Magma (algebra) - Citizendia

In abstract algebra, a magma (or groupoid) is a basic kind of algebraic structure. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, Specifically, a magma consists of a set M equipped with a single binary operation M × M → M. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two A binary operation is closed by definition, but no other axioms are imposed on the operation. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject

The term magma for this kind of structure was introduced by Bourbaki. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore. Øystein Ore ( 7 October 1899 in Oslo, Norway &ndash 13 August 1968 in Oslo was a Norwegian Mathematician However, groupoid also refers to an entirely different algebraic structure described at groupoid. In Mathematics, especially in Category theory and Homotopy theory

1 Types of magmas
2 Morphism of magmas
3 Combinatorics and parentheses
4 Free magma
5 More definitions
6 See also
7 References

Types of magmas

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

quasigroups—nonempty magmas where division is always possible;
loops—quasigroups with identity elements;
semigroups—magmas where the operation is associative;
monoids—semigroups with identity elements;
groups—monoids with inverse elements, or equivalently, associative loops (which are always quasigroups);
abelian groups—groups where the operation is commutative. In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, associativity is a property that a Binary operation can have In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, commutativity is the ability to change the order of something without changing the end result

From magma to group, via two alternative paths. Key:

M = magma, d = divisibility, a = associativity,

Q = quasigroup, S = semigroup, e = identity.

L = loop, i = inversibility, N = monoid, G = group

Note that both divisibility and inversibility imply

the existence of the cancellation property. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible.

Morphism of magmas

A morphism of magmas is a function $f:M\to N$ mapping magma $M$ to magma $N$ , that preserves the binary operation:

$f(x \; *_M \;y) = f(x) \; *_N\; f(y)$

where $* M$ and $* N$ denote the binary operation on $M$ , respectively on $N$ . In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

Combinatorics and parentheses

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. In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols. The set of all possible strings of balanced parenthesis is called the Dyck language. In the theory of Formal languages of Computer science, Mathematics, and Linguistics, the Dyck language (Dyck being pronounced "dike" The total number of different ways of writing n applications of the magma operator is given by the Catalan number $C n$ . In combinatorial mathematics, the Catalan numbers form a Sequence of Natural numbers that occur in various Counting problems often involving Thus, for example, $C 2 = 2$ , which is just the statement that $(a b) c$ and $a (b c)$ are the only two ways of pairing three elements of a magma with two operations.

A shorthand is often used in order to avoid as much parenthesis as possible. This is accomplished by using juxtaposition in place of the operation. For example, if the magma operation is *, then xy*z abbreviates (x * y) * z. Further abbreviations are possible by inserting spaces, for example by writing xy*z * wv in place of ((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, which is, however, counterintuitive. Polish notation, also known as prefix notation, is a form of notation for Logic, Arithmetic, and Algebra.

Free magma

A free magma $M X$ on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object). In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labeled by elements of X. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In Computer science, a binary tree is a tree data structure in which each node has at most two children. The operation is that of joining trees at the root. It therefore has a foundational role in syntax. In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the

A free magma has the universal property such that, if $f:X\to N$ is a function from the set X to any magma N, then there is a unique extension of $f$ to a morphism of magmas $f^\prime$

$f^\prime:M_X \to N.$

See also: free semigroup, free group, Hall set

More definitions

	Totality	Associativity	Identity	Division
Group-like structures
Group	Yes	Yes	Yes	Yes
Monoid	Yes	Yes	Yes	No
Semigroup	Yes	Yes	No	No
Loop	Yes	No	Yes	Yes
Quasigroup	Yes	No	No	Yes
Magma	Yes	No	No	No
Groupoid	No	Yes	Yes	Yes
Category	No	Yes	Yes	No

A magma (S, *) is called

unital if it has an identity element,
medial if it satisfies the identity xy * uz = xu * yz (i. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Abstract algebra, the free monoid on a set A is the Monoid whose elements are all the finite sequences (or strings) of zero or In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be In Mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function In Mathematics, associativity is a property that a Binary operation can have In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, especially in Category theory and Homotopy theory In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i This article is about medial in mathematics For other uses see Medial (disambiguation. 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 it satisfies the identity x = xy,
a semigroup with right zeros if it satisfies the identity x = yx,
a semigroup with zero multiplication if it satisfies the identity xy = uv,
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. In Mathematics, commutativity is the ability to change the order of something without changing the end result Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Mathematics, a unipotent element r of a ring R is one such that r &minus 1 is a Nilpotent element, in In Abstract algebra, a magma G is said to be left alternative if ( xx) y = x ( xy) for all x and y In Abstract algebra, power associativity is a weak form of Associativity. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, associativity is a property that a Binary operation can have Unar ( Urdu: انڑ) is one of the most purest and oldest Sindhi tribe in Sindh, Pakistan. Unar ( Urdu: انڑ) is one of the most purest and oldest Sindhi tribe in Sindh, Pakistan. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Mathematics, the notion of cancellative is a generalization of the notion of Invertible.

If $M \times M \to M$ is instead a partial operation, then M is called a partial magma. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two

References

M. In Mathematics, it can be shown that there exist magmas that are Commutative but not Associative. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, Hazewinkel (2001), “Magma”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
M. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Hazewinkel (2001), “Free magma”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Eric W. Weisstein, Groupoid at MathWorld. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

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