Underlying set

What do a group, a par­tially or­dered set, and a topolog­i­cal space have in com­mon? Each is a set with some struc­ture built on top of it, and in each case, we call the set the un­der­ly­ing set. noteA group is a set with an op­er­a­tion, a poset is a set with an or­der­ing, and a topolog­i­cal space is a set with a col­lec­tion of sub­sets that satisfy a cer­tain prop­erty.

Alge­braic structures

An alge­braic struc­ture is a set equipped with op­er­a­tors that fol­low cer­tain laws, such as a group, which is a pair \((X, \bullet)\) where \(X\) is a set and \(\bullet\) is an op­er­a­tor that fol­lows cer­tain laws. Given an alge­braic struc­ture, we can sim­ply throw away the op­er­a­tors and re­cover the set (\(X\), in this case), which is known as the “un­der­ly­ing set” of the struc­ture.

The un­der­ly­ing set is some­times known as the “car­rier set.” Some alge­braic struc­tures have more than one un­der­ly­ing set; for ex­am­ple, a vec­tor space is an alge­braic struc­ture built out of a field of scalars and a com­mu­ta­tive group, in which case the term “un­der­ly­ing set” is am­bigu­ous.

Parents:

  • Abstract algebra

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