513 research outputs found
Matrices generated by semilattices
AbstractWe give a characterization of 0–1 matrices M which are generated by semilattices in the way that Mij = 0 if and only if xi Λ xj = 0̂ where xi,xj, 0̂ are elements in a semilattice
Approximately multiplicative maps from weighted semilattice algebras
We investigate which weighted convolution algebras , where
is a semilattice, are AMNM in the sense of Johnson (JLMS, 1986). We give an
explicit example where this is not the case. We show that the unweighted
examples are all AMNM, as are all where has either
finite width or finite height. Some of these finite-width examples are
isomorphic to function algebras studied by Feinstein (IJMMS, 1999).
We also investigate when is an AMNM pair in
the sense of Johnson (JLMS, 1988), where denotes the algebra of
2-by-2 complex matrices. In particular, we obtain the following two contrasting
results: (i) for many non-trivial weights on the totally ordered semilattice
, the pair is not
AMNM; (ii) for any semilattice , the pair is AMNM.
The latter result requires a detailed analysis of approximately commuting,
approximately idempotent matrices.Comment: AMS-LaTeX. v3: 31 pages, additional minor corrections to v2. Final
version, to appear in J. Austral. Math. Soc. v4: small correction of
mis-statement at start of Section 4 (this should also be fixed in the journal
version
Stone-type representations and dualities for varieties of bisemilattices
In this article we will focus our attention on the variety of distributive
bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and
involutive bisemilattices. After extending Balbes' representation theorem to
bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn
duality and introduce the categories of 2spaces and 2spaces. The
categories of 2spaces and 2spaces will play with respect to the
categories of distributive bisemilattices and De Morgan bisemilattices,
respectively, a role analogous to the category of Stone spaces with respect to
the category of Boolean algebras. Actually, the aim of this work is to show
that these categories are, in fact, dually equivalent
- …