In [3], the present author used a binary operation as a tool for characterizing geodetic graphs. In this paper a new proof of the main result of the paper cited above is presented. The new proof is shorter and simpler.
Subdirectly irreducible non-idempotent groupoids satisfying and are studied.
Let be a commutative groupoid such that ; ; or . Then is determined uniquely up to isomorphism and if it is finite, then for an integer .
It is an open question whether the variety generated by the left divisible left distributive groupoids coincides with the variety generated by the left distributive left quasigroups. In this paper we prove that every left divisible left distributive groupoid with the mapping surjective lies in the variety generated by the left distributive left quasigroups.
We study the groupoids satisfying both the left distributivity and the left idempotency laws. We show that they possess a canonical congruence admitting an idempotent groupoid as factor. This congruence gives a construction of left idempotent left distributive groupoids from left distributive idempotent groupoids and right constant groupoids.
The aim of the paper is to show that if S(G) is distributive, and also G satisfies some additional condition, then the union of any two subgroupoids of G is also a subgroupoid (intuitively, G has to be in some sense a unary algebra).
The main result of Romanowska A., Roszkowska B., On some groupoid modes, Demonstratio Math. 20 (1987), no. 1–2, 277–290, provides us with an explicit description of the lattice of varieties of differential groupoids. In the present article, we show that this variety is -universal, which means that there is no convenient explicit description for the lattice of quasivarieties of differential groupoids. We also find an example of a subvariety of differential groupoids with a finite number of subquasivarieties....
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.