Extensions of real-valued difference posets
A difference poset (\(D\)-poset) [see \textit{F. Kôpka} and \textit{F. Chovanec}, Math. Slovaca 44, No. 1, 21-34 (1994; Zbl 0789.03048)] is a poset \(L\) with a partial binary operation \(\ominus\) such that \(b\ominus a\) is defined iff \(a\leq b\), and the following axioms hold: (i) \(b\ominus a\leq b\); (ii) \(b\ominus (b\ominus a)= a\); if \(a\leq b\leq c\), then \(c\ominus b\leq c\ominus a\) and \((c\ominus a)\ominus (c\ominus b)= b\ominus a\). The unit interval \([0,1]\) with the usual order and the usual difference is a difference poset. \((L,\ominus)\) is a difference poset iff there is a (unique) generator \(g: [0, 1]\to [0, 1]\), \(g\) is a bijection and increasing, such that \(b\ominus a= g^{-1} (g(b)- g(a))\). The same result holds for a difference poset \((M,\ominus)\), where \(M\) is a dense subset of \([0,1]\).
scientific article; zbMATH DE number 729697
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