meetcurve

Meetcurve is a concept in order theory and lattice theory that describes a parametric curve generated by the meet operation in a partially ordered set.

Formally, let L be a bounded lattice with meet operation ∧. For two functions f, g: [0,1] → L that are continuous with respect to a chosen topology on L, the meet-curve is the map c(t) = f(t) ∧ g(t). The image c([0,1]) traces how the greatest lower bound of f and g evolves with the parameter.

In a chain lattice such as the real numbers with the usual order, if f(t) = a0 + t and g(t) = b0 − t, then c(t) = min(a0 + t, b0 − t), producing a piecewise linear curve with a breakpoint at the parameter where the two lines cross.

In a product lattice, such as R^n with coordinate-wise order, the meet is the coordinate-wise minimum: c(t) = f(t) ∧ g(t) = (min(f1(t), g1(t)), ..., min(fn(t), gn(t))). The resulting curve is obtained by taking the minimum coordinate by coordinate.

Applications and significance: meetcurves appear in visualizations of interactions between criteria in multi-criteria decision analysis, in the study of lower envelopes in optimization, and in certain interpretations of domain theory where meets model information flow.

The term meetcurve is not universally standardized and may be encountered under related terms such as meet-path or lower envelope in different sources. See also lattice, meet, join, poset, lower envelope, domain theory.