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
- quasigroups—nonempty magmas where division is always possible;
- loops—quasigroups with identity elements;
- semigroups—magmas where the operation is associative;
- semilattices—semigroups where the operation is commutative and idempotent;
- monoids—semigroups with identity elements;
- groups—monoids with inverse elements, or equivalently, associative loops or associative quasigroups;
- abelian groups—groups where the operation is commutative.
-
- Note that both divisibility and invertibility
- imply the cancellation property.
Read more about this topic: Magma (algebra)
Famous quotes containing the words types of and/or types:
“Our children evaluate themselves based on the opinions we have of them. When we use harsh words, biting comments, and a sarcastic tone of voice, we plant the seeds of self-doubt in their developing minds.... Children who receive a steady diet of these types of messages end up feeling powerless, inadequate, and unimportant. They start to believe that they are bad, and that they can never do enough.”
—Stephanie Martson (20th century)
“If there is nothing new on the earth, still the traveler always has a resource in the skies. They are constantly turning a new page to view. The wind sets the types on this blue ground, and the inquiring may always read a new truth there.”
—Henry David Thoreau (1817–1862)