Linear Functionals on Idempotent Spaces : An Algebraic Approach

In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis in the sense of [1–3]. Elements of such an approach were used, for example, in [1, 4]. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis , including the theorem on the general form of a linear functional and the Hahn–Banach and Riesz–Fischer theorems. 1. Recall that an additive semigroup S with commutative addition ⊕ is called an idempotent semigroup (IS) if the relation x ⊕ x = x is fulfilled for all elements x ∈ S. If S contains a neutral element, this element is denoted by the symbol 0. Any IS is a partially ordered set with respect to the following standard order: x y if and only if x ⊕ y = y. It is obvious that this order is well defined and x ⊕ y = sup{x, y}. Thus, any IS is an upper semilattice; moreover, the concepts of IS and upper semilattice coincide [5]. An idempotent semigroup S is called a-complete (or algebraically complete) if it is complete as an ordered set, i.e., if any subset X in S has the least upper bound sup(X) denoted by ⊕X and the greatest lower bound inf(X) denoted by ∧X. This semigroup is called b-complete (or boundedly complete), if any bounded above subset X of this semigroup (including the empty subset) has the least upper bound ⊕X (in this case, any nonempty subset Y in S has the greatest lower bound ∧Y and S in a lattice). Note that any a-complete or b-complete IS has the zero element 0 that coincides with ⊕Ø, where Ø is the empty set. Certainly, a-completeness implies the b