Algebraic structure

Roughly speak­ing, an alge­braic struc­ture is a set \(X\), known as the un­der­ly­ing set, paired with a col­lec­tion of op­er­a­tions that obey a given set of laws. For ex­am­ple, a group is a set paired with a sin­gle bi­nary op­er­a­tion that satis­fies the four group ax­ioms, and a ring is a set paired with two bi­nary op­er­a­tions that satisfy the ten ring ax­ioms.

In fact, alge­braic struc­tures can have more than one un­der­ly­ing set. Most have only one (in­clud­ing monoids, groups, rings, fields, lat­tices, and ar­ith­metics), and differ in how their as­so­ci­ated op­er­a­tions work. More com­plex alge­braic struc­tures (such as alge­bras, mod­ules, and vec­tor spaces) have two un­der­ly­ing sets. For ex­am­ple, vec­tor spaces are defined us­ing both an un­der­ly­ing field of scalars and an un­der­ly­ing com­mu­ta­tive group of vec­tors.

For a map of alge­braic struc­tures and how they re­late to each other, see the tree of alge­braic struc­tures.

Children:

  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.

  • Ring
  • Abelian group

    A group where the op­er­a­tion com­mutes. Named af­ter Niels Hen­rik Abel.

  • Monoid
  • Algebraic structure tree

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

Parents:

  • Abstract algebra

    The study of groups, fields, vec­tor spaces, ar­ith­metics, alge­bras, and more.