Related papers: Orthocomplemented weak tensor products
Given two complete atomistic lattices L_1 and L_2, we define a set S(L_1,L_2) of complete atomistic lattices by means of three axioms (natural regarding the description of separated quantum compound systems), or in terms of a universal…
Let H_1 and H_2 be complex Hilbert spaces, L_1=P(H_1) and L_2=P(H_2) the lattices of closed subspaces, and let L be a complete atomistic lattice. We prove under some weak assumptions relating L_i and L, that if L admits an…
A classical tensor product $A \,\otimes\, B$ of complete lattices $A$ and $B$, consisting of all down-sets in $A \times B$ that are join-closed in either coordinate, is isomorphic to the complete lattice $Gal(A,B)$ of Galois maps from $A$…
We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite…
We consider the description of two independent quantum systems by a complete atomistic ortho-lattice (cao-lattice) L. It is known that since the two systems are independent, no Hilbert space description is possible, i.e. $L\ne P(H)$, the…
In this paper, we define and study $S$-Noetherian lattices as a natural generalization of Noetherian rings. We prove that a ring $R$ is $S$-Noetherian if and only if its ideal lattice, $Id(R)$, is $S_L$-Noetherian. Furthermore, we establish…
The notion of a capped tensor product, introduced by G. Gr\"{a}tzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the…
On an arbitrary meet-semilattice S with 0 we define an orthogonality relation and investigate the lattice Cl(S) of all subsets of S closed under this orthogonality. We show that if S is atomic then Cl(S) is a complete atomic Boolean…
Let A and B be lattices with zero. The classical tensor product, $A\otimes B$, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: $A \otimes…
It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic…
For L a finite lattice, let C(L) denote the set of pairs g = (g_0,g_1) such that g_0 is a lower cover of g_1 and order it as follows: g <= d iff g_0 <= d_0, g_1 <= d_1, but not g_1 <= d_0. Let C(L,g) denote the connected component of g in…
In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism $Conc(A \otimes B)\cong Conc A \otimes Conc B$, holds, provided that the tensor product satisfies a very natural condition (of being…
The category of $C^*$-algebras is blessed with many different tensor products. In contrast, virtually the only tensor product ever used in the category of von Neumann algebras is the normal spatial tensor product. We propose a definition of…
Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice $D$ is said to be fully principal congruence representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible…
In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the…
Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and…
Let $A$, $B$, and $S$ be (v,0)-semilattices and let $f: A\to B$ be a (v,0)-embedding. Then the canonical map, $f \otimes \id\_S$, of the tensor product $A \otimes S$ into the tensor product $B \otimes S$ is not necessarily an embedding. The…
Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$…
We investigate the lattice L(V) of subspaces of an m-dimensional vector space V over a finite field GF(q) with q being the n-th power of a prime p. It is well-known that this lattice is modular and that orthogonality is an antitone…
We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects…