R. McKenzie, A. Kostinsky and B. J6nsson have proved the remarkable result that the class of finite sublattices of a free lattice and the class of finite projective lattices coincide [6]. I f we restrict ourselves to modular lattices, the above result is no longer true. Examining the map from the free modular lattice on three generators, FM(3), to the free distributive lattice on three generato...

The is a temporarty set of notes based on several older (and some ancient) notes. In due course they will be re-done. Much of this is extracted from [5], but I don’t recommend reading that. Let Λ be an arbitrary lattice. We always assume the Λ has a top > and a bottom ⊥. Later we need to assume that Λ is modular, and perhaps even later that it is an idiom, but not just yet. Let Cong(Λ) be the f...

The concept of an MV-algebra was introduced by Chang [4] as an algebraic basis for many-valued logic. It turned out that MV-algebras are a subclass of a more general class of effect algebras [7, 6]. Namely, MV-algebras are in one-to-one correspondence with lattice ordered effect algebras satisfying the Riesz decomposition property [2], the latter are called MV-effect algebras. In the study of c...

An element s of an (abstract) algebra A is a single element of A if asb = 0 and a, b ∈ A imply that as = 0 or sb = 0. Let X be a real or complex reflexive Banach space, and let B be a finite atomic Boolean subspace lattice on X, with the property that the vector sum K +L is closed, for every K,L ∈ B. For any subspace lattice D ⊆ B the single elements of Alg D are characterised in terms of a coo...

The goal of this paper is to prove that several variants of deciding whether a poset can be (weakly) embedded into the Boolean lattice, or to a few consecutive levels of it, areNP-complete.

We present classes of algebras which may be viewed as weak relation algebras, where a Boolean part is replaced by a not necessarily distributive lattice. For each of the classes considered in the paper we prove a relational representation theorem.

The correspondence between unmixed bipartite graphs and sublattices of the Boolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.

Let L be a complete orthomodular lattice. There is a one to one correspondence between complete boolean subalgebras of L contained in the center of L and endomorphisms j of L satisfying the Borceux-Van den Bossche conditions.