A monoid structure on the set of all binary operations over a fixed set

In recent years, the word magma has been used to designate a pair of the form (S, ∗) where ∗ is a binary operation on the set S. Inspired by that terminology, we use the notation and terminology M(S) (the magma of S) to denote the set of all binary operations on the set S (i.e. the set of all magmas with underlying set S.) Our study concerns a monoid structure (M(S) , ◃) satisfying that each outset, out(∗)={∘∈M(S)|∗distributesover∘}, is a submonoid. This endowment gives us a possibility to compare the properties of an operation ∗ ∈ M(S) and those of the monoid structure of (out(∗) , ◃). We determine that isomorphic operations yield isomorphic outsets and explore possible converses for that result. Several properties of (M(S) , ◃) are considered, including a complete characterization of its group of units and of a subgroup of its group of automorphisms, induced by permutations, which is a retraction. In addition, we consider various submonoids and ideals; among other results, we obtain a generic decomposition, called the kernel-cokernel decomposition, of arbitrary magmas and ideals. We also characterize those cases when a cokernel-kernel decomposition is also possible as we introduce anticommutative and pseudo-anticommutative operations.