On varieties of left distributive left idempotent groupoids
David Stanovský · Discussiones Mathematicae - General Algebra and Applications · 2004
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 x n ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found. We consider groupoids (i.e. sets equipped with a binary operation) satisfying the following two identities: (LD) (LI) x(yz) ≈ (xy)(xz), (xx)y ≈ xy. We call such groupoids left distributive left idempotent, shortly LDLI. A groupoid is called idempotent, if it satisfies the identity