This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of $p$ -groups from different coclass families that have equivalent Quillen categories.

Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

We present a new construction of crossed-product duality for maximal coactions that uses Fischer’s work on maximalizations. Given a group $G$ and a coaction $(A,\unicode[STIX]{x1D6FF})$ we define a generalized fixed-point algebra as a certain subalgebra of $M(A\rtimes _{\unicode[STIX]{x1D6FF}}G\rtimes _{\,\widehat{\unicode[STIX]{x1D6FF}}}G)$ , and recover the coaction via this double crossed product. Our goal is to formulate this duality in a category-theoretic context, and one advantage of our construction is that it breaks down into parts that are easy to handle in this regard. We first explain this for the category of nondegenerate *-homomorphisms and then, analogously, for the category of $C^{\ast }$ -correspondences. Also, we outline partial results for the ‘outer’ category, which has been studied previously by the authors.

Using a result of Warnaar, we prove a number of single- and multi-sum identities in the spirit of Ramanujan’s partial theta identities, but with partial indefinite binary theta functions in the role of partial theta functions. We also calculate the corresponding residual identities and use a result of Ji and Zhao to recast our identities in terms of indefinite ternary theta functions.

Skew Boolean algebras for which pairs of elements have natural meets, called intersections, are studied from a universal algebraic perspective. Their lattice of varieties is described and shown to coincide with the lattice of quasi-varieties. Some connections of relevance to arbitrary skew Boolean algebras are also established.