SOME FUNDAMENTAL PROPERTIES OF HEAPS

Heap is defined to be a non-empty set H with ternary operation [−, −, −]: H × H × H → H satisfying associativity, that is ([[a, b, c], d, e] = [a, b, [c, d, e]]) for every a, b, c, d, e ∈ H and satisfying Mal’cev identity, that is [a, b, b] = b = [b, b, a] for all a, b ∈ H. There is a connection between heaps and groups. From a given heap, we can construct some groups and vice versa. The binary operation of groups can be built by choosing any fixed element e of heap H and is defined by x ⋅_e y=[x,e,y] for any x, y ∈ H. Otherwise, for given a binary operation of group G, we can make a ternary operation defined by [x, y, z] = xy⁻¹z for every x, y, z ∈ G. On heaps, there are some notions which are inspired by groups, such as sub-heaps, normal subheaps, quotient heaps, and heap morphisms. On this study, we will associate sub-heaps and corresponding subgroups and discuss some properties of heap morphisms.