c = a • holds.

Identity element There exists an element e in S such that for every element a in S, the equations e • a = a • e = a hold. In other words, a monoid is a semigroup with an identity element, it can be thought of as a magma with associativity and identity. The identity element of a monoid is unique. For this reason the identity is regarded as a constant, i. e. 0-ary operation. The monoid therefore is characterized by specification of the triple. Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; this notation does not imply. A monoid in which each element has an inverse is a group. A submonoid of a monoid is a subset N of M, closed under the monoid operation and contains the identity element e of M. Symbolically, N is a submonoid of M if N ⊆ M, x • y ∈ N whenever x, y ∈ N, e ∈ N. N is thus a monoid under the binary operation inherited from M. A subset S of M is said to be a generator of M if M is the smallest set containing S, closed under the monoid operation, or equivalently M is the result of applying the finitary closure operator to S.

If there is a generator of M that has finite cardinality M is said to be finitely generated. Not every set S will generate a monoid. A monoid whose operation is commutative is called a commutative monoid. Commutative monoids are written additively. Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if there exists z such that x + z = y. An order-unit of a commutative monoid M is an element u of M such that for any element x of M, there exists v in the set generated by u such that x ≤ v; this is used in case M is the positive cone of a ordered abelian group G, in which case we say that u is an order-unit of G. A monoid for which the operation is commutative for some, but not all elements is a trace monoid. Out of the 16 possible binary Boolean operators, each of the four that has a two sided identity is commutative and associative and thus makes the set a commutative monoid. Under the standard definitions, AND and XNOR have the identity True while XOR and OR have the identity False.

The monoids from AND and OR are idempotent while those from XOR and XNOR are not. The natural numbers, N, form multiplication. A submonoid of N under addition is called a numerical monoid; the positive integers, N ∖, form a commutative monoid under multiplication. Given a set A, all subsets of A form a commutative monoid under intersection operation. Given a set A, all subsets of A form a commutative monoid under union operation. Generalizing the previous example, every bounded semilattice is an idempotent commutative monoid. In particular, any bounded lattice can be endowed; the identity elements are its bottom, respectively. Being lattices, Heyting algebras and Boolean algebras are endowed with these monoid structures; every singleton set closed under a binary operation • forms the trivial monoid, the trivial group. Every group is a monoid and every abelian group. Any semigroup S may be turned into a monoid by adjoining an element e not in S and defining e • s = s = s • e for all s ∈ S; this conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids.

Thus, an idempotent monoid may be formed by adjoining an identity element e to the left zero semigroup over a set S. The opposite monoid is formed from the right zero semigroup over S. Adjoin an identity e to the left-zero semigroup with two elements; the resulting idempotent monoid models the lexicographical order of a sequence given the orders of its elements, with e representing equality. The elements of

Iqrar Ahmad Khan

Iqrar Ahmad Khan is a Pakistani agricultural scientist and a professor of Horticulture at the University of Agriculture, Faisalabad in Pakistan. Iqrar Ahmad Khan was born on 8 December 1953 in Sheikhupura District, Pakistan, he received his master's degree from the University of Agriculture, Faisalabad in 1976. He left for the United States for higher studies and earned his Ph. D. degree in horticulture from the University of California, Riverside in 1988. Iqrar Ahmad Khan started his teaching career from the University of Faisalabad, he left for the United States for higher studies. Iqrar Ahmad Khan has released a potato variety, has pioneered the research on breeding seedless Kinnow and discovered two new botanical varieties of wheat, he was author of STED funded Citrus Nursery Project launched at UAF. He was instrumental in developing international/regional mango research program to combat the sudden death of mango, he served as a Vice Chancellor for the University of Agriculture, Faisalabad from 2008 to 2017.

He was appointed to this position due to his extensive experience of 31 years in research and development of agriculture and environment. Prior to this appointment in 2008, he had teaching assignments at seven universities, including six foreign universities. In 2009, Iqrar Ahmad Khan expressed his opinion in a newspaper interview stating that during the regime of General Ayub Khan in the 1960s, Government of Pakistan invested in agriculture which resulted in a'green revolution' in Pakistan at that time, he emphasized the need for a similar focus on the Pakistani farmer once again. In the same above interview in 2009, he was quoted as saying, "A university has to be a global player, it must prepare students for a global market. We will have to be the role models. We plan to give admission to students after matriculation, which will be a revolutionary step. I believe agriculture development and rural development can eradicate poverty."In a 2017 interview to a major Pakistani newspaper, he said, "It is a matter of grave concern that farmer's son doesn't want to become a farmer due to low profitability, heavy post-harvest losses and other challenges."

Founder/Director, Center for Agricultural Biochemistry and Biotechnology, Pakistan Director-General, Pakistan Atomic Energy Commission, Pakistan Chief Scientist/Director, Nuclear Institute for Agriculture and Biology, Pakistan Fellow, Pakistan Academy of Sciences Sitara-i-Imtiaz Award by the President of Pakistan in 2012 University of Agriculture, Faisalabad

List of Oricon number-one albums of 2010

The highest-selling albums and mini-albums in Japan are ranked in the Oricon Weekly Chart, published by Oricon Style magazine. The data is compiled by Oricon based on each album's weekly physical sales. In 2010, 42 albums reached number-one. Pop singer-songwriter Hideaki Tokunaga's Vocalist 4 had the longest chart run of 2010; the album remained at the top of the charts from its issue date of April 20 to May 24. The best-selling album overall of 2010 was idol group Arashi's Boku no Miteiru Fūkei, released in mid-2010, which sold 1,053,064 copies; the second-best-selling album was Ikimono-gakari's Ikimonobakari: Members Best Selection, which sold 906.756 copies, followed by J-pop singer Kana Nishino's To Love, with 645,417 copies albums sold. The fourth- and fifth-best-selling albums were Funky Monkey Babys Best and Sense by Funky Monkey Babys and Mr. Children respectively. Funky Monkey Babys Best sold 613,603 copies. 2010 in music