Abstract

Right invariant right holoids H (r.i.r. holoids) are totally (and positively) ordered semigroups such that a > b holds if and only if a = bc for some c ¦ e e the identity of H. These holoids occur as semigroups of the prinicipal right ideals of right invariant right chain rings. We investigate in which way r.i.r. holoids of finite rank are built up from r.i.r. holoids of rank one which are known to be subsemigroups of the non-negative real numbers under addition. This is best described by conditions on f(C,a) where C is a prime segment which is shifted over elements a ϵ H. The functional properties of f(C,a) are studied, especially in the finite-rank-case. These results are then applied to the extension problem. Here, conditions are given under which the extension splits, however even under these assumptions aft additional problem occurs. An element s is called a denominator for b if a solution x exists in H with xs — b. It is crucial to know the denominators sets and solution sets. Under certain condition it is possible to embed H into a r.i.r. holoid H′ with larger sets of denominators.

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