In mathematics, Arf semigroups are certain subsets of the non-negative

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s closed under

addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of ...

, that were studied by . They appeared as the semigroups of values of

Arf ring In mathematics, an Arf ring was defined by to be a 1- dimensional commutative semi-local Macaulay ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and t ...

s. A subset of the integers forms a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

if it includes zero, and if every two elements in the subset have a sum that also belongs to the subset. In this case, it is called a "numerical semigroup". A numerical semigroup is called an Arf semigroup if, for every three elements ''x'', ''y'', and ''z'' with ''z'' = min(''x'', ''y'', and ''z''), the semigroup also contains the element . For instance, the set containing zero and all even numbers greater than 10 is an Arf semigroup.

References

* *. Semigroup theory {{Abstract-algebra-stub