A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [3] completely characterized the commutative local rings R for which Mn(R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring Mn(R) is strongly clean. Here we completely determi...

In the course of providing an (infinitary) axiomatization of the equational theory of the class of context-free languages, Grathwohl, Kozen and Henglein (2013) have introduced the class of μcontinuous Chomsky algebras. These are idempotent semirings where least solutions for systems of polynomial inequations (i.e. context-free grammars) can be computed iteratively and where multiplication is co...

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure μ ∈ M(G) is said to be idempotent if μ ∗ μ = μ, or alternatively if μ̂ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure μ is idempotent if and only if the set {γ ∈ Ĝ : μ̂(γ) = 1} belongs to the cos...

In this article, we discuss the family of cellular automata generated by so-called idempotent cellular automata (CA G such that G 2 = G) on the full shift. We prove a characterization of products of idempotent CA, and show examples of CA which are not easy to directly decompose into a product of idempotents, but which are trivially seen to satisfy the conditions of the characterization. Our pro...

The order of hypersubstitutions, all idempotent elements on the monoid of all hypersubstitutions of type τ 2 were studied by K. Denecke and Sh. L. Wismath and all idempotent elements on the monoid of all hypersubstitutions of type τ 2, 2 were studied by Th. Changpas and K. Denecke. We want to study similar problems for the monoid of all generalized hypersubstitutions of type τ 2 . In this paper...

We consider a common generalization of the embedding and intersection problems for latin squares. Specifically, we extend the definition of embedding to squares whose sides do not meet the necessary condition for embedding and extend the intersection problem to squares of different orders. Results are given for arbitrary latin squares, and those which are idempotent and idempotent symmetric. Th...