Linear inequality implies linearity and order. When combined, the two produce an ordered vector space. Each linear inequality in the simplest environment of the sort is some half-space. Simultaneity implies many instances and so leads to the intersections of half-spaces. These yield polyhedra as well as arbitrary convex sets, identifying the theory of linear inequalities with convexity.
Convexity reigns in the federation of geometry, optimization, and functional analysis. Convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability.
This talk addresses the origin and the state of the art of the relevant areas with a particular emphasis on the Farkas Lemma. Our aim is to demonstrate how Boolean valued analysis may be applied to simultaneous linear inequalities with operators.