International Journal of Algebra and Computation

We survey recent results about asymptotic functions of groups, obtained by the authors in collaboration with J.-C. Birget, V. Guba and E. Rips. We also discuss methods used in the proofs of these results.

We consider the following property for a group G:(RZ_n)ifH₁,…,H_nare finitely generated subgroups of G then the setH₁ H₂⋯ H_n= {h₁ ⋯ h_n| h₁∈ H₁, …,h_n∈ H_n}is closed with respect to the profinite topology of G. It is obvious that finite groups and finitely generated commutative groups have the property (RZ_n). L. Ribes and P. Zalesskiĭ proved that any free group has (RZ_n). We show that the property (RZ_n) is stable under the free product operation. We use techniques developed by B. Herwig and D. Lascar on the one hand, R. Gitik on the other hand.

We show that if G is a fundamental group of a finite k-acylindrical graph of groups where every vertex group is word-hyperbolic and where every edge-monomorphism is a quasi-isometric embedding, then all the vertex groups are quasiconvex in G (the group G is word-hyperbolic by the Combination Theorem of M. Bestvina and M. Feighn). This allows one, in particular, to approximate the word metric on G by normal forms for this graph of groups.

The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilattice S is isomorphic to the semilattice Con_c L of compact congruences of a lattice L.

While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that |S|≤ℵ₁. Furthermore, one can then take Lwith permutable congruences. This contrasts with the case where |S|≥ℵ₂, where there are counterexamples S for which Lcannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well.

We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Con_c functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square2² can be lifted with respect to the Con_c functor.

We prove that the latter results cannot be extended to the cube, 2³. In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Con_c functor, by lattices with permutable congruences.

We also extend many of our results to lattices with almost permutable congruences, that is, α∨β=αβ ∪βα, for all congruences α and β.

We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Con_c functor on finite Boolean semilattices.

This paper presents a study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B₂, denoted V*(B₂). We construct a basis of pseudoidentities for the semidirect powers of the pseudovariety generated by B₂ which leads to the main result, which states that V*(B₂) is decidable.

Independently, using some suggestions given by J. Almeida in his book "Finite Semigroups and Universal Algebra", we constructed an algorithm to solve the membership problem in V* (B₂).

Orthodox transversals were introduced by the first author as a generalization of inverse transversals [Comm. Algebra 27(9) (1999), pp. 4275–4288]. One of our aims in this note is to consider the general case of orthodox transversals. The main results are on the sets I and Λ, two components of regular semigroups with orthodox transversals. We prove that the semibands Ī and generated by I and Λ respectively are bands. Also somewhat interesting generalizations of properties on orthodox semigroups are given; another aim in this note is to give some examples illustrating the situations that the class of regular semigroups with orthodox transversals properly includes the class of regular semigroups with inverse transversals as well as the class of orthodox semigroups.