examples of monoid ,group and field with finite elements
Answers
If a semigroup {M, * } has an identity element with respect to the operation * , then {M, * } is called a monoid. For example, if N is the set of natural numbers, then {N,+} and {N,X} are monoids with the identity elements 0 and 1 respectively. The semigroups {E,+} and {E,X} are not monoids.
Step-by-step explanation:
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element
Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics.
For example, the functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object.
In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing.
In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn–Rhodes theory), and formal language theory (star height problem).
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