Do BiOS agreements allow patenting of improvements?
There is no provision in any BiOS-compatible agreement preventing any party from patenting anything, including improvements to BiOS-licensed technology.
However, those who benefit by non-assertion over BiOS-licensed intellectual property agree to the same obligation of non-assertion against all others who've agreed to the same terms.
The "grantback" is a concept found in many traditional licenses. The Improvements remain the intellectual property of the improver, but the owner of the improvements agrees to allow the licensor to use the improvements without infringing those rights. In the BiOS license, the owners of improvements agree to allow others, who've agreed to the same terms, to use the improvements, but exclusionary rights may still be enforced legally against any entity that does not agree to those terms.
No licensor or licensee under a BiOS license is under any obligation to patent Improvements or maintain patents, but patenting can be requested and supported by the community of contributors. Indeed, strategic patent applications and patents are part of the foundation of the license's enforceability, just as legally enforceable copyright is at the foundation of the UC Berkeley-based open source BSD license (http://www.opensource.org/licenses/bsd-license.php) that's fostered much university research and private sector product development.