# Three new conjectures

See here (especially this draft article) for definition of cross-composition product and quasi-cartesian functions.

Conjecture 1 Cross-composition product (for small indexed families of relations) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation ${\mathfrak{S}_0}$ of binary relations to the quasi-cartesian situation ${\mathfrak{S}_1}$ of pointfree funcoids over posets with least elements.

Conjecture 2 Cross-composition product (for small indexed families of pointfree funcoid between posets with least elements) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation ${\mathfrak{S}_0}$ of pointfree funcoids over posets with least elements to the quasi-cartesian situation ${\mathfrak{S}_1}$ of pointfree funcoids over posets with least elements.

Conjecture 3 Cross-composition product (for small indexed families of reloids) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation ${\mathfrak{S}_0}$ of reloids to the quasi-cartesian situation ${\mathfrak{S}_1}$ of pointfree funcoids over posets with least elements.

Remark 1 The three above conjectures are unsolved even for product of two multipliers.