Green's-like relations on algebras and varieties (Q938493)

On any semigroup \(S\), the five equivalence relations \(\mathcal{L}, \mathcal{R},\mathcal{J}, \mathcal{H}\) and \(\mathcal{D}\) known as Green's relations provide information about the structure of the semigroup. Recall that for any elements \(a,b\) of \(S\), [\(a\mathcal{L}b\), \(a\mathcal{R}b\)] \(a\mathcal{J}b\) iff \(a\) and \(b\) generate the same [left, right] ideal of \(S\). Moreover \(\mathcal{H} = \mathcal{L}\cap \mathcal{R}\) and \(\mathcal{D} =\mathcal{L}\circ \mathcal{R}\), where \(\circ\) is a composition of binary relations \(\mathcal{L}\) and \(\mathcal{R}\). The authors consider how one might extend the definitions of the five Green's relations to algebras of any arbitrary type \(\tau\). They propose some definitions for \(\mathcal{L}\) and \(\mathcal{R}\), and show what properties are needed to make sure that the relations are equivalence relations. Some sort of (super-) associativity is needed for such definitions to work, and the authors consider algebras which are clones of terms of type \(\tau\), where the clone axioms, including superassociativity, hold. This allows them to define for any variety \(V\) of type \(\tau\) two Green's-like relations \(\mathcal{L}_V\) and \(\mathcal{R}_V\) on the term clone of type \(\tau\). The authors prove a number of properties of these two relations, and describe their behaviour when \(V\) is a variety of semigroups.