On \(\wedge\)-subdistributivity and \(\vee\)-superdistributivity with respect to interval map in Kaucher arithmetic (Q5935473)

The authors consider an arbitrary map \({\mathbf x},{\mathbf y}\to {\mathbf x}\otimes {\mathbf y}\) over Kaucher's extended interval space. They prove that the map satisfies \[ ({\mathbf x}_1\wedge {\mathbf x}_2)\otimes ({\mathbf y}_1\wedge {\mathbf y}_2)\subseteq ({\mathbf x}_1\otimes {\mathbf y}_1)\wedge ({\mathbf x}_2\otimes {\mathbf y}_2) \] (superdistributivity) if and only if \(\otimes\) is inclusion monotone, and that this property is also equivalent to \[ ({\mathbf x}_1\otimes {\mathbf y}_1)\vee ({\mathbf x}_2\otimes {\mathbf y}_2) \subseteq ({\mathbf x}_1\vee {\mathbf x}_2)\otimes ({\mathbf y}_1\vee {\mathbf y}_2) \] (subdistributivity).