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A quasigroup is an algebraic structure resembling a group in the sense that it has a binary operation, but it lacks associativity and the requirement for an identity element. Each element in a quasigroup has a unique left and right inverse, making it a generalization of a group with relaxed constraints.
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Non-associative algebra refers to algebraic structures where the associative law (i.e., (ab)c = a(bc) for any elements a, b, and c) does not necessarily hold. These algebras are crucial in various fields, including theoretical physics and cryptography, as they generalize classical algebraic structures like Lie algebras and Jordan algebras.
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Alternative algebra is a non-associative algebraic structure where the associative law is not required to hold universally but does hold when two elements are the same. This structure generalizes associative algebras and includes examples like octonions, which are used in various mathematical and physical theories.
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Associativity is a property of certain binary operations that indicates the grouping of operands does not affect the result. This property is crucial in mathematics and computer science for optimizing computations and ensuring consistency in operations like addition and multiplication.
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Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.
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Loop theory is a branch of abstract algebra focused on the study of loops, which are quasigroups with an identity element. It explores the properties and structures of loops, generalizing group theory by relaxing associativity while maintaining the existence of an identity element.
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An algebraic structure is a set equipped with one or more operations that follow specific axioms, providing a framework to study algebraic systems like groups, rings, and fields. These structures allow mathematicians to abstract and generalize patterns and properties across different mathematical systems, facilitating deeper understanding and applications across various domains.
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Ruth Moufang was a pioneering German mathematician known for her contributions to projective geometry and the theory of non-associative algebraic structures, specifically Moufang loops. Her work laid the foundation for significant developments in algebra and geometry, influencing both theoretical research and practical applications in various scientific fields.
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