\(M\)-valued sets and sheaves over integral commutative \(CL\)-monoids.

*(English)*Zbl 0766.03037
Applications of category theory to fuzzy subsets, Mat. 11th Int. Semin. Fuzzy Set Theory, Linz/Austria 1989, Theory Decis. Libr., Ser. B 14, 34-72 (1992).

[For the entire collection see Zbl 0741.00078.]

The utility of sheaf theory in providing models for intuitionistic logic is well-established [see S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory (1992)]. If \(L\) is a frame, one can also study the notion of \(L\)-valued sets and their morphisms, and the resulting category is equivalent to the topos, sh\((L)\), of sheaves on \(L\) [see M. Fourman and D. Scott, Lect. Notes Math. 753, 302-401 (1979; Zbl 0415.03053)].

The article under review attempts to transport these ideas to the realm of Lukasiewicz logic and fuzzy sets. The first step involves finding an appropriate notion of partially ordered model to play the role of the frame \(L\). This is the notion of \(CL\)-monoid. This is a commutative, unital quantale \(M\) (with operation \(*\)), where the top element of \(M\) is also the identity for \(*\). There is the further requirement that if \(\alpha\leq\beta\) in \(M\), then there exists \(\gamma\in M\) with \(\alpha=\beta*\gamma\). This type of quantale generalizes both frames and the partially ordered models of Lukasiewicz logic. The category \(M\)-SET, of \(M\)-valued sets, is then defined and its categorical properties, such as completeness, cocompleteness, epi-mono factorizations etc., are discussed. The category \(M\)-SET is used as a starting point for defining a category, sh\((M)\), of sheaves over \(M\). Some properties of sh\((M)\) are developed and the author plans to carry out further investigations of this category in the hope that it will play the same kind of fundamental role in fuzzy set theory as localic toposes do in topos theory.

The utility of sheaf theory in providing models for intuitionistic logic is well-established [see S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory (1992)]. If \(L\) is a frame, one can also study the notion of \(L\)-valued sets and their morphisms, and the resulting category is equivalent to the topos, sh\((L)\), of sheaves on \(L\) [see M. Fourman and D. Scott, Lect. Notes Math. 753, 302-401 (1979; Zbl 0415.03053)].

The article under review attempts to transport these ideas to the realm of Lukasiewicz logic and fuzzy sets. The first step involves finding an appropriate notion of partially ordered model to play the role of the frame \(L\). This is the notion of \(CL\)-monoid. This is a commutative, unital quantale \(M\) (with operation \(*\)), where the top element of \(M\) is also the identity for \(*\). There is the further requirement that if \(\alpha\leq\beta\) in \(M\), then there exists \(\gamma\in M\) with \(\alpha=\beta*\gamma\). This type of quantale generalizes both frames and the partially ordered models of Lukasiewicz logic. The category \(M\)-SET, of \(M\)-valued sets, is then defined and its categorical properties, such as completeness, cocompleteness, epi-mono factorizations etc., are discussed. The category \(M\)-SET is used as a starting point for defining a category, sh\((M)\), of sheaves over \(M\). Some properties of sh\((M)\) are developed and the author plans to carry out further investigations of this category in the hope that it will play the same kind of fundamental role in fuzzy set theory as localic toposes do in topos theory.

Reviewer: K.I.Rosenthal (Schenectady)

##### MSC:

03G30 | Categorical logic, topoi |

06F05 | Ordered semigroups and monoids |

03B50 | Many-valued logic |

03B52 | Fuzzy logic; logic of vagueness |

03E72 | Theory of fuzzy sets, etc. |