Inverse element

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This is the default notion of inverse element. There is another, more general notion of inverse element in a semigroup, which does not depend on existence of a neutral element

Definition

Definition with symbols

Given a set S with a binary operation * and a neutral element e for *, and given elements a and b we say that:

  • b is a left inverse to a if b * a = e
  • b is a right inverse to a if a * b = e
  • b is an inverse or two-sided inverse to a if a * b = b * a = e (that is, b is both a left and a right inverse to a)

An element which possesses a (left/right) inverse is termed (left/right) invertible.

Facts

Equality of left and right inverses

If * is an associative binary operation, and an element has both a left and a right inverse with respect to *, then the left and right inverse are equal.

To prove this, let a be an element of S with left inverse b and right inverse c. Then, (b * a) * c = b * (a * c) by associativity. The left side simplifies to e * c  = c while the right side simplifies to b * e = b. Hence, b = c.

Some easy corollaries:

  • If an element has a left inverse, it can have at most one right inverse; moreover, if the right inverse exists, it must be equal to the left inverse, and is thus a two-sided inverse
  • If an element has a right inverse, it can have at most one left inverse; moreover, if the left inverse exists, it must be equal to the right inverse, and is thus a two-sided inverse