This paper continuous the approach of developing an order-theoretic structure theory of one-dimensional continuum structures as elaborated in [Wi07] (see also [Wi83],[Wi03]). The aim is to extend the order-theoretic structure theory by a meaningful algebraization; for this, we concentrate on the real linear continuum structure with its derived concept lattice which gives rise to the so-called "real half-numbers". The algebraization approaches an ordered algebraic structure on the set of all real half-numbers to make the continuum structure of the reals more transparent and tractable.
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