Skeletons on congruence distributive varieties of algebras (Q1342131)

For a variety \(\mathcal M\), let \({\mathcal I}{\mathcal M}\) denote the class of isomorphism types of algebras from \(\mathcal M\). There are two quasi-order relations \(\leq\) and \(\ll\) and a commutative and associative operation \(\times\) on \({\mathcal I}{\mathcal M}\) defined as follows: for \(a,b,c\in{\mathcal I}{\mathcal M}\) being the isomorphism types of the algebras, \(A,B,C\in {\mathcal M}\) respectively, let \(a\leq b\) (\(a\ll b\), \(c= a\times b\)) whenever \(A\) is isomorphic to a subalgebra of \(B\) (\(A\) is a homomorphic image of \(B\), \(C\) is isomorphic to a Cartesian product of \(A\) and \(B\)). The quasi-ordered classes \(\langle{\mathcal I}{\mathcal M},\leq\rangle\) and \(\langle {\mathcal I}{\mathcal M},\ll\rangle\) are called the embedding skeleton and the epimorphism skeleton of the variety \(\mathcal M\) while the monoid \(\langle{\mathcal I}{\mathcal M},\times\rangle\) is called the Cartesian skeleton of \(\mathcal M\). The paper under review briefly surveys results (mostly due to the author) which appeared (with full proofs) in the author's book [Boolean constructions in universal algebras (1993; Zbl 0792.08001)].