1,301 research outputs found
Weak representations of relation algebras and relational bases
Published versio
There is no finite-variable equational axiomatization of representable relation algebras over weakly representable relation algebras
We prove that any equational basis that defines RRA over wRRA must contain
infinitely many variables. The proof uses a construction of arbitrarily large
finite weakly representable but not representable relation algebras whose
"small" subalgebras are representable.Comment: To appear in Review of Symbolic Logi
H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics
A certain class of Frobenius algebras has been used to characterize
orthonormal bases and observables on finite-dimensional Hilbert spaces. The
presence of units in these algebras means that they can only be realized
finite-dimensionally. We seek a suitable generalization, which will allow
arbitrary bases and observables to be described within categorical
axiomatizations of quantum mechanics. We develop a definition of H*-algebra
that can be interpreted in any symmetric monoidal dagger category, reduces to
the classical notion from functional analysis in the category of (possibly
infinite-dimensional) Hilbert spaces, and hence provides a categorical way to
speak about orthonormal bases and quantum observables in arbitrary dimension.
Moreover, these algebras reduce to the usual notion of Frobenius algebra in
compact categories. We then investigate the relations between nonunital
Frobenius algebras and H*-algebras. We give a number of equivalent conditions
to characterize when they coincide in the category of Hilbert spaces. We also
show that they always coincide in categories of generalized relations and
positive matrices.Comment: 29 pages. Final versio
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