Information on the structure of finite paramedial quasigroups, including a classification of finite simple paramedial quasigroups, is given. The problem ``Classify the finite simple paramedial quasigroups'' was posed by J. Ježek and T. Kepka at the conference LOOPS'03, Prague 2003.
Let be a finite group and the cyclic group of order 2. Consider the 8 multiplicative operations , where . Define a new multiplication on by assigning one of the above 8 multiplications to each quarter , for . If the resulting quasigroup is a Bol loop, it is Moufang. When is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops .
The paper deals with quasigroup identities under isotopies. The terminology is taken from [2], [3] and [4]. Stimulated by geometric illustrations, V. D. Belousov in [2] has presented two important identity properties and posed a question for which identities these properties are necessary and sufficient for the identity to be invariant under isotopies. Inspired by V. D. Belousov, G. Monoszová investigated in [6] one special kind of identities for which both Belousov’s properties give necessary and...
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.
We derive necessary and sufficient conditions for there to exist a latin square of order containing two subsquares of order and that intersect in a subsquare of order . We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order cannot have more than subsquares of order , where . Indeed, the number of subsquares of order is bounded by a polynomial of degree at most in . (b) For all there exists a loop of order in which every...