On Klein Surfaces and Dihedral Groups.
We show, with a counterexample, that proposition 3 in [2], as it stands, is not correct; we prove however that by changing the hypothesis the thesis of the proposition remains still valid.
In this paper we prove the following results. Let π be a set of prime numbers and G a finite π-soluble group. Consider U, V ≤ G and such that and . Suppose also is a Hall π-sub-group of some S-permutable subgroup of G. Then and . Therefore,the set of all S-permutably embedded subgroups of a soluble group G into which a given Hall system Σ reduces is a sublattice of the lattice of all Σ-permutable subgroups of G. Moreover any two subgroups of this sublattice of coprimeorders permute.
In this paper the equivalence on a semigroup in terms of a set of idempotents in is defined. A semigroup is called a -liberal semigroup with as the set of projections and denoted by if every -class in it contains an element in . A class of -liberal semigroups is characterized and some special cases are considered.
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.
We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known ‘heapification’ functor to the ambience of Lie groups and principal bundles.