We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units...

Quasi-ideals were introduced by Otto Steinfeld [43] as those non-empty subsets Q of a semigroup T satisfying QTD TQ c Q. When T is regular they are precisely the subsets Q of T which satisfy QTQ = Q ([43, Theorem 9.3]). There are many examples of quasi-ideals in regular semigroup theory. We list below some of the most important: • Every subsemigroup of the form eSe (where e is an idempotent) is...

For a field K, rational function φ ∈ K(z) of degree at least two, and α ∈ P(K), we study the polynomials in K[z] whose roots are given by the solutions in K to φ(z) = α, where φ denotes the nth iterate of φ. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (φ,α) is called eventually stable over K. We conjecture that (φ, α) is eventually stable over K w...

This paper is a very brief introduction to idempotent mathematics and related topics. It appears as an introductory paper in the volume Idempotent Mathematics and Mathematical Physics (G. L. Litvinov and V. P. Maslov, eds; AMS Contemporary Mathematics Proceedings Series, 2005) [73]. This paper is a very brief introduction, without exact theorems and proofs, to the Maslov dequantization and idem...

In this paper, we are mainly concerned with the Cartan determinant conjecture and the no loop conjecture. If A is an artin algebra of finite global dimension, the first conjecture claims that the Cartan determinant of A is equal to 1, while the second one states that every simple A-module admits only the trivial self-extension. Among numerous partial solutions to these conjectures such as those...