magmas

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sortsclosed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra.


Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include
Magma to Group.svg

From magma to group, via two alternative paths. Key:
M = magma, d = divisibility, a = associativity,
Q = quasigroup, S = semigroup, e = identity.
L = loop, i = invertibility, N = monoid, G = group
Note that both divisibility and invertibility imply
the existence of the cancellation property.
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