The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stabilit...

In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, di...

A complete lattice L is constructively completely distributive CCD L if the sup map de ned on down closed subobjects has a left adjoint It was known that in boolean toposes CCD L is equivalent to CCD L We show here that the latter property for all L su ciently for characterizes boolean toposes Research partially supported by grants from NSERC Canada Diagrams typeset using catmac Introduction Th...

A complete lattice L is constructively completely distributive CCD when the sup arrow from down closed subobjects of L to L has a left adjoint The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of CCD lattices and sup preserving arrows There is a restriction to order ideals and totally algebraic lattices Both biequivalences have left exact ver sions As appli...

We devise a condition strictly between the existence of an $n$-ary and $n{+}1$-ary near-unanimity term. evaluate exactly distributivity modularity levels implied by such condition.