Related papers: On sofic monoids
We prove that a monoid is sofic, in the sense recently introduced by Ceccherini-Silberstein and Coornaert, whenever the J-class of the identity is a sofic group, and the quotients of this group by orbit stabilisers in the rest of the monoid…
We give new characterizations of sofic groups: -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups. -- A group $G$ is sofic if and only if any system of equations…
A monoid $M$ is called surjunctive if every injective cellular automata with finite alphabet over $M$ is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all…
We define a notion of relative soficity for countable groups with respect to a family of groups. A group is sofic if and only if it is relative sofic with respect to the family consisting only of the trivial group. If a group is relatively…
A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag--Solitar group, is a…
We give the following characterization of sofic (weakly sofic) groups: a group $G$ is sofic (weakly sofic) if and only if any system of equations solvable in any alternating group (any finite group) is solvable over $G$.
Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic…
Answering some queries of Weiss, we prove that the free product and amenable extensions of sofic groups are sofic as well, and give an example of a finitely generated sofic group that is not residually amenable.
We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We…
We prove that graph products of sofic groups are sofic, as are graphs of groups for which vertex groups are sofic and edge groups are amenable.
We describe elementary examples of finitely presented sofic groups which are not residually amenable (and thus not initially subamenable or LEA, for short). We ask if an amalgam of two amenable groups over a finite subgroup is residually…
The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup $S$ with no isolated nontrivial…
A monoid is aperiodic if all its subgroups are trivial. We completely classify all varieties of aperiodic monoids whose subvariety lattice is distributive.
We give a definition of weakly sofic groups (w-sofic groups). Our definition is rather natural extension of the definition of sofic groups where instead of Hamming metric on symmetric groups we use general bi-invariant metrics on finite…
We study applications of a general approach for arities and arizabilities of theories to group and monoid theories. It is proved that a theory of a group $G$ is aritizable if and only if $G$ is finite. It is shown that this criterion does…
In this short note we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.
We show that free products of sofic groups with amalgamation over monotileably amenable subgroups are sofic. Consequently, so are HNN extensions of sofic groups relative to homomorphisms of monotileably amenable subgroups. We also show that…
We give a classification of noncommutative algebraic monoid structures on normal affine varieties such that the group of invertible elements of the monoid is connected, solvable, and has a one-dimensional unipotent radical. We describe the…
We examine several definitions of soficity for monoids obtained by generalizing various definitions of sofic groups. They are not all equivalent and include the definition recently introduced by Ceccherini-Silberstein and Coornaert. One of…
We introduce the class of strongly sofic monoids. This class of monoids strictly contains the class of sofic groups and is a proper subclass of the class of sofic monoids. We define and investigate sofic topological entropy for actions of…