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Complete Systems Of Subalgebras

In the 1960s, G. Gratzer introduced the notion of the minimal extension property (MEP) of a finite sequence in order to investigate pn-sequences and free spectra of algebras. While there are many particular results on the MEP, stating that some sequences or families of sequences have the MEP, no general result has been obtained so far and the main general problems remain open. In this paper we prove a fact about existence of finite complete systems of subalgebras, which generalizes a method used occasionally to prove the minimal extension property of a sequence. Using this we obtain some general results about minimal extensions and, along the way, a rather unexpected solution of one of the central open problems in the area.