MV-algebra pasting
A difference poset (D-poset) is a partially ordered set \(P\) with a least element 0 and a greatest element 1 equipped with a partial binary difference operation \(\ominus\) such that, for all \(a,b,c\in P\), the following conditions are satisfied: (i) \(a \ominus 0=a\) (ii) \(a\leq b\leq c\) implies \(c \ominus b\leq c \ominus a\) and \((c\ominus a)\ominus (c\ominus b)=b\ominus a\). A Boolean D-poset is a D-poset \(P\) with a binary operation a satisfying the following conditions for all \(a,b,c\in P\): (i) \(a-0=a \) (ii) \(a-(a-b)=b-(b-a)\) (iii) \(a\leq b\) implies \(c-b\leq c-a\) (iv) \((a-b)-c=(a-c)-b\). Boolean D-posets are algebraically equivalent to MV-algebras. It was proved by \textit{Z. Riečanová} [Int. J. Theor. Phys. 39, 231--237 (2000; Zbl 0968.81003)] that every D-lattice is a union of MV-algebras. In the paper under review the construction of a D-poset from a collection of MV algebras is given. This pasting technique is parallel to the construction of quantum logics by pasting a system of Boolean algebras.
scientific article; zbMATH DE number 2036922