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An algebra for this endofunctor is a set X together with a function 1 + N × X → X. To define such a function, we need a point x ∈ X and a function N × X → X. The set of finite lists of natural numbers is an initial algebra for this functor.
The algebra (, [,]) in the above example is an initial algebra. Various finite data structures used in programming, such as lists and trees, can be obtained as initial algebras of specific endofunctors.
Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then An initial object I in C is a universal morphism from • to U.
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
Other synonyms for the notion include absolutely free algebra and anarchic algebra. [ 3 ] From a category theory perspective, a term algebra is the initial object for the category of all X -generated algebras of the same signature , and this object, unique up to isomorphism , is called an initial algebra ; it generates by homomorphic projection ...
It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. [126]
This statement is an initial property of the tensor algebra since it expresses the fact that the pair ((),), where : (()) is the inclusion map, is a universal morphism from the vector space to the functor .
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials [1] —hence the method may be referred to as the FOIL method.The word FOIL is an acronym for the four terms of the product:
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