Reducibility in Finite Posets

All the posets/lattices considered here are finite with element 0. An element x of a poset satisfying certain properties is deletable if P − x is a poset satisfying the same properties. A class of posets is reducible if each poset of this class admits at least one deletable element. When restricted to lattices, a class of lattices is reducible if and only if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step. This notion, however, is different from the notion of dismantalability for lattices; see [6]. It is known that the class of distributive lattices need not be reducible and, thus, also the class of modular lattices. However, the class of pseudocomplemented lattices as well as the classes of semimodular and locally distributive lattices are reducible; see [2]. It would be worthwhile to investigate the notion of reducibility for more general structures such as semilattices and/or posets. In Section 2, we characterize the elements that are deletable in upper semimodular posets. We will show by a counterexample that the class of upper semimodular posets is not reducible. Venkatnarasimhan [7] investigated pseudocomplemented posets. It was then natural to find out whether the class of pseudocomplemented posets also turns out to be reducible or not. In Section 3, we show by a counterexample that this is not so. For a subset A of a poset P , the lower cone Al of the set A is the set given by