Positivity 4: 233–243, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Almost f -algebras: Structure and the Dedekind
and A. VAN ROOIJ
Department of Mathematics, University of Mississippi, Mississippi 38677, USA
Mathematical Institute, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The
Dedicated to the memory of C.B. (“Pay”) Huijsmans.
Abstract. We prove a representation theorem for almost f -algebras, from which we infer the
existence of almost f -algebra multiplications on the Dedekind completions of almost f -algebras.
Over the last decade the importance of f -algebras in the theory of vector lattices
has steadily grown. Various other lattice-ordered algebraic structures have only
recently been getting more attention. Though some of these structures may seem
a little far-fetched, we believe that almost f -algebras, which received their name
from Birkhoff in , offer an interesting side road, in part for clarifying the role
of certain technical aspects in the theory of f -algebras. To remind the reader, f -
algebras and almost f -algebras, are simultaneously real associate algebras as well
as vector lattices. In case of f -algebras the connection between multiplication and
order is given by
if a, b 0, then ab 0, (1)
if a ∧ b = 0andc 0, then ca ∧ b = ac∧ b = 0, (2)
while in almost f-algebras instead of (2) we have
if a ∧ b = 0, then ab = 0, (3)
Archimedean (almost) f -algebras are commutative. (See  or  for a proof.)
The stimulus for our investigations was a question posed by C.B. Huijsmans in
 (last paragraph of section 7): Can the multiplication of an almost f -algebra be
extended to its Dedekind completion? The order continuity of the multiplication in
Supported by NATO Grant CRG #940605.