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.

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

  • Groupoid, sometimes known as a magma. This is the freebie. Have a set and a binary operation? That’s a groupoid.

    • A semigroup is a groupoid where the operation is associative.

      • A monoid is a semigroup where the operation has an identity element.

        • A group is a monoid where every element has an inverse element under the binary operation.

      • A semilattice is a semigroup where the operation is idempotent and commutative.

    • A quasigroup is a groupoid where every element has a left and right quotient under the binary operation (sometimes called the Latin square property).

      • A loop is a quasigroup with identity.

      • A group, as defined above, can also be defined as a (non-empty) quasigroup where the operation is associative (quotients and associativity give a two-sided identity and two-sided inverses, 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!

  • Ringoid assumes only that \(\circ\) distributes over \(*\)—in other words, \(a \circ (b \* c) = (a \circ b) \* (a \circ c)\) and \((a \* b) \circ c = (a \circ c) \* (b \circ c)\).

    • A semiring is a ringoid where both \(*\) and \(\circ\) define semigroups.

      • An additive semiring is a semiring where \(*\) is commutative.

        • A rig is an additive semiring where \(*\) has an identity element. (It’s almost a ring! It’s just missing negatives.)

          • 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 division ring is a ring with unity where every element (except for the identity of \(*\)) has an inverse element under \(\circ\).

                • 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 \(\vee\).)

      • A bounded lattice is a lattice where both operations have identities.