Sentence Analysis

[PaulMatthews]

His lesson cancelled, he went home
Barring accidents, we'll be there by ten.

I think that His lesson cancelled and barring accidents might be termed absolute clauses. They have no subjects.

Your first example has a subject "his lesson" and hence is an absolute clause. Your second example is a supplementary adjunct, though not an absolute clause since it has no overt subject. Its subject is located in the main clause, so there is a clear syntactic link to the main clause.

Here are a couple of links to useful resources that clearly state that an absolute clause contains its own subject:

http://www.oxfordreference.com/view/10.1093/acref/9780192800879.001.0001/acref-9780192800879-e-8

http://www.odlt.org/ballast/absolute_clause.html