This paper investigates the preservation of hopficity and co-hopficity on passing to finite-index subsemigroups and extensions. It was already known that hopficity is not preserved on passing to finite Rees index subsemigroups, even in the finitely generated case. We give a stronger example to show that it is not preserved even in the finitely presented case. It was also known that hopficity is...

For some distinguished properties of semigroups, such as having idempotents, regularity, hopficity etc, we study the following: whether that property or its negation is a Markov property, whether it is decidable for finitely presented semigroups and for one-relation semigroups and monoids. All the results and open problems are summarized in a table.

Proof. It is well-known that the kernels of the powers of φ stabilize (see for example [3]), that is, there exists k > 0 such that ker(φ) = ker(φ) for all n ≥ k. (This easily follows from the stabilization of ranks of the free groups φ(F ) and from Hopficity of finitely generated free groups.) Put N = ker(φ). Then φ factors through to an injective endomorphism φ : F/N → F/N . The group F/N is i...

we prove that if $m$ is a monoid and $a$ a finite set with more than one element‎, ‎then the residual finiteness of $m$ is equivalent to that of the monoid consisting of all cellular automata over $m$ with alphabet $a$‎.

This paper makes a comprehensive survey of results relating to finite Rees index for semigroups. In particular, we survey of the state of knowledge on whether various finiteness properties (such as finite generation, finite presentability, automaticity, and hopficity) are inherited by finite Rees index subsemigroups and extensions. We survey other properties that are invariant under passing to ...