Index of algebraic structures

• Group-like: carrier set + 1 bin op

  • magma: CLOSURE

  • semi-group: CLOSURE, ASSOC

  • monoid: CLOSURE, ASSOC, ID

  • group: CLOSURE, ASSOC, ID, INVERSE

  • abelian group: CLOSURE, ASSOC, ID, INVERSE, COMMUTATIVITY

• ring: 2 bin ops

  • 1st bin op CLOSURE, ASSOC, ID, INVERSE

  • 2nd bin op ID, ASSOC, DISTR over 1st

  • A ring is an ABELIAN GROUP with a second bin op that is ASSOC, DISTRIBUTIVE over the abelian group op and has ID element.

A ring is a set R equipped with 2 binary operations + and · satisfying the following axioms (called the ring axioms):

• R is an abelian group under addition

• R is a monoid under multiplication

• Multiplication is distributive with respect to addition

• Field-like:

  • field: carrier set + 2 bin op

    • CLOSURE, ASSOC, 2xID, 2xINVERSE (additive and multiplicative), DISTR

    • may also be defined as having 4 ops: add (sub), mul (div)

    • addition: a + b, subtraction: a + (-b) using add. inverse element

    • mult: ab, division: a ÷ 1/b using mult. inverse element

    • DISTRIBUTIVITY of mult over addition

• category: ID, ASSOC