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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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