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Publication details

Document type
Journal articles

Document subtype
Full paper

Title
Ehresmann monoids

Participants in the publication
Mário J.J. Branco (Author)
Dep. Matemática
CEMAT-Ciências - Centro de Matemática Computacional e Estocástica-Ciências
Gracinda M. S. Gomes (Author)
Dep. Matemática
CEMAT-Ciências - Centro de Matemática Computacional e Estocástica-Ciências
Victoria Gould (Author)

Scope
International

Refereeing
Yes

Summary
Ehresmann monoids form a variety of bi-unary monoids, that is, monoids equipped with two basic unary operations, the images of which coincide and form a semilattice of projections. The monoid of binary relations B_X on any set X with unary operations of domain and range is Ehresmann. Inverse monoids, regarded as bi-unary submonoids of B_X via the Wagner–Preston representation theorem, are therefore also Ehresmann. At the other extreme, any monoid is Ehresmann, where the unary operations take all elements to the monoid identity. We demonstrate here using semilattices and monoids as building blocks that Ehresmann monoids have a rich structure, fundamentally different from that of inverse monoids and, indeed, from that of the interim class of restriction monoids.\\\\n\\\\nThe article introduces a notion of properness for Ehresmann monoids, that tightly controls structure and is dependent upon sets of generators. We show how to construct an Ehresmann monoid P(T,Y) satisfying our properness condition from a semilattice Y acted upon on both sides by a monoid T via order preserving maps. The free Ehresmann monoid on X is proven to be of the form P(X^*, Y). The next question deals with the existence of proper covers. We answer it in a positive way, proving that any Ehresmann monoid M admits a cover of the form P(X^*, E), where E is the semilattice of projections of M. Here a ‘cover’ is a preimage under a morphism that separates elements in E.

Date of Submisson/Request
2015-01-16
Date of Publication
2015-12-01

Institution
FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA

Where published
Journal of Algebra

Publication Identifiers
ISSN - 0021-8693

Publisher
Elsevier BV

Volume
443

Number of pages
34
Starting page
349
Last page
382

Document Identifiers
DOI - https://doi.org/10.1016/j.jalgebra.2015.06.035
URL - http://dx.doi.org/10.1016/j.jalgebra.2015.06.035

Rankings
SCIMAGO Q1 (2015) - 1.143 - Algebra and Number Theory


Export

APA
Mário J.J. Branco, Gracinda M. S. Gomes, Victoria Gould, (2015). Ehresmann monoids. Journal of Algebra, 443, 349-382. ISSN 0021-8693. eISSN . http://dx.doi.org/10.1016/j.jalgebra.2015.06.035

IEEE
Mário J.J. Branco, Gracinda M. S. Gomes, Victoria Gould, "Ehresmann monoids" in Journal of Algebra, vol. 443, pp. 349-382, 2015. 10.1016/j.jalgebra.2015.06.035

BIBTEX
@article{41348, author = {Mário J.J. Branco and Gracinda M. S. Gomes and Victoria Gould}, title = {Ehresmann monoids}, journal = {Journal of Algebra}, year = 2015, pages = {349-382}, volume = 443 }