Amalgams of inverse semigroups and reversible two-counter machines

We show that the word problem for an amalgam $[S_1,S_2;U,\omega_1,\omega_2]$ of inverse semigroups may be undecidable even if we assume $S_1$ and $S_2$ (and therefore $U$) to have finite $\mathcal{R}$-classes and $\omega_1,\omega_2$ to be computable functions, interrupting a series of positive decidability results on the subject. This is achieved by encoding into an appropriate amalgam of inverse semigroups 2-counter machines with sufficient universality, and relating the nature of certain \sch graphs to sequences of computations in the machine

Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups

We prove that the Schützenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update

Multilinear equations in amalgams of finite inverse semigroups

Let S = S_1 ∗_U S_2 = Inv be the free amalgamated product of the finite inverse semigroups S_1 , S_2 and let Ξ be a finite set of unknowns. We consider the satisfiability problem for multilinear equations over S, i.e. equations w_L ≡ w_R with w_L,w_R ∈ (X ∪ X^{−1} ∪ Ξ ∪ Ξ^{−1})^+ such that each x ∈ Ξ labels at most one edge in the Schutzenberger automaton of either w_L or w_R relative to the presentation . We prove that the satisfiability problem for such equations is decidable using a normal form of the words w_L, w_R and the fact that the language recognized by the Schutzenberger automaton of any word in (X ∪ X^{−1})^+ relative to the presentation is context-free