Factorization Theorems for Morphisms of Ordered Groupoids and Inverse Semigroups

Steinberg, Benjamin

doi:10.1017/s0013091599001017

Cited by 13 publications

(20 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Theorem 6.7, 9 -jfi with j an embedding and £ : M(S) -> T an F-morphism. The congruence associated to /8 is clearly a Billhardt congruence (see [1,10] This result is also proved in [8] and can be deduced from the techniques of [19].…”

Section: Corollary 68 Let 6 : S -• T Be a Surjective Idempotent-purmentioning

confidence: 92%

Expansions of inverse semigroups

Lawson¹,

Margolis²,

Steinberg

2006

J. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of £-unitary inverse semigroups.2000 Mathematics subject classification: primary 20M18, 20M17.

show abstract

Section: Corollary 68 Let 6 : S -• T Be a Surjective Idempotent-purmentioning

confidence: 92%

Expansions of inverse semigroups

Lawson¹,

Margolis²,

Steinberg

2006

J. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then a left action (analogous to the case of an ordered groupoid acting on an ordered groupoid [21]) (π, A) of G on X consists of the following data: First we require an ordered graph morphism π : X (1) → Id(G). Now define an ordered 2-complex (G, X) by:…”

Section: The Schützenberger Complexmentioning

confidence: 99%

“…This map is, in fact, an immersion [20] on the 1-skeleton; also, any based 2-cell has at most one based lift to any vertex. McAlister's P -theorem [10] can easily be shown to be equivalent to stating that I is E-unitary (that is, the natural projection to G is idempotent pure) if and only there is an ordered 2-complex Z containing X and an extension of ψ to Z which is a covering such that Π 1 (Z) is an enlargement of Π 1 (X) in the sense of [7,21]. In fact, all the results of [21] can be restated and proved more generally in the context of ordered 2-complexes where the role of the derived ordered groupoid is replaced by what is called in homotopy theory, the mapping fiber.…”

Section: The Schützenberger Complexmentioning

confidence: 99%

A topological approach to inverse and regular semigroups

Steinberg¹

2003

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

Work of Ehresmann and Schein shows that an inverse semigroup can be viewed as a groupoid with an order structure; this approach was generalized by Nambooripad to apply to arbitrary regular semigroups. This paper introduces the notion of an ordered 2-complex and shows how to represent any ordered groupoid as the fundamental groupoid of an ordered 2-complex. This approach then allows us to construct a standard 2-complex for an inverse semigroup presentation.Our primary applications are to calculating the maximal subgroups of an inverse semigroup which, under our topological approach, turn out to be the fundamental groups of the various connected components of the standard 2-complex. Our main results generalize results of Haatja, Margolis, and Meakin giving a graph of groups decomposition for the maximal subgroups of certain regular semigroup amalgams. We also generalize a theorem of Hall by showing the strong embeddability of certain regular semigroup amalgams as well as structural results of Nambooripad and Pastijn on such amalgams.

show abstract

“…The notions of a groupoid action on a groupoid and the associated construction of the semidirect product seem to have been first defined in [2]. We describe these notions for ordered groupoids: our definitions are equivalent to those given by Steinberg [12]. An action of an ordered groupoid G on an ordered groupoid A is determined by the following data.…”

Section: The Maximum Enlargement Theoremmentioning

confidence: 99%

“…Any one of these properties thus defines p as a fibration. Steinberg [12] adopts star-surjectivity as the definition of a fibration of ordered groupoids. He undertakes a general study of factorization theorems for ordered groupoids, based on the semidirect product construction for one ordered groupoid acting on another (also found in [2]) and a left adjoint Der -the derived ordered groupoid -to the semidirect product.…”

Section: Introductionmentioning

confidence: 99%

Fibrations of Ordered Groupoids and the Factorization of Ordered Functors

AlYamani

Gilbert²,

Miller³

2015

Appl Categor Struct

View full text Add to dashboard Cite

We investigate canonical factorizations of ordered functors of ordered groupoids through star-surjective functors. Our main construction is a quotient ordered groupoid, depending on an ordered version of the notion of normal subgroupoid, that results is the factorization of an ordered functor as a star-surjective functor followed by a star-injective functor. Any star-injective functor possesses a universal factorization through a covering, by Ehresmann's Maximum Enlargement Theorem. We also show that any ordered functor has a canonical factorization through a functor with the ordered homotopy lifting property.

show abstract

Factorization Theorems for Morphisms of Ordered Groupoids and Inverse Semigroups

Cited by 13 publications

References 21 publications

Expansions of inverse semigroups

Expansions of inverse semigroups

A topological approach to inverse and regular semigroups

Fibrations of Ordered Groupoids and the Factorization of Ordered Functors

Contact Info

Product

Resources

About