Some semilattice decompositions of dimonoids

We show that the system of axioms of a dimonoid is independent and prove that every dimonoid with a commutative operation is a semilattice of archimedean subdimonoids, every dimonoid with a commutative periodic semigroup is a semilattice of unipotent subdimonoids, every dimonoid with a commutative operation is a semilattice of

THE IDEMPOTENTS IN A PERIODIC SEMIGROUP

Let [Formula: see text] be the semigroup variety determined by the identity xm=xm+k. For [Formula: see text] we define operations on the set E(S) of idempotents of S and thus obtain the idempotent algebra of S. For any subvariety [Formula: see text] of [Formula: see text] the idempotent algebras of the members of [Formula: see text] form a variety [Formula: see text] and [Formula: see text] yields a complete homomorphism of the lattice [Formula: see text] of subvarieties of [Formula: see text] onto the lattice [Formula: see text] of subvarieties of [Formula: see text]. The lattice [Formula: see text] contains a ∩-semilattice isomorphic to the ∩-semilattice [Formula: see text] of group varieties of exponent dividing k for every m≥1. In particular, for appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to [Formula: see text] is of the power of the continuum. For any [Formula: see text], ρ→ρ|E(S) yields a complete homomorphism of the congruence lattice of S into the lattice of equivalence relations on E(S).