Exploring the Size of Semigroups: How Many Elements Does X^x Have?

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In summary, the problem states that if X is a finite set with n elements, the semigroup X^x (which is isomorphic to the set of functions from X to X) has n^n elements. The notation X^x is not defined, but it can be assumed to represent the set of functions from X to X.
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yaganon
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(Moderator's note: thread moved from "Linear & Abstract Algebra")

problem # 12).

Suppose X is a finite set with n elements. Show that the semigroup X^x has n^n elements.

I'm confused. Isn't semigroup a set of functions? So when it says n elements, it actually means n functions? Also what is X^x defined as?
 
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yaganon said:
problem # 12).

Suppose X is a finite set with n elements. Show that the semigroup X^x has n^n elements.

I'm confused. Isn't semigroup a set of functions? So when it says n elements, it actually means n functions? Also what is X^x defined as?

Well, if this semigroup is isomorphic to the set of functions from X to X, then n^n is clearly the right number. I looked at the semigroup page on Wikipedia and couldn't find that notation, though. Can't find it in your textbook?
 

1. What is a semigroup?

A semigroup is a mathematical structure that consists of a set of elements and an associative binary operation defined on that set. The operation takes any two elements from the set and produces a third element from the set.

2. How is a semigroup different from a group?

A semigroup differs from a group in that it does not necessarily have an identity element or inverse elements for all its elements. This means that not all operations within a semigroup have an inverse or an element that acts as a neutral element when combined with another element.

3. How does a semigroup relate to other algebraic structures?

A semigroup is a more general structure than a monoid, which does have an identity element. It is also a special case of a magma, which is a set with a binary operation defined on it. Additionally, a semigroup can be seen as a generalization of a group, as it relaxes the requirement for an identity element and inverse elements.

4. What is the importance of semigroups in mathematics?

Semigroups have a wide range of applications in mathematics, particularly in algebra, topology, and number theory. They are also useful in computer science, specifically in the field of automata theory and formal languages. Additionally, semigroups have connections to other areas of mathematics, such as functional analysis and dynamical systems.

5. How are semigroups used in practical applications?

Semigroups have practical applications in various fields, including computer science, physics, and engineering. In computer science, they are used to model and analyze the behavior of finite state machines and other computational devices. In physics, semigroups are used to describe the evolution of systems over time. In engineering, they are used in control theory and signal processing.

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