# Ordered groupoid quotients and congruences on inverse semigroups

Ordered groupoid quotients and congruences on inverse semigroups We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ J –relation. The corresponding equivalence relation $$\simeq _N$$ ≃ N is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semigroup Forum Springer Journals

# Ordered groupoid quotients and congruences on inverse semigroups

, Volume 96 (3) – Aug 2, 2017
17 pages

/lp/springer_journal/ordered-groupoid-quotients-and-congruences-on-inverse-semigroups-uoti40hU0X
Publisher
Springer US
Subject
Mathematics; Algebra
ISSN
0037-1912
eISSN
1432-2137
D.O.I.
10.1007/s00233-017-9891-4
Publisher site
See Article on Publisher Site

### Abstract

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ J –relation. The corresponding equivalence relation $$\simeq _N$$ ≃ N is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

### Journal

Semigroup ForumSpringer Journals

Published: Aug 2, 2017

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