Can you give me some concrete examples of magmas?

I’ve seen the following (e.g. here):

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I’ve learned a bit about groups and I could give examples of groups, but when reading the given table, I couldn’t imagine of what a magma would be. It has no associativity, no identity, no divisibility and no commutativity. I can’t imagine what such a thing would be. Can you give a concrete example of a magma?

Answer

A (“strict”) magma you’ve probably heard of is the vector cross-product in R3:
(a,b,c)×(x,y,z)=(bzcy,cxaz,aybx)
R3 is closed under this operation, but it has neither associativity, commutativity, identity nor divisibility.

Kind of in the same way that any square, any rectangle and any parallelogram fulfills the criteria of a trapezoid, and thus are trapezoids, we say that any group, monoid or semigroup is also a magma. All we demand from the structure in order to call it a magma is that it is closed under the binary operation.

And just as any trapezoid in which all angles happens to be right still is a trapezoid even though most people would call it a rectangle, so too will any closed / total algebraic structure with associativity, identity and divisibility be a magma, even though most people would call it a group.

Attribution
Source : Link , Question Author : Red Banana , Answer Author : Arthur

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