António Malheiro

There have been several attempts to extend the notion of conjugacy from groups to monoids.
The aim of this paper is study the decidability and independence of conjugacy problems
for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in
certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids,
$p$-conjugacy is ``almost'' transitive, $\sim_c$ is strictly included in $\sim_p$, and
the $p$- and $c$-conjugacy problems are decidable with linear compexity.
For other classes of monoids, the situation is more complicated.
We show that there exists a monoid $M$ defined by a finite complete
presentation such that the $c$-conjugacy problem for $M$ is undecidable, and
that for finitely presented monoids, the $c$-conjugacy problem and the word
problem are independent, as are the $c$-conjugacy and $p$-conjugacy problems.

n/a

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank n cannot satisfy a non-trivial identity of length less than or equal to n. A new identity is then proven to hold for the monoid of n×n upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank 3 into the direct product of two copies of the monoid of 3×3 upper-triangular tropical matrices, giving a new proof that the plactic monoid of rank 3 satisfies a non-trivial identity.

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite.

A semigroup is amiable if there is exactly one idempotent in each ℛ*-class and in each ℒ*-class. A semigroup is adequate if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups that do not contain isomorphic copies of two particular non-adequate semigroups as subsemigroups.