Quasi-decompositions and quasidirect products of Hilbert algebras

A quasi-decomposition of a Hilbert algebra A is a pair (C, D) of its subalgebras such that (i) every element a A is a meet c d with c C, d D, where c and d are compatible (i.e., c → d = c → (c d)), and (ii) d → c = c (then c is uniquely defined). Quasi-decompositions are intimately related to the so-called triple construction of Hilbert algebras, which we reinterpret as a construction of quasidirect products. We show that it can be viewed as a generalization of the semidirect product construction, that quasidirect products has a certain universal property and that they can be characterised in terms of short exact sequences. We also discuss four classes of Hilbert algebras and give for each of them conditions on a quasi-decomposition of an arbitrary Hilbert algebra A under which A belongs to this class.