We obtain a simple construction for particular subclasses of several varieties of lattice expansions. The construction allows a unified approach to the characterization of the subdirectly irreducible algebras

Every quasigroup (loop, Bol loop, group, respectively) is isomorphic to the factor of a subdirectly irreducible quasigroup (loop, Bol loop, group, respectively) over its monolithic congruence.

A nontrivial algebra with at least one at least binary operation is isomorphic to the factor of a subdirectly irreducible algebra through its monolith if and only if the intersection of all its ideals is nonempty.

This paper is centred around a nice conjecture, known to Wolfgang Rautenberg and myself since 1978: a modal algebra A is subdirectly irreducible if and only if its dual frame A∗ is initial, or generated (see definitions in the text). I here show that in full generality the conjecture is false, but that it becomes true under some mild additional assumptions. Unfortunately, some counterexamples s...

In this paper all subdirectly irreducible pseudocomplemented distributive lattices are found. This result is used to establish a Stone-like representation theorem conjectured by G. Grätzer and to find all equational subclasses of the class of pseudocomplemented distributive lattices.

We prove that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and describe its monolith. The endomorphism semiring is congruence simple if and only if the semilattice has both a least and a largest element.