antiring

Antiring refers to a concept in abstract algebra, specifically within the study of ring theory. A ring is a set equipped with two binary operations, typically called addition and multiplication, satisfying certain axioms analogous to those of integers. An antiring, also known as a semiring with division or a near-semiring, is a structure that is similar to a ring but with a modification to the additive part of the structure.

In a standard ring, the additive operation forms an abelian group, meaning it is associative, commutative, has an identity element (zero), and every element has an additive inverse. In an antiring, the additive operation is typically associative and commutative and has an identity element (zero), but it does not necessarily have additive inverses. This means that for an element 'a' in the antiring, there might not be an element '-a' such that a + (-a) = 0.

The multiplicative operation in an antiring is associative and distributes over the additive operation. Similar to rings and semirings, there may or may not be multiplicative identity elements. The concept of an antiring is often encountered in areas like computer science, particularly in algorithms related to shortest paths (e.g., the $(\min, +)$ algebra) and formal languages. These algebraic structures provide a framework for analyzing problems that can be expressed in terms of summations and multiplications.