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A natural generalization of the concepts of Banach space and Banach algebra is obtained by omitting the condition that the sets under addition form a group, substituting the condition that addition yield a semigroup. Of course, the scalars involved in scalar multiplication must be restricted, otherwise nothing has actually been changed. With the absence of subtraction, the norm becomes useless, and must be replaced by a metric satisfying special conditions. F. F. Bonsall, S. Bourne, and E. J. Barbeau (see the bibliography), among others, have published results concerning this generalization. Bonsall and Barbeau take a semialgebra to be a subset of a Banach algebra which is closed under addition, multiplication, and scalar multiplication by nonnegative reals. Recall here the extensive work on convex cones in locally convex topological vector spaces (Schaefer's book is a good example)-see example (5) of this work. Bourne starts with a semiring, then concentrates on unital left I-semimodules with various restrictions and additional operations (E the nonnegative reals, invariant metric, etc.), among which is semialgebra as defined in this paper. The question arises here whether or not these generalizations are more a difference in point of view than a study of something different; that is, can one always consider a semialgebra, semilinear space, etc., as just a convex cone, semialgebra, etc. in a linear space or algebra. Algebraically, an embedding can be achieved-e.g., mimic the extension of the nonnegative integers to the integers. The real question is whether the metric or topology can be extended. Bourne [8] has something to say about extending norms; Keimel [14] considers the locally compact topological case. It might be said that we have carried some of Bourne's point of view over into a study of the kind of spaces Bonsall and Barbeau have in mind. However, this is not all we have done. We have introduced new concepts (boundary, interior, and linear independence) and studied them in enough detail to demonstrate that they are a significant addition to the literature on semialgebras. The entire field has many gaping holes which need to be filled before outstanding problems can be solved.

Boundaries of semilinear spaces and semialgebras