There has been much research into freeness properties of finitely generated matrix semigroups under various constraints, mainly related to the dimensions of the generator matrices and the semiring over which the matrices are defined. A recent paper has also investigated freeness properties of matrices within a bounded language of matrices, which are of the form M1M2 · · ·Mk ⊆ Fn×n for some semi...

Definition 1. Let G be a group. G is said to be residually finite if the intersection of all normal subgroups of G of finite index in G is trivial. For a survey of results on residual finiteness and related properties, see Mag-nus, Karrass, and Solitar [6, Section 6.5]. We shall present a proof of the following well known theorem, which is important for Kharlampovich [4, 5]. See also O. V. Bele...

In this paper we discuss the splitting or decomposing of finitely generated groups into free products, free products with amalgamation or HNN extensions and we discuss the JSJ decomposition of finitely presented groups.

We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the o...

In this paper we present a method of obtaining finitely based linear representations of possibly infinitely based semigroups. Let R{x1, x2, . . . } be a free associative algebra over a commutative ring R with the countable set of free generators {x1, x2, . . . }. An endomorphism α of R{x1, x2, . . . } is called a semigroup endomorphism if x1α, x2α, . . . are monomials (i.e. finite products of x...

It is well-known that B2 is an inverse 0-simple semigroup that has played a distinguished role in the theory of semigroups. Let B2 be the variety generated by B2, and let B0 be the variety generated by the subsemigroup B0 = {0, d, cd, dc} of B2. Since all semigroups with five or fewer elements are finitely based [5], the variety B2 and its subvariety B0 are finitely based. In fact, it has been ...