Loop (algebra)
Introduction
In abstract algebra, a **loop** is a quasigroup with an identity element. Loops generalize the concept of groups by relaxing some of the axioms that define a group. While every group is a loop, not every loop is a group. The study of loops is a rich and intricate field within algebra, offering insights into various algebraic structures and their properties.
Definition and Basic Properties
A **loop** \( (L, \cdot) \) is a set \( L \) equipped with a binary operation \( \cdot \) such that:
1. **Closure**: For all \( a, b \in L \), the result of the operation \( a \cdot b \) is also in \( L \). 2. **Identity Element**: There exists an element \( e \in L \) such that for every \( a \in L \), \( e \cdot a = a \cdot e = a \). 3. **Inverse Element**: For each \( a \in L \), there exists an element \( b \in L \) such that \( a \cdot b = b \cdot a = e \). 4. **Latin Square Property**: For each pair of elements \( a, b \in L \), there exist unique elements \( x, y \in L \) such that \( a \cdot x = b \) and \( y \cdot a = b \).
The identity element in a loop is unique, and each element has a unique inverse.
Examples of Loops
- Groups as Loops
Every group is a loop. For example, the set of integers under addition, \( (\mathbb{Z}, +) \), forms a loop with 0 as the identity element.
- Non-Associative Loops
Not all loops are associative. A classic example of a non-associative loop is the **Moufang loop**, which satisfies the Moufang identities but not necessarily the associative law.
Types of Loops
- Moufang Loops
A **Moufang loop** is a loop that satisfies any one of the following equivalent identities (Moufang identities): - \( (xy)(zx) = ((xy)z)x \) - \( x(y(xz)) = ((xy)x)z \) - \( (xy)(zx) = x((yz)x) \)
Moufang loops generalize the concept of groups and are named after Ruth Moufang, who studied them extensively.
- Bol Loops
A **Bol loop** is a loop that satisfies the left Bol identity: - \( x(y(xz)) = ((xy)x)z \)
There are also right Bol loops, which satisfy the right Bol identity: - \( ((zx)y)x = z(x(yx)) \)
- Steiner Loops
A **Steiner loop** is a loop that satisfies the identity: - \( (xy)(yx) = x \)
Steiner loops are closely related to Steiner triple systems in combinatorial design theory.
Properties and Theorems
- Isotopy
Two loops \( (L, \cdot) \) and \( (M, \circ) \) are said to be **isotopic** if there exist bijections \( \alpha, \beta, \gamma: L \to M \) such that for all \( a, b \in L \): \[ \alpha(a) \circ \beta(b) = \gamma(a \cdot b) \]
Isotopy is an equivalence relation on the class of loops, and isotopic loops share many algebraic properties.
- Automorphisms
An **automorphism** of a loop \( (L, \cdot) \) is a bijection \( \phi: L \to L \) that preserves the loop operation: \[ \phi(a \cdot b) = \phi(a) \cdot \phi(b) \]
The set of all automorphisms of a loop forms a group under composition, known as the **automorphism group** of the loop.
Applications
- Geometry
Loops, particularly Moufang loops, have applications in projective geometry and the theory of incidence structures. They are used to study properties of geometric configurations that are invariant under certain transformations.
- Cryptography
Certain types of loops, such as quasigroups, are used in cryptographic algorithms for constructing secure communication protocols. The non-associative nature of these structures provides additional security features.
See Also
References
- Bruck, R. H. (1958). "A Survey of Binary Systems". Springer-Verlag.
- Pflugfelder, H. O. (1990). "Quasigroups and Loops: Introduction". Heldermann Verlag.
- Smith, J. D. H. (2007). "An Introduction to Quasigroups and Their Representations". Chapman & Hall/CRC.