## Abstract

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.

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 language | English |
---|---|

Pages (from-to) | 171-195 |

Number of pages | 28 |

Journal | Journal of Algebra |

Volume | 348 |

Issue number | 1 |

DOIs | |

Publication status | Published - 15 Dec 2011 |

## Keywords

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