For every n, we evaluate the smallest k such that the congruence inclusion $\alpha (\beta \circ _n \gamma ) \subseteq \alpha \beta \circ _{k} \alpha \gamma $ holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations.

The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

We show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. As a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. This consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

We prove that if a finite algebra $\mathbf{A}$ generates a congruence distributive variety, then the subalgebras of the powers of $\mathbf{A}$ satisfy a certain kind of intersection property that fails for finite idempotent algebras that locally exhibit affine or unary behaviour. We demonstrate a connection between this property and the constraint satisfaction problem.

We give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even underthe additional assumption that the variety is residually very finite.

It is shown that a variety ν has distributive congruence lattices if and only if the intersection of two principal congruence relations is definable by equations involving terms with parameters. The nature of the terms involved then provides a useful classification of congruence distributive varieties. In particular, the classification puts into proper perspective two stronger properties. A variety is said to have the Principal Intersection Property if the intersection of any two principal congruence relations is principal, or the Compact Intersection Property if the intersection of two compact congruence relations is compact. For non-congruence-distributive varieties, it is shown that some useful constuctions are nevertheless possible.

It is shown that if V is a congruence distributive variety whose members have one element subalgebras, then the class of absolute retracts of V is closed under direct products. If V is residually small, then a characterisation of the amalgamation class of V is given.