Venue: TBA

Venue: TBA

Venue: TBA

More information available here.

Venue: TBA

Room:  1.19 do VII 
 
Abstract: The study and characterization of the subvarieties of W (variety of semigroups whose square is completely regular) when W is a E-semigroup (the set of its idempotents, E(S), is a subsemigroup[1]) has been proposed by Borralho and Kinyon[3] following a question from Araújo et al.[2].
Although in its formulation this problem had been limited to semigroups where the idempotents commute, called E-commutative semigroups[1], ∀e, f ∈ E(S), e.f = f.e, in this work, I will also present results for the E-semigroups in W. These can be defined by the additional condition ∀e, f ∈ E(S), f.e ∈ W(e.f), where W(a) represents the set of weak inverses of a, W(a) = {x : x.a.x = x}.
When idempotents form a subsemigroup, we call it a band. If they commute, E(S) is
a semilattice of idempotents; otherwise, it is a semilattice of rectangular bands. Although the former is a particular case of the latter, this presentation treats both situations distinctly, as they are different structures with different properties.

The point of view we adopt to study the structure of a relatively complex semigroup is to first decompose it into a collection of subsemigroups, each of which has a somewhat
simpler structure, and then compose a complicated and “bigger” semigroup from simpler
ones.
In this work, this is accomplished in 3 steps:
1. Dividing the semigroup S in disjunct closed classes Sα of known structure.
2. Defining a “gross” structure by studying the relation between these classes induced
by the semigroup operation, Sα.Sβ → Sγ.
3. Studying a “fine” structure that relates the positions of two elements from different
classes to the position of their product. For any S ∈ W and any a ∈ S, a2 lies in a group; thus S is an epigroup, also called group-bound in Howie[4]. Then S splits into unipotency classes, Ke, defined by elements whose square lies in Ge, the largest subgroup whose idempotent is e[5]. In W, these unipotency classes are disjoint subsemigroups of S and can be indexed by the elements of the semilattice E(S). Starting from S 2 and taking into account that E(S2) = E(S), we refer to the conditions for being able to define homomorphisms in S, φα,β : Sα → Sβ for all α ≥ β of the semilattice E(S), and so define a multiplication of the elements of the
different unipotency classes. This way, we can study the “fine” structure of S ∈ W when
their idempotents form a subsemigroup.
Except as stated otherwise, the basic concepts used here can be found in Howie[4] and
Shevrin[5].
References
[1] J. Almeida, J.-E. Pin, and P. Weil, Semigroups whose idempotents form a subsemigroup, Math.
Proc. Camb. Phi. Soc. 111 (1992), 241-253.
[2] J. Ara ́ujo, M Kinyon, J. Konieczny, and A. Malheiro, Four notions of conjugacy for abstract
semigroups, in: Proceedings of the Royal Society of Edinburgh Section A: Mathematics 147(6) (2017),
1169-1214.
[3] M. Borralho and M. Kinyon, Variants of epigroups and primary conjugacy, Commun. Algebra
48(12) (2020), 5465-5473.
[4] J.M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs, New Series 12, (Oxford University Press, 1995).
[5] L.N. Shevrin, Epigroups, in Structural theory of automata, semigroups and universal algebra (Springer, 2005) 331-380.