🧮Non-associative Algebra Unit 6 – Quasigroups and loops

Key Concepts and Definitions

• Quasigroup is a set $Q$ with a binary operation $*$ such that for all $a,b \in Q$, there exist unique elements $x,y \in Q$ satisfying $a*x=b$ and $y*a=b$
• Latin square is an $n \times n$ array filled with $n$ different symbols, each occurring exactly once in each row and exactly once in each column
• Loop is a quasigroup with an identity element $e$ such that $a*e=e*a=a$ for all $a \in Q$
  • Identity element in a loop is unique
• Inverse element in a loop for each $a \in Q$, there exists a unique element $a^{-1} \in Q$ such that $a*a^{-1}=a^{-1}*a=e$
• Associativity in a loop $(a*b)*c=a*(b*c)$ for all $a,b,c \in Q$
• Commutativity in a loop $a*b=b*a$ for all $a,b \in Q$
• Homomorphism is a map $f:Q_1 \to Q_2$ between two quasigroups $(Q_1,*_1)$ and $(Q_2,*_2)$ such that $f(a *_1 b)=f(a) *_2 f(b)$ for all $a,b \in Q_1$

Historical Context and Development

• Quasigroups first appeared in the early 20th century in the work of Suschkewitsch and Moufang
• Loops were introduced by Albert in the 1940s as a generalization of groups
• Latin squares, which are closely related to quasigroups, have a longer history dating back to Euler in the 18th century
  • Euler used Latin squares in his work on orthogonal matrices
• Combinatorial aspects of quasigroups and loops were studied extensively in the mid-20th century by Bruck, Belousov, and others
• Connections between quasigroups, loops, and other algebraic structures (groups, rings, fields) were explored in the latter half of the 20th century
• Applications of quasigroups and loops in areas such as cryptography, coding theory, and physics have driven research in recent decades

Properties of Quasigroups

• Cancellation laws hold in a quasigroup for all $a,b,c \in Q$, if $a*b=a*c$ or $b*a=c*a$, then $b=c$
• Isotopy of quasigroups $(Q_1,*_1)$ and $(Q_2,*_2)$ are isotopic if there exist bijections $\alpha, \beta, \gamma: Q_1 \to Q_2$ such that $\alpha(a) *_2 \beta(b) = \gamma(a *_1 b)$ for all $a,b \in Q_1$
  • Isotopy is an equivalence relation on the class of all quasigroups
• Quasigroups can be represented by Latin squares each element of the quasigroup corresponds to a row, column, and symbol in the Latin square
• Orthogonal Latin squares $L_1$ and $L_2$ of order $n$ are orthogonal if, when superimposed, each ordered pair of symbols occurs exactly once
  • Orthogonal Latin squares are related to mutually orthogonal quasigroups
• Quasigroups can be used to construct block designs (Steiner triple systems) and error-correcting codes

Types of Loops

• Moufang loops satisfy the Moufang identities $(xy)(zx)=(x(yz))x$ and $((xy)z)y=x(y(zy))$ for all $x,y,z \in Q$
  • Moufang loops are diassociative $x(yz)=(xy)z$ whenever $x$, $y$, or $z$ are equal
• Bol loops satisfy the Bol identity $((xy)z)y=x((yz)y)$ for all $x,y,z \in Q$
  • Bol loops are left alternative $x(xy)=(xx)y$ for all $x,y \in Q$
• Bruck loops (K-loops) satisfy the identities $(xy)^{-1}=y^{-1}x^{-1}$ and $x(y^{-1}(yz))=(xy)z$ for all $x,y,z \in Q$
• Commutative Moufang loops (C-loops) are Moufang loops that are also commutative
• Lie groups are differentiable manifolds that are also groups the Lie algebra of a Lie group is a loop with additional structure

Relationship to Other Algebraic Structures

• Groups are associative loops every group is a loop, but not every loop is a group
  • Associativity is the key difference between groups and loops
• Moufang loops are generalizations of groups Moufang identities are a weakened form of associativity
• Abelian groups are commutative groups every abelian group is a commutative Moufang loop, but not every commutative Moufang loop is an abelian group
• Rings and fields have two binary operations (addition and multiplication) loops and quasigroups have only one binary operation
  • Some loops (e.g., Lie groups) can be endowed with additional structure to form rings or fields
• Modules over a ring are abelian groups with scalar multiplication by elements of the ring loops can be viewed as non-associative modules

Applications and Examples

• Cryptography quasigroups and loops can be used to construct cryptographic hash functions and block ciphers
  • Quasigroup string transformations are used in the Edon-R hash function
• Coding theory quasigroups and loops are used in the construction of error-correcting codes
  • Quasigroup codes are a class of non-linear codes with good error-correcting properties
• Physics loops and quasigroups appear in the study of non-associative quantum mechanics and alternative algebras
  • Octonions, a non-associative division algebra, form a loop under multiplication
• Combinatorics Latin squares, which are closely related to quasigroups, have numerous applications in experimental design and statistics
  • Orthogonal Latin squares are used in the construction of mutually orthogonal Latin squares (MOLS)
• Genetics Mendel's laws of inheritance can be modeled using quasigroups and loops
  • Genotypes of offspring can be predicted using a quasigroup operation on parental genotypes

Theorems and Proofs

• Lagrange's theorem for loops the order of a subloop divides the order of the loop
  • Proof involves coset decomposition and counting arguments
• Moufang's theorem a loop is a Moufang loop if and only if any two elements generate a subgroup
  • Proof relies on the Moufang identities and the diassociativity of Moufang loops
• Bruck's theorem a loop is a Bol loop if and only if it satisfies the identity $((xy)z)y=x((yz)y)$
  • Proof uses the left alternative property and the Bol identity
• Belousov's theorem a quasigroup is isotopic to a group if and only if it is a loop and satisfies the identity $(xy)(zx)=(xz)(yx)$
  • Proof involves constructing an isotopy between the quasigroup and a group using the identity
• Cauchy's theorem for loops if a prime $p$ divides the order of a finite loop, then the loop has an element of order $p$
  • Proof is similar to the group-theoretic version, using cosets and the Lagrange's theorem for loops

Advanced Topics and Current Research

• Quasigroup representations study of the representation theory of quasigroups and loops, analogous to group representation theory
  • Involves the study of quasigroup modules and loop algebras
• Homotopy theory of quasigroups and loops extension of homotopy theory to the non-associative setting
  • Involves the study of loop spaces and quasigroup cohomology
• Quandles non-associative algebraic structures related to knot theory and low-dimensional topology
  • Quandles are quasigroups satisfying additional identities, used to define knot invariants
• Non-associative geometry study of geometric structures (manifolds, bundles, connections) based on non-associative algebraic structures
  • Involves the study of loop spaces, Lie groups, and alternative algebras
• Computational methods development of algorithms and software for studying quasigroups, loops, and related structures
  • Includes methods for enumerating quasigroups and loops, solving the isomorphism problem, and studying their properties