At the Jam Session on Semigroups and Automata that took place at CMUP in 2011, J. Almeida proposed the following question: Is LG tame? Here LG denotes the pseudovariety of local groups, consisting of all finite semigroups S such that eSe is a group for all idempotents e of S, and tameness is a property of pseudovarieties introduced by J. Almeida and B. Steinberg with the purpose of solving the decidability problem for iterated semidirect products of pseudovarieties. That objective has not yet been reached but tameness has proved to be of interest to solve membership problems involving other types of operators. The tameness property is parameterized by an implicit signature σ and we speak of σ-tameness. Proving the σ-tameness of a pseudovariety V involves proving two properties: that the word problem for σ-terms over V is decidable, and that V is σ-reducible.

This talk is about the word problem for κ-terms over the pseudovariety LG, where κ is the most commonly used signature, known as the canonical signature. A κ-term is a formal expression obtained from letters of an alphabet using two operations: the binary concatenation and the unary (ω-1)-power. A κ-term has a natural interpretation on each finite semigroup S: the concatenation is viewed as the semigroup multiplication while (ω-1)-power is interpreted as the unary operation which sends each element s of S to the inverse of ss^ω in the maximal subgroup containing its unique idempotent power s^ω. The κ-word problem for LG consists in deciding whether two κ-terms have the same interpretation over every finite local group. This problem was solved by Conceição Nogueira, Lurdes Teixeira and the speaker by transforming each arbitrary κ-term into another one in a determined normal form and by showing that different k-terms in normal form have different interpretations over LG.