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 Failed to parse (Missing texvc executable; please see math/README to configure.): M

equipped with a single binary operation Failed to parse (Missing texvc executable; please see math/README to configure.): M \times M \rightarrow M

. A binary operation is closed by definition, but no other axioms are imposed on the operation.

The term magma for this kind of structure was introduced by Nicolas Bourbaki. The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore.

Contents

Definition [link]

A magma is a set Failed to parse (Missing texvc executable; please see math/README to configure.): M

matched with an operation "Failed to parse (Missing texvc executable; please see math/README to configure.): \cdot

" that sends any two elements Failed to parse (Missing texvc executable; please see math/README to configure.): a,b \in M

to another element Failed to parse (Missing texvc executable; please see math/README to configure.): a \cdot b

. The symbol "Failed to parse (Missing texvc executable; please see math/README to configure.): \cdot " is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation Failed to parse (Missing texvc executable; please see math/README to configure.): (M,\cdot)

must satisfy the following requirement (known as the magma axiom):
For all Failed to parse (Missing texvc executable; please see math/README to configure.): a

, Failed to parse (Missing texvc executable; please see math/README to configure.): b

in Failed to parse (Missing texvc executable; please see math/README to configure.): M

, the result of the operation Failed to parse (Missing texvc executable; please see math/README to configure.): a \cdot b

is also in Failed to parse (Missing texvc executable; please see math/README to configure.): M

. And in mathematical notation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall a,b \in M : a \cdot b \in M


Etymology [link]

In French, the word "magma" has multiple common meanings, one of them being "jumble". It is likely that the French Bourbaki group referred to sets with well-defined binary operations as magmas with the "jumble" definition in mind.

Types of magmas [link]

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include

Magma to group2.svg
Note that both divisibility and invertibility imply
the existence of the cancellation property.

Morphism of magmas [link]

A morphism of magmas is a function Failed to parse (Missing texvc executable; please see math/README to configure.): f\colon M\to N

mapping magma Failed to parse (Missing texvc executable; please see math/README to configure.): M
to magma Failed to parse (Missing texvc executable; please see math/README to configure.): N

, that preserves the binary operation:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x \; *_M \;y) = f(x) \; *_N\; f(y)


where Failed to parse (Missing texvc executable; please see math/README to configure.): *_M

and Failed to parse (Missing texvc executable; please see math/README to configure.): *_N
denote the binary operation on Failed to parse (Missing texvc executable; please see math/README to configure.): M
and Failed to parse (Missing texvc executable; please see math/README to configure.): N
respectively.

Combinatorics and parentheses [link]

For the general, non-associative case, the magma operation may be repeatedly iterated. To denote pairings, parentheses are used. The resulting string consists of symbols denoting elements of the magma, and balanced sets of parenthesis. The set of all possible strings of balanced parenthesis is called the Dyck language. The total number of different ways of writing Failed to parse (Missing texvc executable; please see math/README to configure.): n

applications of the magma operator is given by the Catalan number Failed to parse (Missing texvc executable; please see math/README to configure.): C_n

. Thus, for example, Failed to parse (Missing texvc executable; please see math/README to configure.): C_2=2 , which is just the statement that Failed to parse (Missing texvc executable; please see math/README to configure.): (ab)c

and Failed to parse (Missing texvc executable; please see math/README to configure.): a(bc)
are the only two ways of pairing three elements of a magma with two operations.  

A shorthand is often used to reduce the number of parentheses. This is accomplished by using juxtaposition in place of the operation. For example, if the magma operation is Failed to parse (Missing texvc executable; please see math/README to configure.): * , then Failed to parse (Missing texvc executable; please see math/README to configure.): xy * z

abbreviates Failed to parse (Missing texvc executable; please see math/README to configure.): (x * y) * z

. Further abbreviations are possible by inserting spaces, for example by writing Failed to parse (Missing texvc executable; please see math/README to configure.): xy * z * wv

in place of  Failed to parse (Missing texvc executable; please see math/README to configure.): ((x * y) * z) * (w * v)

. Of course, for more complex expressions the use of parenthesis turns out to be inevitable. A way to avoid completely the use of parentheses is prefix notation.

Free magma [link]

A free magma Failed to parse (Missing texvc executable; please see math/README to configure.): M_X

on a set Failed to parse (Missing texvc executable; please see math/README to configure.): X
is the "most general possible" magma generated by the set Failed to parse (Missing texvc executable; please see math/README to configure.): X
(that is there are no relations or axioms imposed on the generators; see free object). It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labeled by elements of Failed to parse (Missing texvc executable; please see math/README to configure.): X

. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

A free magma has the universal property such that, if Failed to parse (Missing texvc executable; please see math/README to configure.): f\colon X\to N

is a function from the set Failed to parse (Missing texvc executable; please see math/README to configure.): X
to any magma Failed to parse (Missing texvc executable; please see math/README to configure.): N

, then there is a unique extension of Failed to parse (Missing texvc executable; please see math/README to configure.): f

to a morphism of magmas Failed to parse (Missing texvc executable; please see math/README to configure.): f^\prime

Failed to parse (Missing texvc executable; please see math/README to configure.): f^\prime\colon M_X \to N.


See also: free semigroup, free group, Hall set

Classification by properties [link]

Group-like structures
Totality Associativity Identity Inverses Commutativity
Magma Required not required not required not required not required
Quasigroup Required not required not required Required not required
Unital Required not required Required not required not required
Loop Required not required Required Required not required
Semigroup Required Required not required not required not required
Monoid Required Required Required not required not required
Group Required Required Required Required not required
Abelian Group Required Required Required Required Required
Groupoid not required Required Required Required not required
Category not required Required Required not required not required
Semicategory not required Required not required not required not required
Note: A quasigroup with associativity (equivalently, a semigroup with inverses)
already has an identity element, and is therefore a group.

A magma (S, *) is called

  • unital if it has an identity element,
  • medial if it satisfies the identity xy * uz = xu * yz (i.e. (x * y) * (u * z) = (x * u) * (y * z) for all x, y, u, z in S),
  • left semimedial if it satisfies the identity xx * yz = xy * xz,
  • right semimedial if it satisfies the identity yz * xx = yx * zx,
  • semimedial if it is both left and right semimedial,
  • left distributive if it satisfies the identity x * yz = xy * xz,
  • right distributive if it satisfies the identity yz * x = yx * zx,
  • autodistributive if it is both left and right distributive,
  • commutative if it satisfies the identity xy = yx,
  • idempotent if it satisfies the identity xx = x,
  • unipotent if it satisfies the identity xx = yy,
  • zeropotent if it satisfies the identity xx * y = yy * x = xx,
  • alternative if it satisfies the identities xx * y = x * xy and x * yy = xy * y,
  • power-associative if the submagma generated by any element is associative,
  • left-cancellative if for all x, y, and z, xy = xz implies y = z
  • right-cancellative if for all x, y, and z, yx = zx implies y = z
  • cancellative if it is both right-cancellative and left-cancellative
  • a semigroup if it satisfies the identity x * yz = xy * z (associativity),
  • a semigroup with left zeros if there are elements x for which the identity x = xy holds,
  • a semigroup with right zeros if there are elements x for which the identity x = yx holds,
  • a semigroup with zero multiplication or a null semigroup if it satisfies the identity xy = uv, for all x,y,u and v
  • a left unar if it satisfies the identity xy = xz,
  • a right unar if it satisfies the identity yx = zx,
  • trimedial if any triple of its (not necessarily distinct) elements generates a medial submagma,
  • entropic if it is a homomorphic image of a medial cancellation magma.

If Failed to parse (Missing texvc executable; please see math/README to configure.): *

is instead a partial operation, then S is called a partial magma.

Generalizations [link]

See n-ary group.

See also [link]

References [link]


https://wn.com/Magma_(algebra)

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 * : AA that is an antiautomorphism and an involution.

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

  • (x + y)* = x* + y*
  • (x y)* = y* x*
  • 1* = 1
  • (x*)* = x
  • 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

  • John A. Holbrook, David W. Kribs, and Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction." Quantum Information Processing. Volume 2, Number 5, p. 381419. Oct 2003.

  • Podcasts:

    PLAYLIST TIME:
    ×