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That reminds me of Vector Spaces. Is there any relation? Are Vector Spaces monoids?
by reese_john 9y ago
That reminds me of Vector Spaces. Is there any relation? Are Vector Spaces monoids?
- tome 9y agoYes, a vector space is a monoid under its addition.
- Jtsummers 9y agotome speaks the truth. More generally, in mathematics there are hierarchies of structures [0]. In this case monoids are both groupoids/magmas (a set with a binary operation returning elements of that set), and semigroups (a set with an associative binary operation returning elements of that set). Monoids add in an identity element. Vector spaces have addition over vectors and multiplication of vectors by scalars. Considering vector spaces and addition only, that is a monoid. It also happens to be further along this hierarchy as you have commutativity, inverses, and other properties. A bit like the statement: all squares are rectangles, not all rectangles are squares. We can say that all vector spaces are monoids (under addition), but not all monoids are vector spaces. [0] https://en.wikipedia.org/wiki/Algebraic_structure https://en.wikipedia.org/wiki/Algebraic_structure