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.