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A sheaf is a presheaf with "something" added allowing us to define things locally. This task is forbidden for presheaves in general. Specifically, a presheaf F on a ...

There are at least two theorems known as Weierstrass's theorem. The first states that the only hypercomplex number systems with commutative multiplication and addition are ...

In nonstandard analysis, the transfer principle is the technical form of the following intuitive idea: "Anything provable about a given superstructure V by passing to a ...

There are at least three definitions of "groupoid" currently in use. The first type of groupoid is an algebraic structure on a set with a binary operator. The only ...

In a set X equipped with a binary operation · called a product, the multiplicative identity is an element e such that e·x=x·e=x for all x in X. It can be, for example, the ...

Consider expressions built up from variables and constants using function symbols. If v_1, ..., v_n are variables and t_1, ..., t_n are expressions, then a set of mappings ...

A map is a way of associating unique objects to every element in a given set. So a map f:A|->B from A to B is a function f such that for every a in A, there is a unique ...

A primitive root of a prime p is an integer g such that g (mod p) has multiplicative order p-1 (Ribenboim 1996, p. 22). More generally, if GCD(g,n)=1 (g and n are relatively ...

A quadratic equation is a second-order polynomial equation in a single variable x ax^2+bx+c=0, (1) with a!=0. Because it is a second-order polynomial equation, the ...

Let K be a number field of extension degree d over Q. Then an order O of K is a subring of the ring of integers of K with d generators over Z, including 1. The ring of ...