BIBLIOS

Document type
Journal articles

Document subtype
Full paper

Title
Left adequate and left Ehresmann monoids

Participants in the publication
Mário J. J. Branco (Author)
Dep. Matemática
CAUL
Gracinda M. S. Gomes (Author)
Dep. Matemática
CEMAT
Victoria Gould (Author)

Summary
This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M. In fact, our notion is that of T-proper, where T is a submonoid of M. We show that any left adequate monoid M has an X*-proper cover for some set X, that is, there is a left adequate monoid N that is X*-proper, and an idempotent separating surjective morphism from N to M of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X*-proper. In a subsequent paper, we show how to construct T-proper left adequate monoids from any monoid T acting via order-preserving maps on a semilattice with identity, and prove that the free left adequate monoid is of this form. An alternative description of the free left adequate monoid will appear in a paper of Kambites. We show how to obtain the labeled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. We also indicate how to obtain some of the analogous results in the two-sided case. This paper and its sequel, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.


Institution
FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA

Where published
International Journal of Algebra and Computation

Publication Identifiers
ISSN - 0218-1967
eISSN - 1793-6500

Publisher
World Scientific Pub Co Pte Lt



Document Identifiers
DOI - https://doi.org/10.1142/s0218196711006935
URL - http://dx.doi.org/10.1142/s0218196711006935

Rankings
SCOPUS Q2 (2011) - 0.648 - General Mathematics