Algebraic structure tree
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.
One set, one binary operation
, sometimes known as a magma. This is the freebie. Have a set and a binary operation? That’s a groupoid.
Ais a groupoid where every element has a under the binary operation (sometimes called the ).
Ais a quasigroup with identity.
A group, as defined above, can also be defined as a (non-empty) quasigroup where the operation is associative (, provided there’s at least one element to be that identity).
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!
\(\circ\) distributes over \(*\)—in other words, \(a \circ (b \* c) = (a \circ b) \* (a \circ c)\) and \((a \* b) \circ c = (a \circ c) \* (b \circ c)\).assumes only that
A \(*\) and \(\circ\) define semigroups.is a ringoid where both
An \(*\) is commutative.is a semiring where
A \(*\) has an identity element. (It’s almost a ring! It’s just missing negatives.)is an additive semiring where
A ring is a rig where every element has an inverse element under \(*\). (Some authors also require \(\circ\) to have an identity to call the structure a ring.)
A ring with unity is a ring where \(\circ\) has an identity. (Some authors just use the word “ring” for this; others use variations like “unit ring”.)
A \(*\)) has an inverse element under \(\circ\).is a ring with unity where every element (except for the identity of
A field is a division ring where \(\circ\) is commutative.
A lattice is a ringoid where both \(*\) and \(\circ\) define semilattices, and satisfy the absorption laws ($a \circ (a * b) = a * (a \circ b) = a$). (While we’ll continue to use \(*\) and \(\circ\) here, the two operations of a lattice are almost always denoted with \(\wedge\) and \(\vee\).)
Ais a lattice where both operations have identities.