Test sets for tautologies in modular quantum logic

As defined by Dunn, Moss, and Wang, an universal test set in an ortholattice $L$ is a subset $T$ such that each term takes value $1$, only, if it does so under all substitutions from $T$. Generalizing their result for ortholattices of subspaces of finite dimensional Hilbert spaces, we show that no infinite modular ortholattice of finite dimension admits a finite universal test set. On the other hand, answering a question of the same authors, we provide a countable universal test set for the ortholattice of projections of any type II$_1$ von Neumann algebra factor as well as for von Neumanns algebraic construction of a continuous geometry. These universal test sets consist of elements having rational normalized dimension with denominator a power of $2$.

A Note on the Third Life of Quantum Logic

The purpose of this note is to discuss some of the questions raised by Dunn, J. Michael; Moss, Lawrence S.; Wang, Zhenghan in Editors introduction: the third life of quantum logic: quantum logic inspired by quantum computing.

Representations of *-regular rings and their ortholattices of projections

We show that a subdirectly irreducible *-regular ring admits a representation within some inner product space provided so does its ortholattice of projections.

Unit-regularity and representability for semiartinian *-regular rings

We show that any semiartinian *-regular ring R is unit-regular; if, in addition, R is subdirectly irreducible then it admits a representation within some inner product space.

(Co)isotropic triples and poset representations

We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we only assume to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is $2 + 2 + 2$ consisting of three independent ordered pairs, with the involution exchanging the members of each pair. A key feature of the classification is that any indecomposable (co)isotropic triple is either split or non-split. The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the split case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a symplectification. In the course of the paper we develop the framework of symplectic poset representations, which can be applied to a range of problems of symplectic linear algebra. The classification of linear Hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples. The framework lends itself equally well to studying poset representations on spaces carrying a non-degenerate symmetric bilinear form; we mainly keep our focus, however, on the symplectic side.

Varieties of *-regular rings

Given a subdirectly irreducible *-regular ring R, we show that R is a homomorphic image of a regular *-subring of an ultraproduct of the (simple) eRe, e in the minimal ideal of R; moreover, R (with unit) is directly finite if all eRe are unit-regular. Finally, unit-regularity is shown for every member of the variety generated by artinian *-regular rings (endowed with unit and pseudo-inversion).

Direct finiteness of representable regular rings with involution

We show that a von Neumann regular ring with involution is directly finite provided that it admits a representation as a ring of endomorphisms (the involution given by taking adjoints) of a vector space endowed with a non-degenerate orthosymmetric sesquilinear form.

On the Finiteness Problem for classes of modular lattices

The Finiteness Problem is shown to be unsolvable for any sufficiently large class of modular lattices.

On the complexity of equational decision problems for finite height(ortho)complemented modular lattices

We study the computational complexity of satisfiability problems for classes of simple finite height (ortho)complemented modular lattices $L$. For single finite $L$, these problems are shown tobe $mc{NP}$-complete; for $L$ of height at least $3$, equivalent to a feasibility problem for the division ring associated with $L$. Moreover, it is shown that the equational theory of the class of subspace ortholattices as well as endomorphism *-rings (with pseudo-inversion) of finite dimensional Hilbert spaces is complete for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This results extends to the category of finite dimensional Hilbert spaces, enriched by pseudo-inversion.

Definable relations in finite dimensional subspace lattices with involution. Part II: Quantifier free and homogeneous descriptions

For finite dimensional hermitean inner product spaces $V$, over $*$-fields $F$, and in the presence of orthogonal bases providing form elements in the prime subfield of $F$, we show that quantifier free definable relations in the subspace lattice $L(V)$ with involution by taking orthogonals, admit quantifier free descriptions within $F$, also in terms of Grassmann-Plucker coordinates.In the latter setting, homogeneous descriptions are obtained if one allows quantification type $Sigma_1$. In absence of involution, these results remain valid.