International Journal of Algebra and Computation

A finite unary algebra of finite type with a constant function 0 that is a one-element subalgebra, and whose operations have range {0, 1}, is called a {0, 1}-valued unary algebra with 0. Such an algebra has a finite basis for its quasi-equations if and only if the relation defined by the rows of the nontrivial functions in the clone form an order ideal.

Let B be a p-block of a finite group G with abelian defect group D such that S ≤ G ≤ Aut(S), S′ = S and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N_G(D) in the sense of Broué. This has been done by Rouquier for principal blocks and it remains to deal with the non-principal blocks.

In this paper we study group actions on hyperbolic Λ-metric spaces, where Λ is an ordered abelian group. Λ-metric spaces were first introduced by Morgan and Shalen in their study of hyperbolic structures and then Chiswell, following Gromov's ideas, introduced the notion of hyperbolicity for such spaces. Only the case of 0-hyperbolic Λ-metric spaces (that is, Λ-trees) was systematically studied, while the theory of general hyperbolic Λ-metric spaces was not developed at all. Hence, one of the goals of the present paper was to fill this gap and translate basic notions and results from the theory of group actions on hyperbolic (in the usual sense) spaces to the case of Λ-metric spaces for an arbitrary Λ. The other goal was to show some principal difficulties which arise in this generalization and the ways to deal with them.

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ ℕ | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is , where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.

We study connections between closure operators on an algebra (A, Ω) and congruences on the extended power algebra defined on the same algebra. We use these connections to give an alternative description of the lattice of all subvarieties of semilattice ordered algebras.

For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring 𝔇(L) constructed by Lichtman. If U(L) is an Ore domain, 𝔇(L) coincides with its ring of fractions. It is well known that the principal involution of L, x ↦ -x, can be extended to an involution of U(L), and Cimpric proved that this involution can be extended to one on 𝔇(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that 𝔇(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U(L) is an Ore domain.