Crown multiplications and a higher order Hopf construction

We describe a ternary function (multiplication) from the product of three 12-point crowns into a 4-point crown in the category of partially ordered sets. The ternary map illustrates a version, in the context of posets, of associative multiplication. A higher order analogue of Hopf's construction applied to σ yields a poset model of a certain homotopy class, which we identify, in π5(S3).