Idempotent elements in binary and s-ary magma monoids

The idempotents elements of the magma monoid (M(S), ) are characterized. The characterization is used to determine, when S has n elements, the number of idempotents in M(S) (also called M(n) ). A combinatorial argument is inferred and then this generalized principle is used in a larger setting, that is, analogs to the magma monoid are introduced for s -ary operations, and the generalized combinatorial principle is used to determine the number of idempotent elements in those new monomials.In addition, the kernel-cokernel decomposition of idempotents in the binary magma monoid is analyzed and its properties and relations with anticommutative and pseudo-anticommutative operations are established.