Weak inverse
In mathematics, the term weak inverse is used with several meanings.
Theory of semigroups
In the theory of semigroups, a weak inverse of an element x in a semigroup (S, •) is an element y such that y • x • y = y. If every element has a weak inverse, the semigroup is called an E-inversive or E-dense semigroup. An E-inversive semigroup may equivalently be defined by requiring that for every element x ∈ S, there exists y ∈ S such that x • y and y • x are idempotents.[1]
An element x of S for which there is an element y of S such that x • y • x = x is called regular. A regular semigroup is a semigroup in which every element is regular. This is a stronger notion than weak inverse. Every regular semigroup is E-inversive, but not vice versa.[1]
If every element x in S has a unique inverse y in S in the sense that x • y • x = x and y • x • y = y then S is called an inverse semigroup.
Category theory
In category theory, a weak inverse of an object A in a monoidal category C with monoidal product ⊗ and unit object I is an object B such that both A ⊗ B and B ⊗ A are isomorphic to the unit object I of C. A monoidal category in which every morphism is invertible and every object has a weak inverse is called a 2-group.
See also
References
- ↑ 1.0 1.1 John Fountain (2002). "An introduction to covers for semigroups". in Gracinda M. S. Gomes. Semigroups, Algorithms, Automata and Languages. World Scientific. pp. 167–168. ISBN 978-981-277-688-4. https://books.google.com/books?id=IL58mAsfXOgC&pg=PA167. preprint
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