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An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field , and an operation called scalar multiplication between elements of the field (called scalars ), and elements of the vector space (called vectors ).
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...
A mathematical proof is a deductive argument ... to prove algebraic propositions ... Modern proof theory treats proofs as inductively defined data structures, ...
The study of mathematical proof is particularly important in logic, and has accumulated to automated theorem proving and formal verification of software. Logical formulas are discrete structures, as are proofs , which form finite trees [ 10 ] or, more generally, directed acyclic graph structures [ 11 ] [ 12 ] (with each inference step combining ...
Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field. A measure: intervals of the real line have a specific length , which can be extended to the Lebesgue measure on many of its subsets .
A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras. [citation ...