We give a negative answer to the question raised by Mart Abel about whether his proposed definition of ${K}_{0} $ and ${K}_{1} $ groups in terms of quasi multiplication is indeed equivalent to the established ones in algebraic $K$-theory.

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.

This paper discusses a flaw in Murasugi–Przytycki’s Memoir ‘An index of a graph with applications to knot theory’ [Mem. Amer. Math. Soc. 106 (1993)]. We point out and partly fix a gap occurring in the proof of Murasugi–Przytycki’s braid index inequalities involving the graph index. We explain why their notion of index fails to precisely reflect the reduction of Seifert circles by their diagram move, and redefine the index to account for that discrepancy.

We prove that when $(a, m)= 1$ and $a$ is a quadratic residue $\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$, there are infinitely many Carmichael numbers in the arithmetic progression $a\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$. Indeed the number of them up to $x$ is at least ${x}^{1/ 5} $ when $x$ is large enough (depending on $m$).

We characterize the abelian groups $G$ for which the functors $\mathrm{Ext} (G, - )$ or $\mathrm{Ext} (- , G)$ commute with or invert certain direct sums or direct products.