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
A triple-semilattice is an algebra with three binary operations, which is a semilattice in respect of each of them. A trice is a triple-semilattice, satisfying so called roundabout absorption laws. In this paper we investigate distributive trices. We prove that the only subdirectly irreducible distributive trices are the trivial one and a two element one. We also discuss finitely generated free distributive trices and prove that a free distributive trice with two generators has 18 elements.
The notion of idempotent modification of an algebra was introduced by Ježek. He proved that the idempotent modification of a group is subdirectly irreducible. For an -algebra we denote by and the idempotent modification, the underlying set or the underlying lattice of , respectively. In the present paper we prove that if is semisimple and is a chain, then is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.
We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity xⁿ ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found.