Mário J. J. Branco

LEFT ADEQUATE AND LEFT EHRESMANN MONOIDS

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 that is X*-proper, and an idempotent separating surjective morphism 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.

Two algebraic approaches to variants of the concatenation product

We extend an existing approach of the bideterministic concatenation product of languages aiming at the study of three other variants: unambiguous, left deterministic and right deterministic. Such an approach is based on monoid expansions. The proofs are purely algebraic and use another approach, based on properties on the kernel category of a monoid relational morphism, without going through the languages. This gives a unified fashion to deal with all these variants and allows us to better understand the connections between these two approaches. Finally, we show that local finiteness of an M-variety is transferred to the M-varieties corresponding to these variants and apply the general results to the M-variety of idempotent and commutative monoids.

On the Pin-Thérien expansion of idempotent monoids

AbstractThe objective of this paper is to pesent a description of the varieties

On generators and relations for unions of semigroups

\bar V

Extensions and covers for semigroups whose idempotents form a left regular band

of finite monoids introduced in the Pin and Thérien’s paper “On the bideterministic concatenation product” [4] in the case thatV is any variety of finite monoids.

Deterministic Concatenation Product of Languages Recognized by Finite Idempotent Monoids

Let a be a letter of an alphabet A. Given a lattice of languages L, we describe the set of ultrafilter inequalities satisfied by the lattice La generated by the languages of the form L or LaA * , where L is a language of L. We also describe the ultrafilter inequalities satisfied by the lattice L1 generated by the lattices La, for a ∈ A. When L is a lattice of regular languages, we first describe the profinite inequalities satisfied by La and L1 and then provide a small basis of inequalities defining L1 when L is a Boolean algebra of regular languages closed under quotient.