Magma (algebra)

In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set, M, equipped with a single binary operation, M × M → M. The binary operation must be closed by definition but no other properties are imposed.

History and terminology

The term groupoid was introduced in 1926 by Heinrich Brandt describing his Brandt groupoid (translated from the German Gruppoid). The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term groupoid is "perhaps most often used in modern mathematics" in the sense given to it in category theory.

Magma

Magma (from Greek μάγμα, "thick unguent") is a mixture of molten or semi-molten rock, volatiles and solids that is found beneath the surface of the Earth, and is expected to exist on other terrestrial planets. Besides molten rock, magma may also contain suspended crystals, dissolved gas and sometimes gas bubbles. Magma often collects in magma chambers that may feed a volcano or solidify underground to form an intrusion. Magma is capable of intrusion into adjacent rocks (forming igneous dikes and sills), extrusion onto the surface as lava, and explosive ejection as tephra to form pyroclastic rock.

Description

Magma is a complex high-temperature fluid substance. Temperatures of most magmas are in the range 700 °C to 1300 °C (or 1300 °F to 2400 °F), but very rare carbonatite magmas may be as cool as 600 °C, and komatiite magmas may have been as hot as 1600 °C. Most magmas are silicate mixtures.

Environments of magma formation and compositions are commonly correlated. Environments include subduction zones, continental rift zones,mid-ocean ridges and hotspots. Despite being found in such widespread locales, the bulk of the Earth's crust and mantle is not molten. Except for the liquid outer core, most of the Earth takes the form of a rheid, a form of solid that can move or deform under pressure. Magma, as liquid, preferentially forms in high temperature, low pressure environments within several kilometers of the Earth's surface.

Magma (Jonathan Darque)

Magma (Jonathan Darque) is a fictional character, a supervillain from Marvel Comics. He first appeared in Marvel Team-Up vol. 1 #110, as an enemy of Spider-Man and Iron Man.

Fictional character biography

Jonathan Darque was the chief executive officer of a mining company investigating new and cheap sources of energy. His investigations were opposed by environmental activists, who held demonstrations at his trial bore sites. His wife died in a car crash when attempting to evade the activists' blockade. Darque used his engineering skills to design a battle suit allowing him to become Magma. He then developed an underground crime organization.

Darque/Magma created a device he called the Long Range Sonic Stata Scanner (LRSSS), which enabled him to discover the epicentres of earthquakes before they erupted. He also used the machine to generate waves causing earthquakes; this enabled him to blackmail the Mayor of New York City. (At the publication time of this story, the historical Mayor was Ed Koch). Magma held a press conference to reveal his plans. Spider-Man and Iron Man joined forces to drill down vertically to reach the source of the earthquake, where they discovered his hidden base. After Magma was defeated by Spider-Man and Iron Man in a battle on the surface, he escaped in a pod into the middle of the Atlantic Ocean. Spider-Man aimed the LRSSS at this location and Magma was engulfed by the resultant waves and he disappeared into the depths of the ocean.

Gorath

Gorath, released in Japan as Calamity Star Gorath (妖星ゴラス, Yōsei Gorasu), is a Japanese science fiction tokusatsu film produced by Toho in 1962. The story for Gorath was by Jojiro Okami.

Plot

The year is 1980, and the film opens with the launch of the JX-1 Hayabusa spaceship into outer space. The ship, originally sent to collect data on Saturn, has its course diverted to investigate the mysterious star Gorath, reported as being 6000 times the size of the Earth. It is feared that the star's path could come dangerously close to Earth. The JX-1 reaches locates Gorath and it's much smaller than earth but with 6000 times the gravity. The JX-1 radio's back any data about the star but gets sucked into the star's gravitational field which drags the ship into Gorath, incinerating it.

Japan and the rest of the world are stunned by the discovery and, after some reluctance, send up the JX-2 Ootori spaceship for a voyage to investigate Gorath. The United Nations band together to discover a solution to the problem, and decide that their only solutions are to either destroy Gorath or move the planet out of the way.

*-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:

for all x, y in A.

This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.

Algebra

Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") 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 and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. 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 more 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 and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).

†-algebra

A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.

References