For a given finite poset (P, ≤), we construct strict completions of P which are models of all finite lattices L such that the set of join-irreducible elements of L is isomorphic to P . This family of lattices, M P , turns out to be itself a lattice, which is lower bounded and lower semimodular. We determine the join-irreducible elements of this lattice. We relate properties of the lattice M P to properties of our given poset P , and in particular we characterize the posets P for which |M P | ≤ 2. Finally we study the case where M P is distributive.
1. Introduction. Given a variety si of lattice-ordered algebras, a lattice L is said to be a relative si-lattice if every closed interval [a, b] of L may be given the structure of an algebra in si (in other words, is the reduct of a member of si-not necessarily unique). This paper discusses the characterisation in terms of forbidden substructures of finite relative .stf-lattices. We treat a large class of varieties si of distributive-lattice-ordered algebras. For these varieties, the finite algebras can be described dually in terms of finite ordered sets, so that order-theoretic results and techniques prove valuable.Our study was prompted by a desire to set in a wider context the following characterisations of relative de Morgan lattices and of relative Stone lattices.
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