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 ...

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...

In this paper we study the lattice of restricted subalgebras a Lie algebra. particular, consider those algebras in which is dually atomistic, lower or upper semimodular, every subalgebra quasi-ideal. The fact that there are one-dimensional not results some these conditions being weaker than for corresponding non-restricted case.

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we characterize the antichain cutsets in semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We ...

A locally modular (resp. locally distributive) lattice is a lattice with a congruence relation and each of whose equivalence class has sufficiently many elements and is a modular (resp. distributive) sublattice. Both the lattice of all closed subspaces of a locally convex space and the lattice of projections of a locally finite von Neumann algebra are locally modular. The lattice of all /^-topo...

Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in an ordered set. They are defined abstractly as closed sets of a closure system on a set V, satisfying certain axioms. Decompositions are partitions of V whose blocks are intervals, and they form an algebraic semimodular lattice. Latticetheoretical properties of decompositions are explored, ...

Whereas the Dedekind-MacNeille completion D(P ) of a poset P is the minimal lattice L such that every element of L is a join of elements of P, the minimal strict completion D(P )∗ is the minimal lattice L such that the poset of join-irreducible elements of L is isomorphic to P. (These two completions are the same if every element of P is joinirreducible). In this paper we study lattices which a...

In [8], P. Jones characterized the regular varieties of semigroups which are congruence semimodular and he partially solved the problem of characterizing the non-regular, congruence semimodular (CSM) varieties of semigroups. For regular varieties his characterization was an equational one, so necessarily a part of his argument was semigroup-theoretic. But some of his argument involved only cong...

For an arbitrary algebra A a new labelling, called the signed labelling, of the Hasse diagram of Con A is described. Under the signed labelling, each edge of the Hasse diagram of Con A receives a label from the set {+,−}. The signed labelling depends completely on a subset of the unary polynomials of A and its inspiration comes from semigroup theory. For finite algebras, the signed labelling co...

The Tamari lattice of order n can be defined by the set Dn of Dyck words endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we study this rotation on the restricted set of Motzkin words. An upper semimodular join semilattice is obtained and a shortest path metric can be defined. We compute the corresponding distance between two Motzkin word...