Poset Loops

Given a ring and a locally finite poset, an incidence loop or poset loop is obtained from a new and natural extended convolution product on the set of functions mapping intervals of the poset to elements of the ring. The paper investigates the interplay between properties of the ring, the poset, and the loop. The annihilation structure of the ring and extremal elements of the poset determine commutative and associative properties for loop elements. Nilpotence of the ring and height restrictions on the poset force the loop to become associative, or even commutative. Constraints on the appearance of nilpotent groups of class 2 as poset loops are given. The main result shows that the incidence loop of a poset of finite height is nilpotent, of nilpotence class bounded in terms of the height of the poset.