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Simplicial structures in MV-algebras and logic. (English) Zbl 1119.03067
An MV-algebra is an algebra $$(A,\oplus ,\odot ,^{*},0,1)$$ of type $$(2,2,1,0,0)$$ such that $$(A,\oplus ,0)$$ , $$(A,\odot ,1)$$ are commutative semigroups with identity, $$^{*}$$ is an involution on $$A$$ such that $$(a\oplus b)^{*}=a^{*}\odot b^{*}$$, $$(a\odot b)^{*}=a^{*}\oplus b^{*}$$ and $$a\oplus (a^{*}\odot b)=b\oplus (b^{*}\odot a)$$, $$0^{*}=1$$. An MV-algebra is semisimple if $$\text{Rad}(A)=0$$, where $$\text{Rad}(A)$$ is the intersection of all maximal ideals of $$A$$. The main results presented in this paper can be summarized as follows:
– An MV-algebra is logically complete if and only if its quotient by a non-zero principal ideal is semisimple.
– Every non-semisimple MV-algebra which is logically complete has exactly one non-zero ideal and is linearly ordered.
– Every non-simple linearly ordered MV-algebra is simplicially complete.
– Every semisimple MV-algebra is simplicially complete.
– Every MV-algebra $$A$$ is simplicially complete if and only if every finite subset $$H$$ of $$A$$ is pseudo-consistent iff it is pseudo-satisfiable.
– Every MV-algebra is homologically complete.
The authors also describe the functorial aspects of these constructions, and furthermore show a connection between the simplicial constructions and the prime ideal spaces of MV-algebras, thus implying a relation between the completeness of an algebra and its prime ideals.

##### MSC:
 03G25 Other algebras related to logic 03B50 Many-valued logic 06D35 MV-algebras
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##### References:
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