Algebraic structure tree

by Ryan Hendrickson Jul 16 2016 updated Jul 16 2016

When is a monoid a semilattice? What's the difference between a semigroup and a groupoid? Find out here!

Some classes of Algebraic structure are given special names based on the properties of their sets and operations. These terms grew organically over the history of modern mathematics, so the overall list of names is a bit arbitrary (and in a few cases, some authors will use slightly different assumptions about certain terms, such as whether a semiring needs to have identity elements). This list is intended to clarify the situation to someone who has some familiarity with what an algebraic structure is, but not a lot of experience with using these specific terms.

%%comment:Tree is the wrong word; this should be more of an algebraic structure collection of disjoint directed acyclic graphs? But this is what other pages seem to have chosen to link to, so here we are!%%

One set, one binary operation

One set, two binary operations

For the below, we'll use $~$*$~$ and $~$\circ$~$ to denote the two binary operations in question. It might help to think of $~$*$~$ as "like addition" and $~$\circ$~$ as "like multiplication", but be careful—in most of these structures, properties of addition and multiplication like commutativity won't be assumed!


Ryan Hendrickson

Here are some things I'd love to be able to add to this page to make it more explanatory (some of these might be technical features, some an invitation to brainstorm more creative visualizations):