Simplicial structures in MV-algebras and logic.

*(English)*Zbl 1119.03067An 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.

– 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.

Reviewer: Florentina Chirteş (Craiova)

##### Keywords:

MV-algebra; simplicial structure; formal consequence; semantical consequence; simplicial completeness; homological completeness; simplicial complexes; prime spectrum
PDF
BibTeX
XML
Cite

\textit{L. P. Belluce} and \textit{A. Di Nola}, J. Symb. Log. 72, No. 2, 584--600 (2007; Zbl 1119.03067)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Transactions of the American Mathematical Society 88 pp 467–490– (1958) |

[2] | Tatra Mountains Mathematical Publications 27 pp 7–22– (2003) |

[3] | A weak completeness theorem for infinite-valued predicate logic 28 pp 43–50– (1963) |

[4] | Algebra Universalis 29 pp 1–9– (1992) |

[5] | Transactions of the American Mathematical Society 93 pp 74–80– (1959) |

[6] | Algebraic topology (1966) · Zbl 0956.55004 |

[7] | Homology theory (1960) |

[8] | Annals of Mathematics 56 pp 84–95– (1952) |

[9] | Foundations of many-valued reasoning (2000) · Zbl 0937.06009 |

[10] | Transactions of the American Mathematical Society 87 pp 1–53– (1958) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.