BIBLIOS

Tipo
Artigos em Revista

Tipo de Documento
Artigo Completo

Título
Ehresmann monoids

Participantes na publicação
Mário J.J. Branco (Author)
Dep. Matemática
CEMAT
Gracinda M. S. Gomes (Author)
Dep. Matemática
CEMAT
Victoria Gould (Author)

Resumo
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.


Instituição
FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA

Suporte
Journal of Algebra

Identificadores da Publicação
ISSN - 0021-8693

Editora
Elsevier BV



Identificadores do Documento
DOI - https://doi.org/10.1016/j.jalgebra.2015.06.035
URL - http://dx.doi.org/10.1016/j.jalgebra.2015.06.035

Identificadores de Qualidade
SCIMAGO Q1 (2015) - 1.143 - Algebra and Number Theory