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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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