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monoid
This term is used exclusively within the specialized registers of abstract algebra, category theory, and theoretical computer science. It describes a specific mathematical object that is less restrictive than a group because it does not require every element to have an inverse.
In programming contexts, particularly in functional languages like Haskell, a monoid is a design pattern used to aggregate data consistently. It is typically paired with adjectives describing its properties, such as "commutative monoid" or "free monoid," to specify the exact nature of the algebraic system being discussed.
Meanings
Examples
A monoid is essentially a semigroup with an identity element.
A monoid is essentially a semigroup with an identity element.
Do you think this set under addition forms a monoid?
I need to prove that the operation is associative to show it is a monoid.
The set of all strings over an alphabet forms a free monoid under concatenation.
Wait, if there is no identity element, then it cannot be a monoid.
Natural numbers under addition constitute a classic example of a monoid.
I wonder if this specific algebraic structure can be classified as a monoid.
The professor explained how every group is also a monoid.