For each variety of bands [Vscr ], we give a formula for ϕ[Vscr ](m,k), which is the largest integer such that for every band B in [Vscr ] generated by m generators and k relations, there is a subset of the generators of size ϕ[Vscr ](m,k) which generates a (relatively) free sub-band of B as a basis. We also determine the semilattice structure of a finitely presented band.

Denote by $(R,\cdot)$ the multiplicative semigroup of an associative algebra $R$ over an infinite field, and let $(R,\circ)$ represent $R$ when viewed as a semigroup via the circle operation $x\circy=x+y+xy$. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of $R$. Namely, we prove that the following conditions on $R$ are equivalent: the semigroup $(R,\circ)$ satisfies an identity; the semigroup $(R,\cdot)$ satisfies a reduced identity; and, the associated Lie algebra of $R$ satisfies the Engel condition. When $R$ is finitely generated these conditions are each equivalent to $R$ being upper Lie nilpotent.

1991 Mathematics Subject Classification 16R40, 20M07, 20M25

An operator $T\in$[Lscr ]$(H)$ is called a square root of a hyponormal operator if $T^2$ is hyponormal. In this paper, we prove the following results: Let $S$ and $T$ be square roots of hyponormal operators.

(1) If $\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then $T$ is isoloid (i.e., every isolated point of $\sigma(T)$ is an eigenvalue of $T$).

(2) If $S$ and $T$ commute, then $ST$ is Weyl if and only if $S$ and $T$ are both Weyl.

(3) If $\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then Weyl's theorem holds for $T$.

(4) If $\sigma(T)\cap[-\sigma(T)]=\phi$, then $T$ is subscalar. As a corollary, we get that $T$ has a nontrivial invariant subspace if $\sigma(T)$ has non-empty interior. (See [3].)

We give the residue class, modulo a certain power of p, for the dimension of a primitive interior G-algebra in terms of the dimension of the source algebra. To illustrate, we improve a theorem of Brauer on the dimension of a block algebra.