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Orthogonal decompositions of MV-spaces. (English) Zbl 0880.06004
This paper continues a previous paper of the authors and the reviewer [Math. Log. Q. 40, No. 3, 331-346 (1994; Zbl 0815.06010)]. Let \(A\) be an MV-algebra and \(S\) be a maximal disjoint subset of \(A\). If \(\{x\in A:x\leq a\}\) is linearly ordered for all \(a\in S\), then \(S\) is called a basis of \(A\). Let \(\text{Spec} A\) be the set of the prime ideals of \(A\) with the hull-kernel topology and \(B(A)\) be the Boolean subalgebra of \(A\). A decomposition \(\text{Spec} A =(\bigcup_{i\in I} T_i) \cup X\) is called orthogonal if \(T_i= V(a_i)= \{P\in\text{Spec} A: a_i\notin P\}\) for some \(a_i\in A\), i.e. \(T_i\) is compact open, \(X\cap T_i= \emptyset \) for any \(i\in I\) and \(S=\{a_i:i\in I\}\) is a maximal disjoint subset of \(A\). Such decomposition is unrefinable, i.e. no \(T_i= \Theta\cup Y\) with \(\Theta\) open, \(\text{int} Y\neq \emptyset\) and \( \Theta\cap Y=\emptyset\), iff \(S\) is a basis. Many results are given for semisimple MV-algebras, deeply studied by the first author [Can. J. Math. 38, 1356-1379 (1986; Zbl 0625.03009)]. For instance, \(A\) semisimple with a basis implies \(B(A)\) atomic.
Annihilator ideals and basis are related via the “Finite Basis Theorem”, as happens similarly in lattice-ordered groups.
Reviewer: A.Di Nola (Napoli)

MSC:
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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