Copulas stable under univariate conditioning are studied. Limit approach to construction of conditioning stable copulas is introduced and illustrated. In the class of Archimedean copulas, Clayton copulas are shown to be the only conditioning stable copulas. Conditioning stable singular copulas are also discussed and examples of non-Archimedean absolutely continuous copulas which are conditioning stable are given.
The univariate conditioning of copulas is studied, yielding a construction method for copulas based on an a priori given copula. Based on the gluing method, g-ordinal sum of copulas is introduced and a representation of copulas by means of g-ordinal sums is given. Though different right conditionings commute, this is not the case of right and left conditioning, with a special exception of Archimedean copulas. Several interesting examples are given. Especially, any Ali-Mikhail-Haq copula with a given parameter λ > 0 allows to construct via conditioning any Ali-Mikhail-Haq copula with parameter μ \in [0,λ].