Gaggle theory: An abstraction of Galois connections and residuation, with applications to negation, implication, and various logical operators.

*(English)*Zbl 0814.03044
Eijck, Jan van (ed.), Logics in AI. European workshop JELIA ’90, Amsterdam, Netherlands, September 10-14, 1990. Proceedings. Berlin etc.: Springer-Verlag. Lect. Notes Comput. Sci. 478, 31-51 (1991).

The paper introduces some algebraic structures for studying residuation and Galois connections in an abstract way. These studies are motivated by researches on various kinds of logics: modal logics, relevance logics, orthologics, dynamic logics, Lambek calculus, etc.

Namely, a distributoid is a distributive lattice with additional finitary operations. These operations should be distributive in each of their places over \(\wedge\) or \(\vee\) (possibly with changing \(\wedge\) or \(\vee\) to its dual). Two operations are called relatives if they satisfy an “abstract residuation law”. A gaggle is a distributoid in which all the additional operations are relatives to some fixed one (the “head” of the gaggle).

Representation theorems for distributoids and gaggles are proved; the result for distributoids generalizes the well-known theorem of Jónsson- Tarski for Boolean algebras with operators. It is also shown that distributoids and gaggles are equationally definable.

For the entire collection see [Zbl 0768.00012].

Namely, a distributoid is a distributive lattice with additional finitary operations. These operations should be distributive in each of their places over \(\wedge\) or \(\vee\) (possibly with changing \(\wedge\) or \(\vee\) to its dual). Two operations are called relatives if they satisfy an “abstract residuation law”. A gaggle is a distributoid in which all the additional operations are relatives to some fixed one (the “head” of the gaggle).

Representation theorems for distributoids and gaggles are proved; the result for distributoids generalizes the well-known theorem of Jónsson- Tarski for Boolean algebras with operators. It is also shown that distributoids and gaggles are equationally definable.

For the entire collection see [Zbl 0768.00012].

Reviewer: V.Shekhtman (Moskva)

##### MSC:

03G25 | Other algebras related to logic |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

06D05 | Structure and representation theory of distributive lattices |

03B45 | Modal logic (including the logic of norms) |

06E25 | Boolean algebras with additional operations (diagonalizable algebras, etc.) |