Verbless Clause/Construction or Absolute Construction

A grammatical unit which resembles a clause, except that it lacks a verb phrase. Verbless clauses are often clauses from which the verb be has been omitted by ellipsis (Leech, 2006)

Verbless clause contain no verb element, and often no subject. Verbless clause are regarded as clauses because they function like finite and non-finite clauses and because they can be analysed in terms of one or more clause element. We can usually assume that a form of the verb has been omitted.

Traditionally structure of this kind have been called free adjuncts without verbal form. 

Verbless Clause as Complement of PP

with/without --->  non-finite/verbless

although/if/once/while ---> finite/non finite/verbless

Finite           : While I was working in Jakarta, I lived with them

Participial    : While working in Jakarta, I lived with them

Verbless       : While in Jakarta, I lived with them

Finite         : Although he was not unattractive, he was cut in a rougher mould than his father.

Verbless     : Although not unattractive, he was cut in a rougher mould than his father.

Finite         : These are now on order and will be circulated when they become available.

Verbless     : These are now on order and will be circulated when available.

The oranges, when ripe, are picked and sorted

Whether right or wrong, Alex always comes off worst in an argument

Verbless Clause Functioning Directly as Supplement

The meeting finally over, they all adjourned to the local cafe

A large crowd of refugees, many of them women and children, were imprisoned in the football stadium 

Dozens of tourists were stranded, many of them children

......other grammar books

Absolute Construction / Nominative Absolute

The absolute phrase also known as the nominative absolute is a structure independent from the main sentence; in form the absolute phrase is a noun phrase that include postnoun modifier (Kolln, 1994)

The weather being warm and clear, we decided to have a picnic 

The train having started, we went back to the hotel.

Reference :
Huddleston, R., Pullum, G.K., Reynolds, B., 2022, A Student's Introduction to English Grammar Second Edition, Cambridge : Cambridge University Press.

Kolln, Martha, 1994, Understanding English Grammar 4th Ed, New York : Maxwell Macmillan International

Leech, Geoffrey, 2006, A Glossary of English Grammar, Edinburg : Edinburg University Press. 

Leech, Geoffrey and Svartvik, Jan, 2013, A Communicative Grammar of English Third Edition. New York, NY : Routledge Taylor and Francis Group

Maiklejohn, JMD., 1895, A short grammar of the English Tongue With Three Hundred and Thirty Exercises, London : AM Holden