The monogenic free inverse semigroup $FI_1$ is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of $FI_1$ is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of $FI_1$ is finitely presented as a semigroup if and only if it is a finite semilattice.

Let $\mathbb {Z}$ be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product $\mathbb {Z}\times S$ contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that $\mathbb {Z}\times S$ has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.

The direct product $\mathbb{N}\times \mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of $S$ has a relative left or right identity element.

We consider several familiar varieties of completely regular semigroups such as groups and completely simple semigroups. For each of them, we characterize their members in terms of absence of certain kinds of subsemigroups, as well as absence of certain divisors, and in terms of a homomorphism of a concrete semigroup into the semigroup itself. For each of these varieties $ \mathcal{V} $ we determine minimal non-$ \mathcal{V} $ varieties, provide a basis for their identities, determine their join and give a basis for its identities. Most of this is complete; one of the items missing is a basis for identities for minimal nonlocal orthogroups. Three tables and a figure illustrate the results obtained.

It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.)