On the varieties of ai-semirings satisfying $${x^{3}\approx x}$$

The aim of this paper is to study the varieties of ai-semirings satisfying \({x^{3}\approx x}\) . It is shown that the collection of all such varieties forms a distributive lattice of order 179. Also, all of them are finitely based and finitely generated. This generalizes and extends the main results obtained by Ghosh et al., Pastijn and Ren and Zhao.     

On free semilattice-ordered semigroups satisfying x n =x

Let B(k,0,n) denote the group with k generators which is free in the group variety defined by the identity x ⁿ =1. Let B _slo (k,1,n) denote the semilattice-ordered semigroup with k generators which is free in the semilattice-ordered semigroup variety defined by the identity x ⁿ =x. We prove a generalization of the Green-Rees theorem: B _slo (k,1,n) is finite for all k≥1 if and only if B(k,0,n−1) is finite for all k≥1. We find a formula for card(B _slo (1,1,n)). We construct B _slo (k,1,n) for some concrete values of k and n.     

On a theorem of Burnside on fixed-point-free automorphisms

We prove that a finite group having a fixed-point-free automorphism in the Fitting subgroup of its automorphism group must be abelian of rather restricted structure. As a consequence, no finite nonabelian group could have a fixed-point-free automorphism in the Frattini subgroup of its automorphism group. Received: 21 April 2007, Revised: 18 May 2007     

On algebras satisfying $x^2x^2=N(x)x$

The commutative algebras satisfying the “adjoint identity”: , where N is a cubic form, are shown to be related to a class of generically algebraic Jordan algebras of degree at most 4 and to the pseudo-composition algebras. They are classified under a nondegeneracy condition. As byproducts, the associativity of the norm of any pseudo-composition algebra is proven and the unital commutative and power-associative algebras of degree are shown to be Jordan algebras. Received January 26, 1999; in final form August 26, 1999 / Published online July 3, 2000     

Absolute valued algebras satisfying (x,x,x2) = 0

Let A be an absolute valued algebra satisfying the identity (x,x,x²) = 0. We give some conditions which imply that A is isomorphic to R, \mathbbC \mathbb{C} , H or D. These results enable us to show that if A is an algebra with involution then A is one of those classical algebras. We construct an example of A having dimension two and is not isomorphic to \mathbbC \mathbb{C} .     

On graphs satisfying a local ore-type condition

For an integer i, a graph is called an L_i-graph if, for each triple of vertices u, v, w with d(u, v) = 2 and w (element of) N(u) (intersection) N(v), d(u) + d(v) ≥ | N(u) (union) N(v) (union) N(w)| —i. Asratian and Khachatrian proved that connected L_o-graphs of order at least 3 are hamiltonian, thus improving Ore's Theorem. All K_1,3-free graphs are L₁-graphs, whence recognizing hamiltonian L₁-graphs is an NP-complete problem. The following results about L₁-graphs, unifying known results of Ore-type and known results on K_1,3-free graphs, are obtained. Set K = lcub;G|K_p,p+1 (contained within) G (contained within) K_p V K_p+1 for some ρ ≥ } (v denotes join). If G is a 2-connected L₁-graph, then G is 1-tough unless G (element of) K. Furthermore, if G is as connected L₁-graph of order at least 3 such that |N(u) (intersection) N(v)| ≥ 2 for every pair of vertices u, v with d(u, v) = 2, then G is hamiltonian unless G ϵ K, and every pair of vertices x, y with d(x, y) ≥ 3 is connected by a Hamilton path. This result implies that of Asratian and Khachatrian. Finally, if G is a connected L₁-graph of even order, then G has a perfect matching. © 1996 John Wiley & Sons, Inc.     

On a class of groups satisfying Bass' conjecture

In this paper, we use the Connes-Karoubi character liftings of the Hattori-Stallings rank in order to obtain examples of groups satisfying Bass' conjecture. We consider the nilpotence of the periodicity operator in cyclic homology and verify the conjecture for a certain class of groups which contains properly the ones studied by Eckmann. We also give an application of this method to the idempotent conjecture. Oblatum 21-III-1997 & 12-VI-1997