nLab
maximal partial function

A maximal partial function is a partial function, say from XX to YY, which is maximal in the poset of partial functions from XX to YY. Explicitly, f:XYf\colon X \nrightarrow Y is maximal if, given any g:XYg\colon X \nrightarrow Y, if domfdomg\dom f \subseteq \dom g and f(a)=g(a)f(a) = g(a) whenever adomfa \in \dom f, then domf=domg\dom f = \dom g (and so f=gf = g).

All total functions are maximal, as are all partial functions whose codomain YY is the empty set. Assuming the law of excluded middle, these are the only examples. Outside of the context of constructive mathematics, therefore, (and usually in constructive math too) we are concerned with maximal partial functions subject to some restrictions, such as the maximal partial sections of some given function from YY to XX, the maximal continuous partial functions given some topological structures on XX and YY, the maximal local sections of some given continuous function from YY to XX given some topological structures on XX and YY, etc. Then we are not working in the poset of all partial functions from XX to YY but in a given subposet?.

More abstractly, we may also consider the maximal partial morphisms in a general category. This includes, for example, the maximal continuous partial functions between two topological spaces.

Created on July 12, 2015 at 20:21:45. See the history of this page for a list of all contributions to it.