Left adequate and left Ehresmann monoids II

Gracinda Gomes, Victoria Gould

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This article is the second of two presenting a new approach to left
adequate monoids. In the first, we introduced the notion of being T -proper,
where T is a submonoid of a left adequate monoid M. We showed that the
free left adequate monoid on a set X is X-proper. Further, any left adequate
monoid M has an X-proper cover for some set X, that is, there is an X-
proper left adequate monoid cM and an idempotent separating epimorphism
: cM ¿ M of the appropriate signature.
We now show how to construct a T -proper left adequate monoid P(T, Y )
from a monoid T acting via order preserving maps on a semilattice Y with
identity. Our construction plays the role for left adequate monoids that the
semidirect product of a group and a semilattice plays for inverse monoids. A
left adequate monoid M with semilattice E has an X-proper cover P(X,E).
Hence, by choosing a suitable semilattice EX and an action of X on EX,
we prove that the free left adequate monoid is of the form P(X,EX). An
alternative description of the free left adequate monoid appears in a recent
preprint of Kambites. We show how to obtain the labelled 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. Indeed this is the right setting: the class of
left Ehresmann monoids is the variety generated by the quasi-variety of left
adequate monoids. This paper, 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.
Original languageEnglish
Pages (from-to)171-195
Number of pages28
JournalJournal of Algebra
Issue number1
Publication statusPublished - 15 Dec 2011


  • (left) adequate monoid,
  • Ehresmann,
  • proper,
  • free objects

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