skewbands

A skewband is a type of algebraic structure that generalizes the concept of a group. It consists of a set G, a binary operation, and an element e, which satisfies certain axioms. These axioms are similar to those of a group, but they relax the requirement that every element must have an inverse. Specifically, a skewband is a set S with a binary operation * and a left identity element 1 such that for all a, b, c in S: (1) a * (b * c) = (a * b) * c (associativity) and (2) 1 * a = a (left identity). Some definitions also include a right identity axiom (a * 1 = a), in which case it is called a monoid.

The term "skewband" is often used in contexts where the operation might not be commutative, and where the existence of inverses is not guaranteed. This contrasts with a group, where every element has a unique inverse. Skewbands can be viewed as a weakening of the group structure. They find applications in various areas of abstract algebra and theoretical computer science, particularly in the study of semigroups and related algebraic systems. The "skew" in skewband can refer to the fact that the operation might not be commutative, or that certain properties typically associated with groups are "skewed" or modified.