variant on ring objects in the category of complete lattices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:39:04Z http://mathoverflow.net/feeds/question/109888 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109888/variant-on-ring-objects-in-the-category-of-complete-lattices variant on ring objects in the category of complete lattices Ittay Weiss 2012-10-17T09:12:07Z 2012-10-17T09:12:07Z <p>Let $L$ be a complete lattice and denote its top and bottom elements by $0$ and $\infty$ respectively. Consider two binary operations $+$ and $\times$ defined on $L$ such that $(L^{op},+,0)$ is a commutative unital quantale, $(L,\times)$ is a (not-necessarily unital) quantale, and so that a suitable (i.e., there is some freedom for play) distributivity condition holds.</p> <p>Is there a name for such a structure? </p> <p>The motivation for the question is that I am interested in categorifying Lipschitz mappings. Thus, enriching over the structure above I can use the multiplicative structure to talk about a Lipschitz condition. </p> <p>For clarify, the additive structure $(L^{op},+,0)$ being a commutative unital quantale means that $a+0=a$ and that $a+\bigwedge S=\bigwedge(a+S)$, and so, taking $S=\emptyset$ we obtain $a+\infty =\infty$. In contrast, the multiplicative structure $(L,\times)$ being a quantale means that $a \times \bigvee S = \bigvee (a\times S)$ and so $a\times 0=0$. Hence this structure is a bit like a ring object in the category of complete lattices. </p>