We investigate different notions of recognizability for a free monoid morphism σ : A∗ → B∗. Full recognizability occurs when each (aperiodic) point in BZ admits at most one tiling with words σ(a), a ∈ A. This is stronger than the classical notion of recognizability of a substitution σ : A∗ → A∗, where the tiling must be compatible with the language of the substitution. We show that if |A| = 2, ...

let $r$ be an associative ring with identity and $z^*(r)$ be its set of non-zero zero divisors.  the zero-divisor graph of $r$, denoted by $gamma(r)$, is the graph whose vertices are the non-zero  zero-divisors of  $r$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  in this paper, we bring some results about undirected zero-divisor graph of a monoid ring ov...

If A is a stable basis algebra of rank n, then the set Sn−1 of endomorphisms of rank at most n−1 is a subsemigroup of the endomorphism monoid of A. This paper gives a number of necessary and sufficient conditions for Sn−1 to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left T -sets of finite rank where T is can...

We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating finite case by various structural observations counterexamples, also highlighting role played positive orderability. Moreover, we construct a non-left-orderable, positively right-orderable submonoid of free product cyclic group order 7 with on 4 generators. Any extension that mon...

A zero-one language L is a regular language whose asymptotic probability converges to either zero or one. In this case, we say that L obeys the zero-one law. We prove that a regular language obeys the zero-one law if and only if its syntactic monoid has a zero element, by means of Eilenberg’s variety theoretic approach. Our proof gives an effective automata characterisation of the zero-one law ...