Abstract. Let ${\msbm W}$ be a proper subvariety of a variety ${\msbm V}$. We say that a functor $F\colon{\cal K}\to{\msbm V}$ is a ${\msbm W}$-relatively full embedding if $F$ is faithful, $\mathop{\rm Im} (Ff)\notin{\msbm W}$ for any ${\cal K}$-morphism $f$, and if $f\colon Fa\to Fb$ is a homomorphism for ${\cal K}$-objects $a$ and $b$ then either $\mathop{\rm Im} (f)\in{\msbm W}$ or $f=Fg$ for some ${\cal K}$-morphism $g\colon a\to b$. A variety of algebras ${\msbm V}$ is called var-relatively universal if there exist a proper subvariety ${\msbm W}$ of ${\msbm V}$ and a ${\msbm W}$-relatively full embedding from the category of all graphs and compatible mappings into ${\msbm V}$. We prove that a variety ${\msbm V}$ of bands is var-relatively universal if and only if ${\msbm V}$ contains the variety of all left semi-normal bands or the variety of all right semi-normal bands.

AMS Subject Classification (1991): 18B15, 20M07, 20M15

Keyword(s): full embedding, lattice of varieties of bands, determinacy

Received July 27, 1999, and in revised form April 4, 2000. (Registered under 2748/2009.)