C(X) as a real ℓ-group

C(X)-objects in the Category of Semi-affine Lattices

“…Its proof is the same as that in [9] (Proposition 2.7)-unlike the proof in [8] no ultrafilters are used.…”

“…Analyzing procedures in [9] we found out that one needs partial binary operation only, namely f + r for f ∈ L and r ∈ R (in fact even less, for r from a subset of R-for simplicity we shall use the whole set R). The group operation induces a multiplication r f for r ∈ Z and f ∈ L. Having a partial binary operation only, we must assume a kind of multiplication added to requested properties of L.…”

“…In the procedures of [9] the multiplication nx was needed for n ∈ Z and f ∈ L. That was possible because of an Abelian group structure on L. Using an additive action, such a multiplication is no more available and we must require it separately. It is not necessary to use n ∈ Z, it suffices to assume n ∈ R for some R ⊂ R having both ±∞ as accumulation points.…”

“…It seems that the formulation of functionally separated sets by means of Urysohn's method is simpler than the other formulations. It was used also in [11] and in our previous paper [9]. The next procedure uses a pure lattice formulation.…”

“…In [3], C(X) is characterized as an f -ring, and in [11] or [13] as a -algebra. In our previous paper [9] we added group structures to lattices and now we add much less, namely the so called semi-affine structure.…”