In this paper, we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers N. We describe lattice its ideals for an arbitrary field K and study derivations over any commutative, unital ring R.

Let g be a simple Lie algebra over C. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). Write θ for the highest root in∆. An ideal of b is called ad-nilpotent, if it is contained in [b, b]. Consequently, such a subspace is fully determined by the corresponding set of positive roots, and this set is called an ideal (in ...

One basic activity in combinatorics is to establish combinatorial identities by so-called ‘bijective proofs,’ which consists in constructing explicit bijections between two types of the combinatorial objects under consideration. We show how such bijective proofs can be established in a systematic way from the ‘lattice properties’ of partition ideals, and how the desired bijections are computed ...

For a Banach space X, let B(X) denote the Banach algebra of all continuous linear operators on X. First, we study the lattice of closed ideals in B(Jp), where 1 < p < ∞ and Jp is the pth James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in B(Jp). Applications of this result include the following. (i) The Brown–McCoy radical of B(X), which by ...

We consider a very simple Mealy machine (two non-trivial states over a two-symbol alphabet), and derive some properties of the semigroup it generates. It is an infinite, finitely generated semigroup, and we show that the growth function of its balls behaves asymptotically like nα, for α = 1 + log 2/log 1+ √ 5 2 ; that the semigroup satisfies the identity g = g; and that its lattice of two-sided...

We consider a very simple Mealy machine (three states over a twosymbol alphabet), and derive some properties of the semigroup it generates. In particular, this is an infinite, finitely generated semigroup; we show that the growth function of its balls behaves asymptotically like nα, for α = 1 + log 2/log 1+ √ 5 2 ; that the semigroup satisfies the identity g = g; and that its lattice of two-sid...

In the case of the group SL(3,C), we describe how the BernsteinGel′fand-Gel′fand complex can be used to construct an element of Kasparov’s equivariant K-homology. The resulting construction is a model for the γ-element. The key to the construction is the introduction of a lattice of operator ideals associated to the natural fibrations of the complete flag variety for SL(3,C).

A complemented modular lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jónsson, are first-order. Nevertheless, we prove that coordinatizability of complemented modular lattices is not firstorder, even for countable 2-...

It is well known that given a poset, X, the lattice of order ideals of X, 〈I(X),⊆〉, is a completion of X via the order-embedding φ : X ↪→ I(X) where φ(x) = ↓x. Herein we define a lattice of antichains in X, 〈A(X),4〉, and prove it is isomorphic to 〈I(X),⊆〉. We establish the “join” and “meet” operations of the lattice, and present results for 〈A(X),4〉 analogous to standard results for 〈I(X),⊆〉, i...

We provide a concise axiomatization of a broad class of generalized quantiiers in many-valued logic, so-called distribution quantiiers. Although sound and complete axiomatizations for such quantiiers exist, their size renders them virtually useless for practical purposes. We show that for certain lattice-based quan-tiiers relatively small axiomatizations can be obtained in a schematic way. This...