By an equivalence system is meant a couple $𝒜=(A,\theta )$ where $A$ is a non-void set and $\theta$ is an equivalence on $A$. A mapping $h$ of an equivalence system $𝒜$ into $ℬ$ is called a class preserving mapping if $h\left({\left[a\right]}_{\theta }\right)={\left[h\left(a\right)\right]}_{\theta {}^{\text{'}}}$ for each $a\in A$. We will characterize class preserving mappings by means of permutability of $\theta$ with the equivalence ${\Phi }_h$ induced by $h$.

In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a “well-behaved core”, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this “core” is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term $t\left(x\right)$) in every member of the given variety. Here, we try to give a unified account of this phenomenon....

An M-Set is a unary algebra $\langle X,M\rangle$ whose set $M$ of operations is a monoid of transformations of $X$; $\langle X,M\rangle$ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $\langle X,M\rangle$ if the congruence lattice of $\langle X,M\rangle$ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\Pi \left(L\right)$ is introduced here. $\Pi \left(L\right)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\Pi \left(L\right)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi$-product lattice. A $\Pi$-product...